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Somemobile phonessupport use of twoSIM cards, described asdual SIMoperation. When a second SIM card is installed, the phone may allow users to switch between two separatemobile networkservices manually, have hardware support for keeping both connections in a "standby" state for automatic switching, or have twotransceiversto maintain both network connections at once.
Dual SIM phones are mainstream in many countries where phones are normally sold unlocked. Dual SIMs are popular for separating personal and business calls, in locations where lower prices apply to calls between clients of the same provider, where a single network may lack comprehensive coverage, and for travel across national and regional borders.[1][2]In countries where dual SIM phones are the norm, people who require only one SIM leave the second SIM slot empty. Dual SIM phones usually have two uniqueIMEInumbers, one for each SIM slot.
Devices that use more than two SIM cards have also been developed and released, notably the LG A290 triple SIM phone,[3]and even handsets that support four SIMs,[4][5]such as theCherry Mobile Quad Q70.[6]
The first phone to include dual SIM functionality was the Benefon Twin, released byBenefonin 2000.[7]More dual SIM phones were introduced in about 2007, most of them coming from small Chinese firms producing phones usingMediateksystems-on-a-chip. They started to attract mainstream attention.[8][9]
Such phones were initially eschewed by major manufacturers due to potential pressure from telecommunications companies,[10]but from about 2010Nokia,Samsung,Sonyand several others followed suit, with theNokia C2-00,Nokia C1-00andNokia C2-03and most notably theNokia X,[11][12][13]phones fromSamsung's Duos series,[14]and theSony Xperia Z3 Dual,Sony Xperia C[15]andtipo dual.[16][17]Appleadded dual SIM support in its 2018iPhone XSmodels, with models sold inChinacontaining two physical SIM slots, and models sold elsewhere supporting dual SIM by means of(Embedded) eSIMalongside a single physical SIM.[18][19]
For originating communications via the mobile phone network, the way to choose which SIM is used may vary on different phones. For example, one can be selected asprimaryor default for making calls, and one (which could be the same one) for data. Apple phones supporting dual SIMs can be set up to automatically use a specific SIM for each contact or the same one used for the last call to the contact, for iMessage, and for FaceTime.[20]Typically when dialling or sending a message an option to select a SIM is displayed.
Prior to the introduction of dual SIM phones, adapters that fit in the SIM card slot and hold two SIMs, with provision to switch between them when required.[10][21]
In dual SIMswitchphones, such as theNokia C1-00, only one SIM, selected by the user, is active at any time; it is not possible to receive or make calls on the inactive SIM.[22]
Dual SIMstandbyphones allow both SIMs to be accessed by usingtime multiplexing. When one SIM is in active use, for example on a call, the modem locks to it, leaving the other SIM unavailable. Older examples of dual-SIM standby phones include theSamsung Galaxy S Duos,[23]theSony Xperia M2Dual,[24]and theiPhone XS,XS MaxandiPhone XR.[25]
Dual SIM dualactive(DSDA) phones have two transceivers, and can receive calls on both SIM cards, at the cost of increased battery consumption and more complex hardware.[26][27]One example is theHTC Desire 600.[28]
Some telephones have a primary and a secondary SIM slot that support different generations of connectivity. For example,4Gand3Gprimary, and 3G and2Gsecondary,[29]or5Gand 5G, or 5G and 4G.[30]Selecting either of the SIMs as primary is usually possible without physically swapping the SIMs.
Some phone models utilize a "hybrid" SIM tray, which can hold either two SIM cards, or one SIM card and oneMicroSDmemory card.[31][32]TheHuawei Mate 20range introduced a proprietary memory card format calledNano Memory, exactly the size and shape of a nano SIM card.[33]
Some devices accept dual SIMs of different form factors. The Xiaomi Redmi Note 4 has a hybrid dual SIM tray that accepts one micro SIM card and one nano SIM card, the latter of which can be swapped for a MicroSD card.[29]
Dual SIM phones have become popular especially with business users[10][34]due to reduced costs by being able to use two different networks, with one possibly for personal use or based on signal strength or cost, without requiring several phones.
Some sub-contract Chinese companies supply inexpensive dual SIM handsets, mainly inAsiancountries. The phones, which also usually includetouch screeninterfaces and other modern features, typically retail for a much lower price than branded models. While some such phones are sold under generic names or arerebadgedby smaller companies under their own brand,[9]numerous manufacturers, especially in China, produce phones, including dual SIM models, undercounterfeittrademarks such as those of Nokia or Samsung,[35]either as cosmetically-identical clones of the originals, or in completely different designs, with the logo of a notable manufacturer present in order to take advantage of brand recognition or brand image.[8]
Dual SIM phones are common indeveloping countries, especially inChina,Southeast Asiaand theIndian subcontinent, with local firms likeKarbonn Mobiles,LYF,MicromaxandCherry Mobilereleasingfeature phonesandsmartphonesincorporating multiple SIM slots.[36][37]
The FrenchWikoMobile is also an example of rebadged Chinese Dual-SIM phones sold in few European countries as well as in North-West Africa.
Dual SIM phones have been rare in countries where phones have been usually sold on contract, as the carriers selling those phones prevent SIMs from competing carriers from being used with the phones. However, dual SIMs have been popular in locations where people normally buy phones directly from manufacturers. In such places there is little lock-in to carrier networks, and the costs of having two phone numbers are much lower.
Dual SIM phones allow separate numbers for personal and business calls on the same handset. Access to multiple networks is useful for people living in places where a single network's coverage may prove inadequate or unreliable. They are also useful in places where lower prices apply to calls between clients of the same provider.[38]
Dual SIM phones allow users to keep separate contact lists on each SIM, and allow easier roaming by being able to access a foreign network while keeping the existing local card.[39]
Vendors of foreign SIMs for travel often promote dual-SIM operation, with a home country and local SIM in the same handset.
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ASIM cardorSIM(subscriber identity module) is anintegrated circuit(IC) intended to securely store aninternational mobile subscriber identity(IMSI) number and its related key, which are used to identify and authenticate subscribers onmobile telephonedevices (such asmobile phones,tablets, andlaptops). SIMs are also able to storeaddress bookcontacts information,[1]and may be protected using aPIN codeto prevent unauthorized use.
SIMs are always used onGSMphones; forCDMAphones, they are needed only forLTE-capable handsets. SIM cards are also used in varioussatellite phones, smart watches, computers, or cameras.[2]The first SIM cards were the size ofcredit and bank cards; sizes were reduced several times over the years, usually keeping electrical contacts the same, to fit smaller-sized devices.[3]SIMs are transferable between different mobile devices by removing the card itself.
Technically, the actual physical card is known as auniversal integrated circuit card(UICC); thissmart cardis usually made ofPVCwith embedded contacts andsemiconductors, with the SIM as its primary component. In practice the term "SIM card" is still used to refer to the entire unit and not simply the IC. A SIM contains a unique serial number, integrated circuit card identification (ICCID), international mobile subscriber identity (IMSI) number, security authentication and ciphering information, temporary information related to the local network, a list of the services the user has access to, and four passwords: apersonal identification number(PIN) for ordinary use, and apersonal unblocking key(PUK) for PIN unlocking as well as a second pair (called PIN2 and PUK2 respectively) which are used for managingfixed dialing numberand some other functionality.[4][5]In Europe, the serial SIM number (SSN) is also sometimes accompanied by aninternational article number(IAN) or aEuropean article number(EAN) required when registering online for the subscription of a prepaid card.
As of 2020,eSIMis superseding physical SIM cards in some domains, including cellular telephony. eSIM uses a software-based SIM embedded into an irremovableeUICC.
The SIM card is a type ofsmart card,[2]the basis for which is thesiliconintegrated circuit(IC) chip.[6]The idea of incorporating a silicon IC chip onto a plastic card originates from the late 1960s.[6]Smart cards have since usedMOS integrated circuitchips, along withMOS memorytechnologies such asflash memoryandEEPROM(electricallyEPROM).[7]
The SIM was initially specified by theETSIin the specification TS 11.11. This describes the physical and logical behaviour of the SIM. With the development ofUMTS, the specification work was partially transferred to3GPP. 3GPP is now responsible for the further development of applications like SIM (TS 51.011[8]) and USIM (TS 31.102[9]) and ETSI for the further development of the physical cardUICC.
The first SIM card was manufactured in 1991 byMunichsmart-card makerGiesecke+Devrient, who sold the first 300 SIM cards to the Finnishwireless network operatorRadiolinja,[10][11]who launched the world's first commercial2GGSMcell network that year.[12]
Today, SIM cards are considered ubiquitous, allowing over 8 billion devices to connect to cellular networks around the world daily. According to the International Card Manufacturers Association (ICMA), there were 5.4 billion SIM cards manufactured globally in 2016 creating over $6.5 billion in revenue for traditional SIM card vendors.[13]The rise of cellular IoT and 5G networks was predicted by Ericsson to drive the growth of the addressable market for SIM cards to over 20 billion devices by 2020.[14]The introduction ofembedded-SIM(eSIM) andremote SIM provisioning(RSP) from the GSMA[15]may disrupt the traditional SIM card ecosystem with the entrance of new players specializing in "digital" SIM card provisioning and other value-added services for mobile network operators.[7]
There are three operating voltages for SIM cards:5 V,3 Vand1.8 V(ISO/IEC 7816-3 classes A, B and C, respectively). The operating voltage of the majority of SIM cards launched before 1998 was5 V. SIM cards produced subsequently are compatible with3 Vand5 V. Modern cards support5 V,3 Vand1.8 V.[7]
Modern SIM cards allow applications to load when the SIM is in use by the subscriber. These applications communicate with the handset or a server usingSIM Application Toolkit, which was initially specified by3GPPin TS 11.14. (There is an identical ETSI specification with different numbering.) ETSI and 3GPP maintain the SIM specifications. The main specifications are: ETSI TS 102 223 (the toolkit for smart cards), ETSI TS 102 241 (API), ETSI TS 102 588 (application invocation), and ETSI TS 131 111 (toolkit for more SIM-likes). SIM toolkit applications were initially written in native code using proprietary APIs. To provide interoperability of the applications, ETSI choseJava Card.[16]A multi-company collaboration calledGlobalPlatformdefines some extensions on the cards, with additional APIs and features like more cryptographic security andRFIDcontactless use added.[17]
SIM cards store network-specific information used to authenticate and identify subscribers on the network. The most important of these are the ICCID, IMSI,authentication key (Ki), local area identity (LAI) and operator-specific emergency number. The SIM also stores other carrier-specific data such as the SMSC (Short Message service center) number, service provider name (SPN), service dialing numbers (SDN), advice-of-charge parameters and value-added service (VAS) applications. (Refer to GSM 11.11.[18])
SIM cards can come in various data capacities, from8 KBto at least256 KB.[11]All can store a maximum of 250 contacts on the SIM, but while the32 KBhas room for 33Mobile country code(MCCs) ornetwork identifiers, the64 KBversion has room for 80 MNCs.[1]This is used by network operators to store data on preferred networks, mostly used when the SIM is not in its home network but isroaming. The network operator that issued the SIM card can use this to have a phone connect to a preferred network that is more economic for the provider instead of having to pay the network operator that the phone discovered first. This does not mean that a phone containing this SIM card can connect to a maximum of only 33 or 80 networks, instead it means that the SIM card issuer can specify only up to that number of preferred networks. If a SIM is outside these preferred networks, it uses the first or best available network.[14]
Each SIM is internationally identified by itsintegrated circuit card identifier(ICCID). Nowadays ICCID numbers are also used to identify eSIM profiles, not only physical SIM cards. ICCIDs are stored in the SIM cards and are also engraved or printed on the SIM card body during a process called personalisation.
The ICCID is defined by the ITU-T recommendationE.118as theprimary account number.[19]Its layout is based onISO/IEC 7812. According to E.118, the number can be up to 19 digits long, including a single check digit calculated using theLuhn algorithm. However, the GSM Phase 1[20]defined the ICCID length as an opaque data field, 10 octets (20 digits) in length, whose structure is specific to amobile network operator.
The number is composed of three subparts:
Their format is as follows.
Issuer identification number (IIN)
Individual account identification
Check digit
With the GSM Phase 1 specification using 10octetsinto which ICCID is stored as packed BCD[clarification needed], the data field has room for 20 digits with hexadecimal digit "F" being used as filler when necessary. In practice, this means that on GSM cards there are 20-digit (19+1) and 19-digit (18+1) ICCIDs in use, depending upon the issuer. However, a single issuer always uses the same size for its ICCIDs.
As required by E.118, the ITU-T updates a list of all current internationally assigned IIN codes in its Operational Bulletins which are published twice a month (the last as of January 2019 was No. 1163 from 1 January 2019).[22]ITU-T also publishes complete lists: as of August 2023, the list issued on 1 December 2018 was current, having all issuer identifier numbers before 1 December 2018.[23]
SIM cards are identified on their individual operator networks by a uniqueinternational mobile subscriber identity(IMSI).Mobile network operatorsconnect mobile phone calls and communicate with their market SIM cards using their IMSIs. The format is:
The Kiis a 128-bit value used in authenticating the SIMs on aGSMmobile network (for USIM network, the Kiis still needed but other parameters are also needed). Each SIM holds a unique Kiassigned to it by the operator during the personalisation process. The Kiis also stored in a database (termedauthentication centeror AuC) on the carrier's network.
The SIM card is designed to prevent someone from getting the Kiby using thesmart-card interface. Instead, the SIM card provides a function,Run GSM Algorithm, that the phone uses to pass data to the SIM card to be signed with the Ki. This, by design, makes using the SIM card mandatory unless the Kican be extracted from the SIM card, or the carrier is willing to reveal the Ki. In practice, the GSM cryptographic algorithm for computing a signed response (SRES_1/SRES_2: see steps 3 and 4, below) from the Kihas certain vulnerabilities[1]that can allow the extraction of the Kifrom a SIM card and the making of aduplicate SIM card.
Authentication process:
The SIM stores network state information, which is received from thelocation area identity(LAI). Operator networks are divided into location areas, each having a unique LAI number. When the device changes locations, it stores the new LAI to the SIM and sends it back to the operator network with its new location. If the device is power cycled, it takes data off the SIM, and searches for the prior LAI.
Most SIM cards store a number of SMS messages and phone book contacts. It stores the contacts in simple "name and number" pairs. Entries that contain multiple phone numbers and additional phone numbers are usually not stored on the SIM card. When a user tries to copy such entries to a SIM, the handset's software breaks them into multiple entries, discarding information that is not a phone number. The number of contacts and messages stored depends on the SIM; early models stored as few as five messages and 20 contacts, while modern SIM cards can usually store over 250 contacts.[24]
SIM cards have been made smaller over the years; functionality is independent of format. Full-size SIM was followed by mini-SIM, micro-SIM, and nano-SIM. SIM cards are also made to embed in devices.
JEDECDesign Guide 4.8, SON-8GSMA SGP.22 V1.0
All versions of the non-embedded SIM cards share the sameISO/IEC 7816pin arrangement.
Themini-SIMor (2FF , 2nd form factor) card has the same contact arrangement as the full-size SIM card and is normally supplied within a full-size card carrier, attached by a number of linking pieces. This arrangement (defined inISO/IEC 7810asID-1/000) lets such a card be used in a device that requires a full-size card – or in a device that requires a mini-SIM card, after breaking the linking pieces. As the full-size SIM is obsolete, some suppliers refer to the mini-SIM as a "standard SIM" or "regular SIM".
Themicro-SIM(or 3FF) card has the same thickness and contact arrangements, but reduced length and width as shown in the table above.[25]
The micro-SIM was introduced by theEuropean Telecommunications Standards Institute(ETSI) along with SCP,3GPP(UTRAN/GERAN),3GPP2(CDMA2000),ARIB,GSM Association(GSMA SCaG and GSMNA), GlobalPlatform,Liberty Alliance, and theOpen Mobile Alliance(OMA) for the purpose of fitting into devices too small for a mini-SIM card.[21][26]
The form factor was mentioned in the December 1998 3GPP SMG9UMTSWorking Party, which is the standards-setting body for GSM SIM cards,[24]and the form factor was agreed upon in late 2003.[27]
The micro-SIM was designed for backward compatibility. The major issue for backward compatibility was the contact area of the chip. Retaining the same contact area makes the micro-SIM compatible with the prior, larger SIM readers through the use of plastic cutout surrounds. The SIM was also designed to run at the same speed (5 MHz) as the prior version. The same size and positions of pins resulted in numerous "How-to" tutorials and YouTube videos with detailed instructions how to cut a mini-SIM card to micro-SIM size.
The chairman of EP SCP, Klaus Vedder, said[27]
ETSI has responded to a market need from ETSI customers, but additionally there is a strong desire not to invalidate, overnight, the existing interface, nor reduce the performance of the cards.
Micro-SIM cards were introduced by various mobile service providers for the launch of the original iPad, and later for smartphones, from April 2010. TheiPhone 4was the first smartphone to use a micro-SIM card in June 2010, followed by many others.[28]
After a debate in early 2012 between a few designs created by Apple,NokiaandRIM, Apple's design for an even smaller SIM card was accepted by the ETSI.[29][30]Thenano-SIM(or 4FF) card was introduced in June 2012, when mobile service providers in various countries first supplied it for phones that supported the format. The nano-SIM measures 12.3 mm × 8.8 mm × 0.67 mm (0.484 in × 0.346 in × 0.026 in) and reduces the previous format to the contact area while maintaining the existing contact arrangements.[31]A small rim of isolating material is left around the contact area to avoid short circuits with the socket. The nano-SIM can be put into adapters for use with devices designed for 2FF or 3FF SIMs, and is made thinner for that purpose,[32]and telephone companies give due warning about this.[33]4FF is 0.67 mm (0.026 in) thick, compared to the 0.76 mm (0.030 in) of its predecessors.
TheiPhone 5, released in September 2012, was the first device to use a nano-SIM card,[34]followed by other handsets.
In July 2013, Karsten Nohl, a security researcher from SRLabs, described[35][36]vulnerabilities in some SIM cards that supportedDES, which, despite its age, is still used by some operators.[36]The attack could lead to the phone being remotelyclonedor let someone steal payment credentials from the SIM.[36]Further details of the research were provided atBlackHaton 31 July 2013.[36][37]In response, theInternational Telecommunication Unionsaid that the development was "hugely significant" and that it would be contacting its members.[38]
In February 2015,The Interceptreported that theNSAandGCHQhad stolen the encryption keys (Ki's) used byGemalto(now known asThales DIS, manufacturer of 2 billion SIM cards annually)[39]), enabling these intelligence agencies to monitor voice and data communications without the knowledge or approval of cellular network providers or judicial oversight.[40]Having finished its investigation, Gemalto claimed that it has “reasonable grounds” to believe that the NSA and GCHQ carried out an operation to hack its network in 2010 and 2011, but says the number of possibly stolen keys would not have been massive.[41]
In September 2019, Cathal Mc Daid, a security researcher from Adaptive Mobile Security, described[42][43]how vulnerabilities in some SIM cards that contained the S@T Browser library were being actively exploited. This vulnerability was namedSimjacker. Attackers were using the vulnerability to track the location of thousands of mobile phone users in several countries.[44]Further details of the research were provided atVirusBulletinon 3 October 2019.[45][46]
When GSM was already in use, the specifications were further developed and enhanced with functionality such asSMSandGPRS. These development steps are referred as releases by ETSI. Within these development cycles, the SIM specification was enhanced as well: new voltage classes, formats and files were introduced.
In GSM-only times, the SIM consisted of the hardware and the software. With the advent of UMTS, this naming was split: the SIM was now an application and hence only software. The hardware part was called UICC. This split was necessary because UMTS introduced a new application, the universal subscriber identity module (USIM). The USIM brought, among other things, security improvements like mutual authentication and longer encryption keys, and an improved address book.
"SIM cards" in developed countries today are usuallyUICCscontaining at least a SIM application and a USIM application. This configuration is necessary because older GSM only handsets are solely compatible with the SIM application and some UMTS security enhancements rely on the USIM application.
OncdmaOnenetworks, the equivalent of the SIM card is theR-UIMand the equivalent of the SIM application is theCSIM.
Avirtual SIMis a mobile phone number provided by amobile network operatorthat does not require a SIM card to connect phone calls to a user's mobile phone.
An embedded SIM (eSIM) is a form of programmable SIM that is embedded directly into a device.[47]The surface mount format provides the same electrical interface as the full size, 2FF and 3FF SIM cards, but is soldered to a circuit board as part of the manufacturing process. In M2M applications where there is no requirement[15]to change the SIM card, this avoids the requirement for a connector, improving reliability and security.[citation needed]An eSIM can beprovisioned remotely; end-users can add or remove operators without the need to physically swap a SIM from the device or use multiple eSIM profiles at the same time.[48][49]
The eSIM standard, initially introduced in 2016, has progressively supplanted traditional physical SIM cards across various sectors, notably in cellular telephony.[50][51][52]In September 2017, Apple introduced the Apple Watch Series 3 featuring eSIM.[53]In October 2018, Apple introduced theiPad Pro (3rd generation),[54]which was the first iPad to support eSIM. In September 2022, Apple introduced the iPhone 14 series which was the first eSIM exclusive iPhone in the United States.[55]
An integrated SIM (iSIM) is a form of SIM directly integrated into the modem chip or main processor of the device itself. As a consequence they are smaller, cheaper and more reliable than eSIMs, they can improve security and ease the logistics and production of small devices i.e. forIoTapplications. In 2021,Deutsche Telekomintroduced thenuSIM, an "Integrated SIM for IoT".[56][57][58]
The use of SIM cards is mandatory inGSMdevices.[59][60]
Thesatellite phonenetworksIridium,ThurayaandInmarsat'sBGANalso use SIM cards. Sometimes, these SIM cards work in regular GSM phones and also allow GSM customers to roam in satellite networks by using their own SIM cards in a satellite phone.
Japan's 2GPDCsystem (which was shut down in 2012;SoftBank Mobileshut down PDC from 31 March 2010) also specified a SIM, but this has never been implemented commercially. The specification of the interface between the Mobile Equipment and the SIM is given in theRCRSTD-27 annexe 4. The Subscriber Identity Module Expert Group was a committee of specialists assembled by the European Telecommunications Standards Institute (ETSI) to draw up the specifications (GSM11.11) for interfacing between smart cards and mobile telephones. In 1994, the name SIMEG was changed to SMG9.
Japan's current and next-generation cellular systems are based on W-CDMA (UMTS) andCDMA2000and all use SIM cards. However, Japanese CDMA2000-based phones are locked to the R-UIM they are associated with and thus, the cards are not interchangeable with other Japanese CDMA2000 handsets (though they may be inserted into GSM/WCDMA handsets for roaming purposes outside Japan).
CDMA-based devices originally did not use a removable card, and the service for these phones is bound to a unique identifier contained in the handset itself. This is most prevalent in operators in the Americas. The first publication of the TIA-820 standard (also known as 3GPP2 C.S0023) in 2000 defined the Removable User Identity Module (R-UIM). Card-based CDMA devices are most prevalent in Asia.
The equivalent of a SIM inUMTSis called the universal integrated circuit card (UICC), which runs a USIM application. The UICC is still colloquially called aSIM card.[61]
The SIM card introduced a new and significant business opportunity forMVNOswho lease capacity from one of the network operators rather than owning or operating a cellular telecoms network and only provide a SIM card to their customers. MVNOs first appeared in Denmark, Hong Kong, Finland and the UK. By 2011 they existed in over 50 countries, including most of Europe, the United States, Canada, Mexico, Australia and parts of Asia, and accounted for approximately 10% of all mobile phone subscribers around the world.[62]
On some networks, the mobile phone islocked to its carrier SIM card, meaning that the phone only works with SIM cards from the specific carrier. This is more common in markets where mobile phones are heavily subsidised by the carriers, and the business model depends on the customer staying with the service provider for a minimum term (typically 12, 18 or 24 months). SIM cards that are issued by providers with an associated contract, but where the carrier does not provide a mobile device (such as a mobile phone) are calledSIM-onlydeals. Common examples are the GSM networks in the United States, Canada, Australia, and Poland. UK mobile networks ended SIM lock practices in December 2021. Many businesses offer the ability to remove the SIM lock from a phone, effectively making it possible to then use the phone on any network by inserting a different SIM card. Mostly, GSM and 3G mobile handsets can easily be unlocked and used on any suitable network with any SIM card.
In countries where the phones are not subsidised, e.g., India, Israel and Belgium, all phones are unlocked. Where the phone is not locked to its SIM card, the users can easily switch networks by simply replacing the SIM card of one network with that of another while using only one phone. This is typical, for example, among users who may want to optimise their carrier's traffic by different tariffs to different friends on different networks, or when travelling internationally.
In 2016, carriers started using the concept of automatic SIM reactivation[63]whereby they let users reuse expired SIM cards instead of purchasing new ones when they wish to re-subscribe to that operator. This is particularly useful in countries whereprepaid callsdominate and where competition drives highchurn rates, as users had to return to a carrier shop to purchase a new SIM each time they wanted to churn back to an operator.
Commonly sold as a product by mobiletelecommunicationscompanies, "SIM-only" refers to a type oflegally liabilitycontract between a mobile network provider and a customer. The contract itself takes the form of a credit agreement and is subject to a credit check.
SIM-only contracts can bepre-pay- where the subscriber buyscreditbefore use (often called pay as you go, abbreviated to PAYG), orpost-pay, where the subscriber pays in arrears, typically monthly.
Within a SIM-only contract, the mobile network provider supplies their customer with just one piece of hardware, a SIM card, which includes an agreed amount of network usage in exchange for a monthly payment. Network usage within a SIM-only contract can be measured in minutes, text, data or any combination of these. The duration of a SIM-only contract varies depending on the deal selected by the customer, but in the UK they are typically available over 1, 3, 6, 12 or 24-month periods.
SIM-only contracts differ from mobile phone contracts in that they do not include any hardware other than a SIM card. In terms of network usage, SIM-only is typically more cost-effective than other contracts because the provider does not charge more to offset the cost of a mobile device over the contract period. The short contract length is one of the key features of SIM-only – made possible by the absence of a mobile device.
SIM-only is increasing in popularity very quickly.[64]In 2010 pay monthly based mobile phone subscriptions grew from 41 percent to 49 percent of all UK mobile phone subscriptions.[65]According to German research companyGfK, 250,000 SIM-only mobile contracts were taken up in the UK during July 2012 alone, the highest figure since GfK began keeping records.
Increasing smartphone penetration combined with financial concerns is leading customers to save money by moving onto a SIM-only when their initial contract term is over.
Dual SIMdevices have two SIM card slots for the use of two SIM cards, from one or multiple carriers. Multiple SIM devices are commonplace in developing markets such as inAfrica,East Asia,South AsiaandSoutheast Asia, where variable billing rates, network coverage and speed make it desirable for consumers to use multiple SIMs from competing networks. Dual-SIM phones are also useful to separate one's personal phone number from a business phone number, without having to carry multiple devices. Some popular devices, such as theBlackBerry KeyOne, have dual-SIM variants; however, dual-SIM devices were not common in the US or Europe due to lack of demand. This has changed with mainline products from Apple and Google featuring either two SIM slots or a combination of a physical SIM slot and an eSIM.
In September 2018,AppleintroducediPhone XS,iPhone XS Max, andiPhone XRfeaturing Dual SIM (nano-SIM andeSIM) andApple Watch Series 4featuring DualeSIM.
Athin SIM(oroverlay SIMorSIM overlay) is a very thin device shaped like a SIM card, approximately 120 microns (1⁄200inch) thick. It has contacts on its front and back. It is used by placing it on top of a regular SIM card. It provides its own functionality while passing through the functionality of the SIM card underneath. It can be used to bypass the mobile operating network and run custom applications, particularly on non-programmable cell phones.[66]
Its top surface is a connector that connects to the phone in place of the normal SIM. Its bottom surface is a connector that connects to the SIM in place of the phone. With electronics, it can modify signals in either direction, thus presenting a modified SIM to the phone, and/or presenting a modified phone to the SIM. (It is a similar concept to theGame Genie, which connects between a game console and a game cartridge, creating a modified game). Similar devices have also been developed for iPhones to circumvent SIM card restrictions on carrier-locked models.[67]
In 2014,Equitel, an MVNO operated by Kenya'sEquity Bank, announced its intention to begin issuing thin SIMs to customers, raising security concerns by competition, particularly concerning the safety of mobile money accounts. However, after months of security testing and legal hearings before the country's Parliamentary Committee on Energy, Information and Communications, theCommunications Authority of Kenya(CAK) gave the bank the green light to roll out its thin SIM cards.[68]
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Phone cloningis the copying of acellular device's identity to another.
Analogue mobile telephones were notorious for their lack of security.[1]Casual listeners easily heard conversations as plainnarrowband FM; eavesdroppers with specialized equipment readily intercepted handsetElectronic Serial Numbers(ESN) and Mobile Directory Numbers (MDN or CTN, the Cellular Telephone Number) over the air. The intercepted ESN/MDN pairs would be cloned onto another handset and used in other regions for making calls. Due to widespread fraud, some carriers required aPINbefore making calls or used a system ofradio fingerprintingto detect the clones.
Code-Division Multiple Access(CDMA) mobile telephone cloning involves gaining access to the device's embeddedfile system/nvm/num directory via specialized software or placing a modifiedEEPROMinto the target mobile telephone, allowing theElectronic Serial Number(ESN) and/orMobile Equipment Identifier(MEID) of the mobile phone to be changed. To obtain the MEID of your phone, simply open your phone's dialler and type *#06# to get its MEID number.[2]The ESN or MEID is typically transmitted to the cellular company'sMobile Telephone Switching Office(MTSO) in order to authenticate a device onto the mobile network. Modifying these, as well as the phone'sPreferred Roaming List(PRL) and themobile identification number, or MIN, can pave the way for fraudulent calls, as the target telephone is now a clone of the telephone from which the original ESN and MIN data were obtained.
GSMcloning occurs by copying a secret key from the victimSIM card,[3]typically not requiring any internal data from the handset (the phone itself). GSM handsets do not have ESN or MIN, only anInternational Mobile Equipment Identity(IMEI) number. There are various methods used to obtain the IMEI. The most common method is to eavesdrop on a cellular network.
Older GSM SIM cards can be cloned by performing a cryptographic attack against theCOMP128authentication algorithm used by these older SIM cards.[4]By connecting the SIM card to a computer, the authentication procedure can be repeated many times in order to slowly leak information about the secret key. If this procedure is repeated enough times, it is possible to derive theKikey.[5][6]Later GSM SIMs have various mitigations built in, either by limiting the number of authentications performed in a power on session, or by the manufacturer choosing resistant Kikeys. However if it is known that a resistant key was used, it is possible to speed up the attack by eliminating weak Kikeys from the pool of possible keys.
Phone cloning is outlawed in theUnited Statesby the Wireless Telephone Protection Act of 1998, which prohibits "knowingly using, producing, trafficking in, having control or custody of, or possessing hardware or software knowing that it has been configured to insert or modify telecommunication identifying information associated with or contained in a telecommunications instrument so that such instrument may be used to obtain telecommunications service without authorization."[7]
The effectiveness of phone cloning is limited. Every mobile phone contains aradio fingerprintin its transmission signal which remains unique to that mobile despite changes to the phone's ESN, IMEI, or MIN. Thus, cellular companies are often able to catch cloned phones when there are discrepancies between the fingerprint and the ESN, IMEI, or MIN.[citation needed]
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https://en.wikipedia.org/wiki/SIM_cloning
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Theuniversal integrated circuit card(UICC) is the physicalsmart card(integrated circuitcard) used inmobile terminalsin 2G (GSM), 3G (UMTS), 4G (LTE), and5Gnetworks. The UICC ensures the integrity and security of all kinds of personal data, and it typically holds a few hundred kilobytes.[1]
The official definition for UICC is found inETSITR 102 216, where it is defined as a "smart cardthat conforms to the specifications written and maintained by the ETSI Smart Card Platform project". In addition, the definition has a note that states that "UICC is neither an abbreviation nor an acronym".[2]
NIST SP 800-101 Rev. 1 and NIST Computer Security Resource Center Glossary state that, "A UICC may be referred to as a SIM, USIM, RUIM or CSIM, and is used interchangeably with those terms",[3][4]though this is an over-simplification. The primary component of a UICC is aSIM card.[citation needed]
A UICC consists of aCPU,ROM,RAM,EEPROMandI/Ocircuits. Early versions consisted of the whole full-size (85 × 54 mm,ISO/IEC 7810ID-1) smart card. Soon the race for smaller telephones called for a smaller version of the card. The card was cropped down to 25 × 15 mm (ISO/IEC 7810 ID-000), as illustrated.
In 2G networks, the SIM card and SIM application were bound together, so that "SIM card" could mean the physical card, or any physical card with the SIM application.
In aGSMnetwork, the UICC contains a SIM application and in aUMTSnetwork, it contains aUSIMapplication. A UICC may contain several applications, making it possible for the same smart card to give access to both GSM and UMTS networks, and also provide storage of a phone book and other applications. It is also possible to access a GSM network using a USIM application and it is possible to access UMTS networks using a SIM application with mobile terminals prepared for this. With the UMTS release 5 a new application, the IP multimedia Services Identity Module (ISIM) is required for services in theIMS. The telephone book is a separate application and not part of either subscriber identity module.
In acdmaOne/CDMA2000("CDMA") network, the UICC contains aCSIMapplication, in addition to 3GPP USIM and SIM applications. A card with all 3 features is called a removable user identity card, orR-UIM. Thus, the R-UIM card can be inserted into CDMA, GSM, or UMTS handsets, and will work in all three cases.
In 3G networks, it is a mistake to speak of a USIM, CSIM, or SIM card, as all three are applications running on a UICC card.
Since the card slot is standardized, a subscriber can easily move their wireless account and phone number from one handset to another. This will also transfer their phone book and text messages. Similarly, usually a subscriber can change carriers by inserting a new carrier's UICC card into their existing handset. However, it is not always possible because some carriers (e.g., in U.S.)SIM-lockthe phones that they sell, preventing rival carriers' cards from being used.
The use and content of the card can be protected by use ofPINcodes. One code, PIN1, can be defined to control normal use of the phone. Another code, PIN2, can be set, to allow the use of special functions (likelimiting outbound telephone calls to a list of numbers).PUK1and PUK2 is used to reset PIN1 and PIN2 respectively.
The integration of the ETSI framework and the Application management framework ofGlobalPlatformis standardized in the UICC configuration.[5]
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ASIM cardorSIM(subscriber identity module) is anintegrated circuit(IC) intended to securely store aninternational mobile subscriber identity(IMSI) number and its related key, which are used to identify and authenticate subscribers onmobile telephonedevices (such asmobile phones,tablets, andlaptops). SIMs are also able to storeaddress bookcontacts information,[1]and may be protected using aPIN codeto prevent unauthorized use.
SIMs are always used onGSMphones; forCDMAphones, they are needed only forLTE-capable handsets. SIM cards are also used in varioussatellite phones, smart watches, computers, or cameras.[2]The first SIM cards were the size ofcredit and bank cards; sizes were reduced several times over the years, usually keeping electrical contacts the same, to fit smaller-sized devices.[3]SIMs are transferable between different mobile devices by removing the card itself.
Technically, the actual physical card is known as auniversal integrated circuit card(UICC); thissmart cardis usually made ofPVCwith embedded contacts andsemiconductors, with the SIM as its primary component. In practice the term "SIM card" is still used to refer to the entire unit and not simply the IC. A SIM contains a unique serial number, integrated circuit card identification (ICCID), international mobile subscriber identity (IMSI) number, security authentication and ciphering information, temporary information related to the local network, a list of the services the user has access to, and four passwords: apersonal identification number(PIN) for ordinary use, and apersonal unblocking key(PUK) for PIN unlocking as well as a second pair (called PIN2 and PUK2 respectively) which are used for managingfixed dialing numberand some other functionality.[4][5]In Europe, the serial SIM number (SSN) is also sometimes accompanied by aninternational article number(IAN) or aEuropean article number(EAN) required when registering online for the subscription of a prepaid card.
As of 2020,eSIMis superseding physical SIM cards in some domains, including cellular telephony. eSIM uses a software-based SIM embedded into an irremovableeUICC.
The SIM card is a type ofsmart card,[2]the basis for which is thesiliconintegrated circuit(IC) chip.[6]The idea of incorporating a silicon IC chip onto a plastic card originates from the late 1960s.[6]Smart cards have since usedMOS integrated circuitchips, along withMOS memorytechnologies such asflash memoryandEEPROM(electricallyEPROM).[7]
The SIM was initially specified by theETSIin the specification TS 11.11. This describes the physical and logical behaviour of the SIM. With the development ofUMTS, the specification work was partially transferred to3GPP. 3GPP is now responsible for the further development of applications like SIM (TS 51.011[8]) and USIM (TS 31.102[9]) and ETSI for the further development of the physical cardUICC.
The first SIM card was manufactured in 1991 byMunichsmart-card makerGiesecke+Devrient, who sold the first 300 SIM cards to the Finnishwireless network operatorRadiolinja,[10][11]who launched the world's first commercial2GGSMcell network that year.[12]
Today, SIM cards are considered ubiquitous, allowing over 8 billion devices to connect to cellular networks around the world daily. According to the International Card Manufacturers Association (ICMA), there were 5.4 billion SIM cards manufactured globally in 2016 creating over $6.5 billion in revenue for traditional SIM card vendors.[13]The rise of cellular IoT and 5G networks was predicted by Ericsson to drive the growth of the addressable market for SIM cards to over 20 billion devices by 2020.[14]The introduction ofembedded-SIM(eSIM) andremote SIM provisioning(RSP) from the GSMA[15]may disrupt the traditional SIM card ecosystem with the entrance of new players specializing in "digital" SIM card provisioning and other value-added services for mobile network operators.[7]
There are three operating voltages for SIM cards:5 V,3 Vand1.8 V(ISO/IEC 7816-3 classes A, B and C, respectively). The operating voltage of the majority of SIM cards launched before 1998 was5 V. SIM cards produced subsequently are compatible with3 Vand5 V. Modern cards support5 V,3 Vand1.8 V.[7]
Modern SIM cards allow applications to load when the SIM is in use by the subscriber. These applications communicate with the handset or a server usingSIM Application Toolkit, which was initially specified by3GPPin TS 11.14. (There is an identical ETSI specification with different numbering.) ETSI and 3GPP maintain the SIM specifications. The main specifications are: ETSI TS 102 223 (the toolkit for smart cards), ETSI TS 102 241 (API), ETSI TS 102 588 (application invocation), and ETSI TS 131 111 (toolkit for more SIM-likes). SIM toolkit applications were initially written in native code using proprietary APIs. To provide interoperability of the applications, ETSI choseJava Card.[16]A multi-company collaboration calledGlobalPlatformdefines some extensions on the cards, with additional APIs and features like more cryptographic security andRFIDcontactless use added.[17]
SIM cards store network-specific information used to authenticate and identify subscribers on the network. The most important of these are the ICCID, IMSI,authentication key (Ki), local area identity (LAI) and operator-specific emergency number. The SIM also stores other carrier-specific data such as the SMSC (Short Message service center) number, service provider name (SPN), service dialing numbers (SDN), advice-of-charge parameters and value-added service (VAS) applications. (Refer to GSM 11.11.[18])
SIM cards can come in various data capacities, from8 KBto at least256 KB.[11]All can store a maximum of 250 contacts on the SIM, but while the32 KBhas room for 33Mobile country code(MCCs) ornetwork identifiers, the64 KBversion has room for 80 MNCs.[1]This is used by network operators to store data on preferred networks, mostly used when the SIM is not in its home network but isroaming. The network operator that issued the SIM card can use this to have a phone connect to a preferred network that is more economic for the provider instead of having to pay the network operator that the phone discovered first. This does not mean that a phone containing this SIM card can connect to a maximum of only 33 or 80 networks, instead it means that the SIM card issuer can specify only up to that number of preferred networks. If a SIM is outside these preferred networks, it uses the first or best available network.[14]
Each SIM is internationally identified by itsintegrated circuit card identifier(ICCID). Nowadays ICCID numbers are also used to identify eSIM profiles, not only physical SIM cards. ICCIDs are stored in the SIM cards and are also engraved or printed on the SIM card body during a process called personalisation.
The ICCID is defined by the ITU-T recommendationE.118as theprimary account number.[19]Its layout is based onISO/IEC 7812. According to E.118, the number can be up to 19 digits long, including a single check digit calculated using theLuhn algorithm. However, the GSM Phase 1[20]defined the ICCID length as an opaque data field, 10 octets (20 digits) in length, whose structure is specific to amobile network operator.
The number is composed of three subparts:
Their format is as follows.
Issuer identification number (IIN)
Individual account identification
Check digit
With the GSM Phase 1 specification using 10octetsinto which ICCID is stored as packed BCD[clarification needed], the data field has room for 20 digits with hexadecimal digit "F" being used as filler when necessary. In practice, this means that on GSM cards there are 20-digit (19+1) and 19-digit (18+1) ICCIDs in use, depending upon the issuer. However, a single issuer always uses the same size for its ICCIDs.
As required by E.118, the ITU-T updates a list of all current internationally assigned IIN codes in its Operational Bulletins which are published twice a month (the last as of January 2019 was No. 1163 from 1 January 2019).[22]ITU-T also publishes complete lists: as of August 2023, the list issued on 1 December 2018 was current, having all issuer identifier numbers before 1 December 2018.[23]
SIM cards are identified on their individual operator networks by a uniqueinternational mobile subscriber identity(IMSI).Mobile network operatorsconnect mobile phone calls and communicate with their market SIM cards using their IMSIs. The format is:
The Kiis a 128-bit value used in authenticating the SIMs on aGSMmobile network (for USIM network, the Kiis still needed but other parameters are also needed). Each SIM holds a unique Kiassigned to it by the operator during the personalisation process. The Kiis also stored in a database (termedauthentication centeror AuC) on the carrier's network.
The SIM card is designed to prevent someone from getting the Kiby using thesmart-card interface. Instead, the SIM card provides a function,Run GSM Algorithm, that the phone uses to pass data to the SIM card to be signed with the Ki. This, by design, makes using the SIM card mandatory unless the Kican be extracted from the SIM card, or the carrier is willing to reveal the Ki. In practice, the GSM cryptographic algorithm for computing a signed response (SRES_1/SRES_2: see steps 3 and 4, below) from the Kihas certain vulnerabilities[1]that can allow the extraction of the Kifrom a SIM card and the making of aduplicate SIM card.
Authentication process:
The SIM stores network state information, which is received from thelocation area identity(LAI). Operator networks are divided into location areas, each having a unique LAI number. When the device changes locations, it stores the new LAI to the SIM and sends it back to the operator network with its new location. If the device is power cycled, it takes data off the SIM, and searches for the prior LAI.
Most SIM cards store a number of SMS messages and phone book contacts. It stores the contacts in simple "name and number" pairs. Entries that contain multiple phone numbers and additional phone numbers are usually not stored on the SIM card. When a user tries to copy such entries to a SIM, the handset's software breaks them into multiple entries, discarding information that is not a phone number. The number of contacts and messages stored depends on the SIM; early models stored as few as five messages and 20 contacts, while modern SIM cards can usually store over 250 contacts.[24]
SIM cards have been made smaller over the years; functionality is independent of format. Full-size SIM was followed by mini-SIM, micro-SIM, and nano-SIM. SIM cards are also made to embed in devices.
JEDECDesign Guide 4.8, SON-8GSMA SGP.22 V1.0
All versions of the non-embedded SIM cards share the sameISO/IEC 7816pin arrangement.
Themini-SIMor (2FF , 2nd form factor) card has the same contact arrangement as the full-size SIM card and is normally supplied within a full-size card carrier, attached by a number of linking pieces. This arrangement (defined inISO/IEC 7810asID-1/000) lets such a card be used in a device that requires a full-size card – or in a device that requires a mini-SIM card, after breaking the linking pieces. As the full-size SIM is obsolete, some suppliers refer to the mini-SIM as a "standard SIM" or "regular SIM".
Themicro-SIM(or 3FF) card has the same thickness and contact arrangements, but reduced length and width as shown in the table above.[25]
The micro-SIM was introduced by theEuropean Telecommunications Standards Institute(ETSI) along with SCP,3GPP(UTRAN/GERAN),3GPP2(CDMA2000),ARIB,GSM Association(GSMA SCaG and GSMNA), GlobalPlatform,Liberty Alliance, and theOpen Mobile Alliance(OMA) for the purpose of fitting into devices too small for a mini-SIM card.[21][26]
The form factor was mentioned in the December 1998 3GPP SMG9UMTSWorking Party, which is the standards-setting body for GSM SIM cards,[24]and the form factor was agreed upon in late 2003.[27]
The micro-SIM was designed for backward compatibility. The major issue for backward compatibility was the contact area of the chip. Retaining the same contact area makes the micro-SIM compatible with the prior, larger SIM readers through the use of plastic cutout surrounds. The SIM was also designed to run at the same speed (5 MHz) as the prior version. The same size and positions of pins resulted in numerous "How-to" tutorials and YouTube videos with detailed instructions how to cut a mini-SIM card to micro-SIM size.
The chairman of EP SCP, Klaus Vedder, said[27]
ETSI has responded to a market need from ETSI customers, but additionally there is a strong desire not to invalidate, overnight, the existing interface, nor reduce the performance of the cards.
Micro-SIM cards were introduced by various mobile service providers for the launch of the original iPad, and later for smartphones, from April 2010. TheiPhone 4was the first smartphone to use a micro-SIM card in June 2010, followed by many others.[28]
After a debate in early 2012 between a few designs created by Apple,NokiaandRIM, Apple's design for an even smaller SIM card was accepted by the ETSI.[29][30]Thenano-SIM(or 4FF) card was introduced in June 2012, when mobile service providers in various countries first supplied it for phones that supported the format. The nano-SIM measures 12.3 mm × 8.8 mm × 0.67 mm (0.484 in × 0.346 in × 0.026 in) and reduces the previous format to the contact area while maintaining the existing contact arrangements.[31]A small rim of isolating material is left around the contact area to avoid short circuits with the socket. The nano-SIM can be put into adapters for use with devices designed for 2FF or 3FF SIMs, and is made thinner for that purpose,[32]and telephone companies give due warning about this.[33]4FF is 0.67 mm (0.026 in) thick, compared to the 0.76 mm (0.030 in) of its predecessors.
TheiPhone 5, released in September 2012, was the first device to use a nano-SIM card,[34]followed by other handsets.
In July 2013, Karsten Nohl, a security researcher from SRLabs, described[35][36]vulnerabilities in some SIM cards that supportedDES, which, despite its age, is still used by some operators.[36]The attack could lead to the phone being remotelyclonedor let someone steal payment credentials from the SIM.[36]Further details of the research were provided atBlackHaton 31 July 2013.[36][37]In response, theInternational Telecommunication Unionsaid that the development was "hugely significant" and that it would be contacting its members.[38]
In February 2015,The Interceptreported that theNSAandGCHQhad stolen the encryption keys (Ki's) used byGemalto(now known asThales DIS, manufacturer of 2 billion SIM cards annually)[39]), enabling these intelligence agencies to monitor voice and data communications without the knowledge or approval of cellular network providers or judicial oversight.[40]Having finished its investigation, Gemalto claimed that it has “reasonable grounds” to believe that the NSA and GCHQ carried out an operation to hack its network in 2010 and 2011, but says the number of possibly stolen keys would not have been massive.[41]
In September 2019, Cathal Mc Daid, a security researcher from Adaptive Mobile Security, described[42][43]how vulnerabilities in some SIM cards that contained the S@T Browser library were being actively exploited. This vulnerability was namedSimjacker. Attackers were using the vulnerability to track the location of thousands of mobile phone users in several countries.[44]Further details of the research were provided atVirusBulletinon 3 October 2019.[45][46]
When GSM was already in use, the specifications were further developed and enhanced with functionality such asSMSandGPRS. These development steps are referred as releases by ETSI. Within these development cycles, the SIM specification was enhanced as well: new voltage classes, formats and files were introduced.
In GSM-only times, the SIM consisted of the hardware and the software. With the advent of UMTS, this naming was split: the SIM was now an application and hence only software. The hardware part was called UICC. This split was necessary because UMTS introduced a new application, the universal subscriber identity module (USIM). The USIM brought, among other things, security improvements like mutual authentication and longer encryption keys, and an improved address book.
"SIM cards" in developed countries today are usuallyUICCscontaining at least a SIM application and a USIM application. This configuration is necessary because older GSM only handsets are solely compatible with the SIM application and some UMTS security enhancements rely on the USIM application.
OncdmaOnenetworks, the equivalent of the SIM card is theR-UIMand the equivalent of the SIM application is theCSIM.
Avirtual SIMis a mobile phone number provided by amobile network operatorthat does not require a SIM card to connect phone calls to a user's mobile phone.
An embedded SIM (eSIM) is a form of programmable SIM that is embedded directly into a device.[47]The surface mount format provides the same electrical interface as the full size, 2FF and 3FF SIM cards, but is soldered to a circuit board as part of the manufacturing process. In M2M applications where there is no requirement[15]to change the SIM card, this avoids the requirement for a connector, improving reliability and security.[citation needed]An eSIM can beprovisioned remotely; end-users can add or remove operators without the need to physically swap a SIM from the device or use multiple eSIM profiles at the same time.[48][49]
The eSIM standard, initially introduced in 2016, has progressively supplanted traditional physical SIM cards across various sectors, notably in cellular telephony.[50][51][52]In September 2017, Apple introduced the Apple Watch Series 3 featuring eSIM.[53]In October 2018, Apple introduced theiPad Pro (3rd generation),[54]which was the first iPad to support eSIM. In September 2022, Apple introduced the iPhone 14 series which was the first eSIM exclusive iPhone in the United States.[55]
An integrated SIM (iSIM) is a form of SIM directly integrated into the modem chip or main processor of the device itself. As a consequence they are smaller, cheaper and more reliable than eSIMs, they can improve security and ease the logistics and production of small devices i.e. forIoTapplications. In 2021,Deutsche Telekomintroduced thenuSIM, an "Integrated SIM for IoT".[56][57][58]
The use of SIM cards is mandatory inGSMdevices.[59][60]
Thesatellite phonenetworksIridium,ThurayaandInmarsat'sBGANalso use SIM cards. Sometimes, these SIM cards work in regular GSM phones and also allow GSM customers to roam in satellite networks by using their own SIM cards in a satellite phone.
Japan's 2GPDCsystem (which was shut down in 2012;SoftBank Mobileshut down PDC from 31 March 2010) also specified a SIM, but this has never been implemented commercially. The specification of the interface between the Mobile Equipment and the SIM is given in theRCRSTD-27 annexe 4. The Subscriber Identity Module Expert Group was a committee of specialists assembled by the European Telecommunications Standards Institute (ETSI) to draw up the specifications (GSM11.11) for interfacing between smart cards and mobile telephones. In 1994, the name SIMEG was changed to SMG9.
Japan's current and next-generation cellular systems are based on W-CDMA (UMTS) andCDMA2000and all use SIM cards. However, Japanese CDMA2000-based phones are locked to the R-UIM they are associated with and thus, the cards are not interchangeable with other Japanese CDMA2000 handsets (though they may be inserted into GSM/WCDMA handsets for roaming purposes outside Japan).
CDMA-based devices originally did not use a removable card, and the service for these phones is bound to a unique identifier contained in the handset itself. This is most prevalent in operators in the Americas. The first publication of the TIA-820 standard (also known as 3GPP2 C.S0023) in 2000 defined the Removable User Identity Module (R-UIM). Card-based CDMA devices are most prevalent in Asia.
The equivalent of a SIM inUMTSis called the universal integrated circuit card (UICC), which runs a USIM application. The UICC is still colloquially called aSIM card.[61]
The SIM card introduced a new and significant business opportunity forMVNOswho lease capacity from one of the network operators rather than owning or operating a cellular telecoms network and only provide a SIM card to their customers. MVNOs first appeared in Denmark, Hong Kong, Finland and the UK. By 2011 they existed in over 50 countries, including most of Europe, the United States, Canada, Mexico, Australia and parts of Asia, and accounted for approximately 10% of all mobile phone subscribers around the world.[62]
On some networks, the mobile phone islocked to its carrier SIM card, meaning that the phone only works with SIM cards from the specific carrier. This is more common in markets where mobile phones are heavily subsidised by the carriers, and the business model depends on the customer staying with the service provider for a minimum term (typically 12, 18 or 24 months). SIM cards that are issued by providers with an associated contract, but where the carrier does not provide a mobile device (such as a mobile phone) are calledSIM-onlydeals. Common examples are the GSM networks in the United States, Canada, Australia, and Poland. UK mobile networks ended SIM lock practices in December 2021. Many businesses offer the ability to remove the SIM lock from a phone, effectively making it possible to then use the phone on any network by inserting a different SIM card. Mostly, GSM and 3G mobile handsets can easily be unlocked and used on any suitable network with any SIM card.
In countries where the phones are not subsidised, e.g., India, Israel and Belgium, all phones are unlocked. Where the phone is not locked to its SIM card, the users can easily switch networks by simply replacing the SIM card of one network with that of another while using only one phone. This is typical, for example, among users who may want to optimise their carrier's traffic by different tariffs to different friends on different networks, or when travelling internationally.
In 2016, carriers started using the concept of automatic SIM reactivation[63]whereby they let users reuse expired SIM cards instead of purchasing new ones when they wish to re-subscribe to that operator. This is particularly useful in countries whereprepaid callsdominate and where competition drives highchurn rates, as users had to return to a carrier shop to purchase a new SIM each time they wanted to churn back to an operator.
Commonly sold as a product by mobiletelecommunicationscompanies, "SIM-only" refers to a type oflegally liabilitycontract between a mobile network provider and a customer. The contract itself takes the form of a credit agreement and is subject to a credit check.
SIM-only contracts can bepre-pay- where the subscriber buyscreditbefore use (often called pay as you go, abbreviated to PAYG), orpost-pay, where the subscriber pays in arrears, typically monthly.
Within a SIM-only contract, the mobile network provider supplies their customer with just one piece of hardware, a SIM card, which includes an agreed amount of network usage in exchange for a monthly payment. Network usage within a SIM-only contract can be measured in minutes, text, data or any combination of these. The duration of a SIM-only contract varies depending on the deal selected by the customer, but in the UK they are typically available over 1, 3, 6, 12 or 24-month periods.
SIM-only contracts differ from mobile phone contracts in that they do not include any hardware other than a SIM card. In terms of network usage, SIM-only is typically more cost-effective than other contracts because the provider does not charge more to offset the cost of a mobile device over the contract period. The short contract length is one of the key features of SIM-only – made possible by the absence of a mobile device.
SIM-only is increasing in popularity very quickly.[64]In 2010 pay monthly based mobile phone subscriptions grew from 41 percent to 49 percent of all UK mobile phone subscriptions.[65]According to German research companyGfK, 250,000 SIM-only mobile contracts were taken up in the UK during July 2012 alone, the highest figure since GfK began keeping records.
Increasing smartphone penetration combined with financial concerns is leading customers to save money by moving onto a SIM-only when their initial contract term is over.
Dual SIMdevices have two SIM card slots for the use of two SIM cards, from one or multiple carriers. Multiple SIM devices are commonplace in developing markets such as inAfrica,East Asia,South AsiaandSoutheast Asia, where variable billing rates, network coverage and speed make it desirable for consumers to use multiple SIMs from competing networks. Dual-SIM phones are also useful to separate one's personal phone number from a business phone number, without having to carry multiple devices. Some popular devices, such as theBlackBerry KeyOne, have dual-SIM variants; however, dual-SIM devices were not common in the US or Europe due to lack of demand. This has changed with mainline products from Apple and Google featuring either two SIM slots or a combination of a physical SIM slot and an eSIM.
In September 2018,AppleintroducediPhone XS,iPhone XS Max, andiPhone XRfeaturing Dual SIM (nano-SIM andeSIM) andApple Watch Series 4featuring DualeSIM.
Athin SIM(oroverlay SIMorSIM overlay) is a very thin device shaped like a SIM card, approximately 120 microns (1⁄200inch) thick. It has contacts on its front and back. It is used by placing it on top of a regular SIM card. It provides its own functionality while passing through the functionality of the SIM card underneath. It can be used to bypass the mobile operating network and run custom applications, particularly on non-programmable cell phones.[66]
Its top surface is a connector that connects to the phone in place of the normal SIM. Its bottom surface is a connector that connects to the SIM in place of the phone. With electronics, it can modify signals in either direction, thus presenting a modified SIM to the phone, and/or presenting a modified phone to the SIM. (It is a similar concept to theGame Genie, which connects between a game console and a game cartridge, creating a modified game). Similar devices have also been developed for iPhones to circumvent SIM card restrictions on carrier-locked models.[67]
In 2014,Equitel, an MVNO operated by Kenya'sEquity Bank, announced its intention to begin issuing thin SIMs to customers, raising security concerns by competition, particularly concerning the safety of mobile money accounts. However, after months of security testing and legal hearings before the country's Parliamentary Committee on Energy, Information and Communications, theCommunications Authority of Kenya(CAK) gave the bank the green light to roll out its thin SIM cards.[68]
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Removable User Identity Module(R-UIM, usually pronounced as "R-yuim") is a card developed forcdmaOne/CDMA2000("CDMA") handsets that extends theGSMSIMcard to CDMA phones and networks. To work in CDMA networks, the R-UIM contains an early version of theCSIMapplication. The card also contains SIM (GSM) application, so it can work on both networks. It is physically compatible with GSM SIMs and can fit into existing GSM phones as it is an extension of the GSM 11.11 standard.[1]
Thisinterfacebrings one of the main advantages of GSM to CDMA network phones. By having a removable identity card, CDMA users can change phones while keeping their phone numbers by simply swapping the cards. This simplifies many situations such as phone upgrades, phone replacements due to damage, or using the same phone on a different provider's CDMA network.
The R-UIM card has been superseded byCSIMonUICC. This technique allows all three applications (SIM, CSIM, andUSIM) to coexist on a single smartcard, allowing the card to be used in virtually any phone worldwide that supports smart cards.
The CSIM application, a port of R-UIM functionality to the UICC, is defined in standard.[2]
This form of card is widely used inChinaunder the CDMA service ofChina Telecom(which was acquired byChina Unicomin 2008). However, it is also used elsewhere such asIndia,Indonesia,Japan,Taiwan,Thailand, and theUS.
This article related totelecommunicationsis astub. You can help Wikipedia byexpanding it.
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1-Wireis awiredhalf-duplexserialbusdesigned byDallas Semiconductorthat provides low-speed (16.3 kbit/s[1])data communicationandsupply voltageover a singleconductor.[2]
1-Wire is similar in concept toI2C, but with lower data rates and longer range. It is typically used to communicate with small inexpensivedevicessuch as digitalthermometersand weather instruments. A network of 1-Wire devices with an associatedmasterdevice is called aMicroLAN. The protocol is also used in small electronic keys known as aDallas keyoriButton.
One distinctive feature of the bus is the possibility of using only two conductors — data and ground. To accomplish this, 1-Wire devices integrate a smallcapacitor(~800pF) to store charge, which powers the device during periods when the data line is active.
1-Wire devices are available in different packages:integrated circuits, aTO-92-style package (as typically used for transistors), and a portable form called an iButton or Dallas key which is a small stainless-steel package that resembles awatch battery. Manufacturers also produce devices more complex than a single component that use the 1-Wire bus to communicate.
1-Wire devices can fit in different places in a system. It might be one of many components on a circuit board within a product. It also might be a single component within a device such as a temperature probe. It could be attached to a device being monitored. Some laboratory systems connect to 1-Wire devices using cables withmodular connectorsorCAT-5cable. In such systems,RJ11(6P2C or 6P4Cmodular plugs, commonly used for telephones) are popular.
Systems of sensors and actuators can be built by wiring together many 1-Wire components. Each 1-Wire component contains all of the logic needed to operate on the 1-Wire bus. Examples includetemperatureloggers, timers,voltageand current sensors, battery monitors, andmemory. These can be connected to a PC using a bus converter.USB,RS-232serial, andparallel portinterfaces are popular solutions for connecting a MicroLan to the host PC. 1-Wire devices can also be interfaced directly to microcontrollers from various vendors.
iButtons are connected to 1-Wire bus systems by means of sockets with contacts that touch the "lid" and "base" of the canister. Alternatively, the connection can be semi-permanent with a socket into which the iButton clips, but from which it is easily removed.
Each 1-Wire chip has a unique identifier code. This feature makes the chips, especially iButtons, suitable electronic keys. Some uses include locks, burglar alarms, computer systems, manufacturer-approved accessories, time clocks and courier and maintenance keys for smart safes. iButtons have been used asAkbil smart ticketsfor thepublic transport in Istanbul.
AppleMagSafe- and MagSafe-2-connector–equipped power supplies, displays, and Mac laptops use the 1-Wire protocol to send and receive data to and from the connected Mac laptop, via the middle pin of the connector. Data include power supply model, wattage, and serial number; and laptop commands to send full power, and illuminate the red or greenlight-emitting diodesin the connector.[3]
GenuineDelllaptop power supplies use the 1-Wire protocol to send data via the third wire to thelaptop computerabout power, current and voltage ratings. The laptop will then refuse charging if the adapter does not meet requirements.[4]
In any MicroLan, there is always onemasterin overall charge, which may be apersonal computeror amicrocontroller. The master initiates activity on the bus, simplifying the avoidance of collisions on the bus. Protocols are built into the master's software to detect collisions. After a collision, the master retries the required communication.
A 1-Wire network is a singleopen drainwire with a singlepull-up resistor. The pull-up resistor pulls the wire up to 3 or 5 volts. The master device and all the slaves each have a single open-drain connection to drive the wire, and a way to sense the state of the wire. Despite the "1-Wire" name, all devices must also have a second conductor for agroundconnection to permit a return current to flow through the data wire.[5]Communication occurs when a master or slave briefly pulls the bus low,i.e., connects the pull-up resistor to ground through its output MOSFET. The data wire is high when idle, and so it can also power a limited number of slave devices. Data rates of 16.3 kbit/s can be achieved. There is also an overdrive mode that speeds up the communication by a factor of 10.
A short 1-Wire bus can be driven from a single digital I/O pin on a microcontroller. Auniversal asynchronous receiver-transmitter(UART) can also be used.[6]Specific 1-Wiredriverandbridgechips are available.Universal Serial Bus"bridge" chips are also available. Bridge chips are particularly useful to drive cables longer than 100 m. Up to 300-metertwisted pairs,i.e., telephone cables, have been tested by the manufacturer. These extreme lengths require adjustments to the pull-up resistances from5 to 1 kΩ.
The master starts a transmission with aresetpulse, which pulls the wire to 0 volts for at least 480μs. This resets every slave device on the bus. After that, any slave device, if present, shows that it exists with a "presence" pulse: it holds the bus low for at least 60 μs after the master releases the bus.
To send abinary number"1", the bus master sends a very brief (1–15 μs) low pulse. To send a binary number "0", the master sends a 60 μs low pulse. The falling (negative) edge of the pulse is used to start amonostable multivibratorin the slave device. The multivibrator in the slave reads the data line about 30 μs after the falling edge. The slave's internal timer is an inexpensive analog timer. It has analog tolerances that affect its timing accuracy. Therefore, the pulses are calculated to be within margins. Therefore, the "0" pulses have to be 60 μs long, and the "1" pulses can't be longer than 15 μs.
When receiving data, the master sends a1–15 μs0 voltpulse to start each bit. If the transmitting slave unit wants to send a "1", it does nothing, and the bus goes to the pulled-up voltage. If the transmitting slave wants to send a "0", it pulls the data line to ground for60 μs.
The basic sequence is a reset pulse followed by an eight-bit command, and then data are sent or received in groups of eight bits.
When a sequence of data is being transferred, errors can be detected with an eight-bitCRC(weak data protection).
Many devices can share the same bus. Each device on the bus has a 64-bit serial number, of which eight bits are used as a checksum, thus allowing a "universe" of 256(over 7.2 × 1016) unique device identities. Theleast significant byteof the serial number is an eight-bit number that tells the type of the device. Themost significant byteis a standard (for the 1-Wire bus) eight-bit CRC.[7]
There are several standard broadcast commands, as well as commands used to address a particular device. The master can send a selection command, then the address of a particular device. The next command is executed only by the addressed device.
The 1-Wire bus enumeration protocol, like othersingulationprotocols, is an algorithm the master uses to read the address of every device on the bus. Since the address includes the device type and a CRC, recovering the roster of addresses also produces a reliable inventory of the devices on the bus. To find the devices, the master broadcasts anenumerationcommand, and then an address, "listening" after each bit of an address. If a slave's address matches all the address bits sent so far, it returns a 0. The master uses this simple behavior to search systematically for valid sequences of address bits. The process is much faster than a brute force search of all possible 56-bit numbers, because as soon as an invalid bit is detected, all subsequent address bits are known to be invalid. The 56-bit address space is searched as a binary tree, allowing up to 75 devices to be found per second. The order in which device addresses are discovered by this enumeration protocol is deterministic and depends only on the device type and serial number. Bit-reversing these 56 bits yields the order of discovery for devices using Maxim's published algorithm (algorithm defined in Application Note 187[8]). The search algorithm can be implemented in an alternative form, initially searching paths with address bits equal to 1, rather than 0. In this case, inverting the 56 address bits and then reversing them yields the order of discovery.
The location of devices on the bus is sometimes significant. For these situations, a microcontroller can use several pins, or the manufacturer has a 1-Wire device that can switch the bus off or pass it on. Software can therefore explore sequentialbusdomains.[7]
Every 1-Wire device’s 64-bit ROM ID ends with an 8-bit family code.
In most cases this byte is assigned to a single part number, so reading it from the bus is usually enough to identify the device—for example, 0x10 (DS18S20 thermometer),[9]0x01 (DS2401 silicon serial number),[10]or 0x2D (DS2431 1 kbit EEPROM).[11]
Various people have created online databases of family codes from the broad range of 1-Wire memory, authenticator, ID, and battery-monitor devices.[12]
The following signals were generated by anFPGA, which was the master for the communication with a DS2432 (EEPROM) chip, and measured with a logic analyzer. A logic high on the 1-Wire output, means the output of the FPGA is in tri-state mode and the 1-Wire device can pull the bus low. A low means the FPGA pulls down the bus. The 1-Wire input is the measured bus signal. On input sample time high, the FPGA samples the input for detecting the device response and receiving bits.
When developing and/or troubleshooting the 1-Wire bus, examination of hardware signals can be very important.Logic analyzersandbus analyzersare tools that collect, analyze, decode, and store signals to simplify viewing the high-speed waveforms.
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NFC-WIisNFCwiredinterfacehaving 2 wires SIGIN (signal-in) and SIGOUT (signal-out).[1]It is also called S2C (SignalIn/SignalOut Connection) interface.[2]In 2006,ECMAstandardized the NFC wired interface with specificationECMA-373(ECMA, 2006).[3]
It has three modes of operation: off, wired and virtual mode. In off mode, there is no communication with the SE. In wired mode, the SE is visible to the internal NFC controller.[4]In virtual mode, the SE is visible to external RF readers. These modes are naturally mutually exclusive.
This article related totelecommunicationsis astub. You can help Wikipedia byexpanding it.
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Host Controller InterfaceorHost controller interfacemay refer to:
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Below are abbreviations used inaviation,avionics,aerospace, andaeronautics.
CAT I Enhanced: Allows for lower minimums than CAT I in some cases to CAT 2 minimumsCAT II: Operational performance Category IICAT IIIa: Operational performance Category IIIaCAT IIIb: Operational performance Category IIIbCAT IIIc: Operational performance Category IIIc
Sources
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Animmobiliserorimmobilizeris an electronic security device fitted to amotor vehiclethat prevents the engine from being started unless the correct key (transponderorsmart key) is present. This prevents the vehicle from being "hot wired" after entry has been achieved and thus reducesmotor vehicle theft. Research shows that the uniform application of immobilisers reduced the rate of car theft by 40%.[1]
The electric immobiliser/alarm system was invented by St. George Evans and Edward Birkenbeuel and patented in 1919.[2]They developed a 3x3 grid of double-contactswitcheson a panel mounted inside the car so when the ignition switch was activated, current from the battery (ormagneto) went to the spark plugs allowing the engine to start, or immobilizing the vehicle andsoundingthehorn.[3]The system settings could be changed each time the car was driven.[3]Modern immobiliser systems are automatic, meaning the owner does not have to remember to activate it.[4][5]
Early models used a static code in theignition key(orkey fob) which was recognised by anRFIDloop (transponder) around the lock barrel and checked against the vehicle'sengine control unit(ECU) for a match. If the code is unrecognised, the ECU will not allow fuel to flow and ignition to take place.
Later models userolling codesor advancedcryptographyto defeat copying of the code from the key or ECU (smart key).[citation needed]The microcircuit inside the key is activated by a small electromagnetic field which induces current to flow inside the key body, which in turn broadcasts a uniquebinary code, which is read by the automobile's ECU. When the ECU determines that the coded key is both current and valid, the ECU activates the fuel-injection sequence.[citation needed]
In some vehicles, attempts to use an unauthorised or "non-sequenced" key cause the vehicle to activate a timed "no-start condition" and in some highly advanced systems, even use satellite or mobile phone communication to alert asecurity firmthat an unauthorised attempt was made to code a key.[citation needed]
Coincidentally, this information is often recorded in modern automobile ECUs as part of theiron-board diagnosticswhich may record many other variables including speed, temperature, driver weight, geographic location,throttle positionandyaw angle. This information can be used during insurance investigations, warranty claims or technical troubleshooting.[citation needed]
Immobilisers have been mandatory in all new cars sold inGermanysince 1 January 1998, in theUnited Kingdomsince 1 October 1998, inFinlandsince 1998, inAustraliasince 2001.[citation needed]
In September 2007, aTransport Canadaregulation mandated the installation of engine immobilisers in all new lightweight vehicles and trucks manufactured in Canada.[6]
Hondawas the firstmotorcyclemanufacturer to include immobilisers on its products in the 1990s.[7]Add-on immobilisers are available for older cars or vehicles that do not come equipped with factory immobilisers. The insurance approval for a self-arming immobiliser is known as "Thatcham 2" after the Motor Insurance Repair Research Centre inThatcham,England. Approved immobilisers must intercept at least two circuits; typically the low-voltage ignition circuit and the fuel pump circuit. Some may also intercept the low-current starter motor circuit from the key switch to therelay.
Lack of immobilizers in manyKiaandHyundaiU.S. models after 2010 and before mid-2021 made these cars targets for theft in the early 2020s, especially inMilwaukee County, WisconsinandColumbus, Ohio.[8]TheKia ChallengeTikTok trend was linked to series of Hyundai/Kia vehicle thefts in 2022.
Numerous vulnerabilities have been found in the immobilisers designed to protect modern cars from theft.[9]Many vehicle immobilisers use the Megamos chip, which has been proven to be crackable.[10]The Megamos transponder is one of many different transponders found in today's immobiliser systems and also comes in many different versions. Hacking of an immobiliser in the real world would be performed on the vehicle, not on the key. It would be faster to program a new key to the vehicle than to try to clone the existing key, especially on modern vehicles.[11]
Some immobiliser systems tend to remember the last key code for so long that they may even accept a non-transponder key even after the original key has been removed from the ignition for a few minutes.[12]
A 2016 study in theEconomic Journalfound that the immobiliser lowered the overall rate of car theft by about 40% between 1995 and 2008.[1]The benefits in terms of prevented thefts were at least three times higher than the costs of installing the device.[1]
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Inradio communication, atransceiveris an electronic device which is a combination of a radiotransmitterand areceiver, hence the name. It can both transmit and receiveradio wavesusing anantenna, for communication purposes. These two related functions are often combined in a single device to reduce manufacturing costs. The term is also used for other devices which can both transmit and receive through acommunications channel, such asoptical transceiverswhich transmit and receive light inoptical fibersystems, andbus transceiverswhich transmit and receivedigital datain computerdata buses.
Radio transceivers are widely used inwireless devices. One large use is intwo-way radios, which areaudiotransceivers used for bidirectional person-to-person voice communication. Examples arecell phones, which transmit and receive the two sides of a phone conversation using radio waves to acell tower,cordless phonesin which both the phone handset and the base station have transceivers to communicate both sides of the conversation, andland mobile radio systemslikewalkie-talkiesandCB radios. Another large use is inwireless modemsin mobile networked computer devices suchlaptops, pads, and cellphones, which both transmit digital data to and receive data from awireless router. Aircraft carry automatedmicrowavetransceivers calledtransponderswhich, when they are triggered by microwaves from anair traffic control radar, transmit a coded signal back to the radar to identify the aircraft. Satellite transponders incommunication satellitesreceive digital telecommunication data from asatellite ground station, and retransmit it to another ground station.
The transceiver first appeared in the 1920s.[citation needed]Before then, receivers and transmitters were manufactured separately and devices that wanted to receive and transmit data required both components. Almost all amateur radio equipment today[when?]uses transceivers, but there is an active market for pure radio receivers, which are mainly used byshortwave listeningoperators.[citation needed]
Analog transceivers usefrequency modulationto send and receive data. Although this technique limits the complexity of the data that can be broadcast, analog transceivers operate very reliably and are used in many emergency communication systems. They are also cheaper than digital transceivers, which makes them popular with theCBandHAM radiocommunities.
Digital transceivers send and receivebinary dataover radio waves. This allows more types of data to be broadcast, including video and encrypted communication, which is commonly used by police and fire departments. Digital transmissions tend to be clearer and more detailed than their analog counterparts. Many modern wireless devices operate on digital transmissions.
In a wiredtelephone, the handset contains the transmitter (for speaking) and receiver (for listening). Despite being able to transmit and receive data, the whole unit is colloquially referred to as a "receiver". On amobile telephoneor otherradiotelephone, the entire unit is a transceiver for both audio and radio.
Acordless telephoneuses an audio and radio transceiver for the handset, and a radio transceiver for thebase station. If aspeakerphoneis included in a wired telephone base or in a cordless base station, the base also becomes an audio transceiver.
Amodemis similar to a transceiver in that it sends and receives a signal, but a modem uses modulation and demodulation. It modulates the signal being transmitted and demodulates the signal being received.
Transceivers are calledMediumAttachment Units (MAUs) inIEEE 802.3documents and were widely used in10BASE2and10BASE5Ethernetnetworks.Fiber-opticgigabit,10 Gigabit Ethernet,40 Gigabit Ethernet, and100 Gigabit EthernetutilizeGBIC,SFP,SFP+,QSFP,XFP,XAUI,CXP, andCFPtransceiver systems.
Because transceivers are capable of broadcasting information over airwaves, they are required to adhere to various regulations. In the United States, theFederal Communications Commissionoversees their use. Transceivers must meet certain standards and capabilities depending on their intended use, and manufacturers must comply with these requirements. However, transceivers can be modified by users to violate FCC regulations. For instance, they might be used to broadcast on a frequency or channel that they should not have access to. For this reason, the FCC monitors not only the production but also the use of these devices.
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Infiber-optic communications, amuxponderis the element that sends and receives the optical signal on afiberin much the same way as atransponderexcept that the muxponder has the additional functionality ofmultiplexingmultiple sub-rate client interfaces onto the line interface.[1][2]
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TheRebecca/Eureka transponding radarwas a short-rangeradio navigationsystem used for the dropping ofairborne forcesand their supplies. It consisted of two parts, theRebeccaairbornetransceiverand antenna system, and theEurekaground-basedtransponder. Rebecca calculated the range to the Eureka based on the timing of the return signals, and its relative position using a highly directional antenna. The 'Rebecca' name comes from the phrase "Recognition of beacons". The 'Eureka' name comes from the Greek word meaning "I have found it!".
The system was developed in the UK at theTelecommunications Research EstablishmentbyRobert Hanbury BrownandJohn William Sutton Pringle. Rebecca was essentially anASV radarfit to a new broadcaster unit, while the Eureka system was all-new. Initial production began in 1943, and the system was used for dropping supplies to resistance fighters in occupied Europe, after delivery of the portable Eureka unit. TheUS Army Air Forcestarted production in the US as well, and both examples could be used interchangeably. Over time, the Rebecca/Eureka found a number of other uses, including blind-bombing, airfield approach, and as a blind-landing aid in theBABS(Beam Approach Beacon Signal) form.
As many of the war-era systems used similar display units, theLucerosystem was introduced to send the proper signals to interrogate any of these systems, allowing a single display unit of any type to be used forH2S, ASV,AI, Rebecca and BABS.
Rebecca/Eureka owes its existence largely to the efforts ofRobert Hanbury Brown, an astronomer and physicist who worked with theAir Ministry'sAMESgroup on the development of radar. During 1940, Brown had led development of a new version of theAI Mk. IV radarthat included apilot's indicator, better known today as aC-scope. This display directly represented the relative position between the fighter and its target, as if the pilot were looking through a gunsight. It was hoped that this would greatly ease the problems that the radar operators had trying to relay instructions from their instruments to the pilot, especially at closer range.[1]
Prototype sets became available in late 1940, with the first production examples arriving in January 1941. During a test flight in February, the aircraft was flying at 20,000 feet (6.1 km) when Hanbury Brown's oxygen supply failed and he passed out. The test pilot, Peter Chamberlain, realized what had happened and quickly landed the plane. Brown awoke in an ambulance.[2]This accident, along with the many previous flights at high altitude, aggravated an ear injury he had received atRAF Martlesham Heathin 1939, and during the spring he was hospitalized for amastoidectomyoperation inBrighton. The operation was successful, but a post-operation infection caused him to go deaf in both ears and several follow-up visits were required.[2]
By the time he returned to the AMES research center, now inWorth Matraversand renamed theTelecommunications Research Establishment(TRE), major research on the early AI sets had ended in favour of new systems working atmicrowavefrequencies using the recently inventedcavity magnetron. Brown had missed most of the development of this system, and he was no longer allowed to fly at high altitudes, so his work on AI ended. He was instead placed in a new group led byJohn Pringle, a zoologist fromCambridge University, and the two began to study new applications for radar technologies.[3]
In June 1941, Brown visited the Army headquarters atOld Sarum Airfieldto see if theRAF Army Co-operation Command'sSchool of Army Co-operationmight put radar to good use. The Army Co-operation squadrons carried out a variety of missions including artillery spotting, general reconnaissance, andground attack. He found the group was only mildly interested in radar, thinking it might make a useful device for warning of the approach of enemy fighters, but were perfectly happy using flags and smoke signals for navigation and communications.[3]Brown then visited amilitary exerciseinvolving ground attacks in close coordination with the Army, and was convinced that radar systems could be used to improve these results. However, he also came to realize that almost all such missions would be carried out by aircraft of other forces, notably the RAF, so any system they proposed would have to be mounted in those aircraft.[4]
Pringle then arranged for Brown and himself to meet with the Commander in Chief (C-in-C) for Army Co-operation, SirArthur Barratt. In a long conversation, the two outlined the possibilities of radar for bombing, navigation and return-to-base roles, all of which proved to be interesting to Barratt. Barratt then stated that any system they did adopt would have to fit in single seat aircraft like theTomahawk, which eliminated most of these possibilities. Both Pringle and Brown then focused on the use of atranspondersystem combined with existing radars to allow accurate bombing or delivery of supplies or troops by parachute, a role that would almost always be carried out by twin-engine aircraft or larger. If this broadcast on the 200 MHz frequency then being used by many British radars, any aircraft withaircraft interception (AI)orair-to-surface-vessel (ASV)radarcould pick it up.[4]
To illustrate the concept, Brown gave them a small transponder and told them to hide it anywhere within 15 miles (24 km) of Army Co-operation headquarters inBracknell. One of the TRE aircraft fromRAF Christchurchwould attempt to find it and fire a smoke signal within 100 yards (91 m) of its location. The test was carried out on 28 July 1941, and while they waited for their aircraft to arrive, another aircraft approached the hiding spot and flew around several times before flying off again. The Army suspected that they had sent this aircraft to spy out the location. Just as Brown managed to convince them they were not spies, their own aircraft, aBristol Blenheim, arrived and fired a smoke signal only 50 yards from the transponder. To be sure, the Army was told to hide it again in another location, this time choosing to place under a tree on the lawn of their headquarters. Their Blenheim once again easily found it. It was later learned that the first aircraft was from theFighter Interception Unitwho saw the oddblipon their radar and decided to investigate.[5]
In spite of the successful demonstrations, and enthusiastic support from Barratt and others, no orders for a transponder system were immediately forthcoming. A visit to the airborne headquarters atRAF Ringwaywould ultimately result in orders for both the UK and US, but those would be some time in the making.[5]
A more immediate outcome of the visit to Ringway was an invitation for Brown to meet a secret group known as theSpecial Operations Executive(SOE) atWhitehall. Brown arrived to find this was not actually their office, and had to prove his identity before being told the real address was quite a distance away. When he finally arrived atBaker Streethe was not impressed.[6]
SOE then explained the problem they were having dropping supplies topartisansoperating across Europe, as far away asPoland. Brown explained that their beacon could be seen as far as 50 miles (80 km) under good conditions, but that might drop to as short as 5 miles if it was under trees or otherwise blocked. SOE stated that they could be reasonably certain to place it on open ground. However, they also stated that they could only navigate to perhaps ten miles of the beacon, which would be only marginally satisfactory. Brown asked why they could not use theGeenavigation system to address this, and when they admitted they had no idea what this was, he had the satisfaction of saying he could not explain it to them because it was secret.[6]
Ultimately, Brown was taken to meet the C-in-C of the SOE and arranged a demonstration similar to the one given earlier for the Army. SOE was given the transponder and told to hide it anywhere within a large area, and their aircraft would not attempt to find it until a week later. On 11 February 1942 one of the TRE'sAvro Ansonstook off fromRAF Hurnand picked it up at a range of 37 miles (60 km), approaching and dropping two containers within 200 yards. An order was placed immediately.[7]
One of the major problems with the original AI radars was that the transmissions spread out over the entire front hemisphere of the aircraft. Shorter wavelengths, like those used in AI, tended toscatterfrom the ground, sending a portion of the signal back towards the aircraft, the "ground reflection" or "ground return". For the simple reason that the ground is much larger than a target aircraft, the scattered signal overwhelmed any target return, and made it impossible to see any target further away than the aircraft's current altitude. For the supply mission, which was carried out at very low altitudes, this was clearly not going to work.[7]
As the SOE had their own aircraft, and there was no need to make the system work with an existing production radar design, the solution was relatively simple. Instead of the transponder replying on the same frequency, and thus being lost in the ground reflections, it would receive the signal from the radar and then re-broadcast it on a second frequency. In the aircraft, the receiver would be tuned not to the radar's broadcasts, but the transponder's. This way the ground reflection would simply be ignored, although at the cost of requiring the transponder to have two separate antenna systems.[7]
Powering the transponder was a more serious problem. The system had to operate in any weather, which made conventionallead-acid batteriesunsuitable due to their poor cold-weather performance. The system also had to be stored for long periods of time before being activated, which again argued against lead-acid. The solution was found to be smallnickel-iron batteriesthat could be repeatedly and rapidly recharged in the field, and operated across a wide range of temperatures. To protect the system from capture, it was fitted with small explosives that would destroy enough of the circuitry to make it impossible to determine the exact frequencies being used. The transponders were mounted in suitcases, selected on a case-by-case basis to match common types in use in that area.[8]
A dozen transponders were supplied to the SOE in 1942, at which point the TRE's involvement largely ended. Brown later learned that these were used extensively throughout the war. One example, dropped in Norway, was used on seven separate drops, in spite of being buried for the better part of a year in abiscuit tin.[8]
The Airborne Forces Equipment Committee took up development of the system in the summer of 1942, funding low-priority development of a Mark II system intended for use on glider tugs and paratroop aircraft. At the time, it was decided that each Eureka should be able to handle interrogation from up to 40 aircraft at a time. They also selected a design based on several sub-units that would allow the equipment to be changed simply by swapping sub-units from a common chassis.
Both Rebecca II and Eureka II were developed byMurphy Radio, with early pre-production of Rebecca II byDynatron Radio. A system using a selection of tunedcapacitorswas used to select the operational frequency from a set of four. Looking for a controller, Murphy selected aGeneral Post Office5-position electromechanical system used in theirtelephone exchangesystems. A similar selection of four channels was available in the Eureka units, but these were selected manually. Rebecca was powered off the aircraft mains, whilst Eureka was battery powered with a lifetime of about six hours.
In testing, Eureka II proved to be too heavy for practical use, soA.C. Cossorwas selected to build a Mk III version. They used US miniature 9000-series tubes for this version and a much smaller battery with a three-hour life.
In December 1942 Brown was flown to the US viaPan Am Clipperto meet with the USI Troop Carrier Command.[9]They started production of a number of versions of the Mk. III as the AN/PPN-1 (Eureka), AN/PPN-2 (Portable Eureka) and AN/TPN-1 (Transportable Eureka). The AN/APN-2 (Rebecca), also known as the SCR-729, used a display that saw use for a number of purposes.
When many Britishmilitary glidersfailed to reach their landing zones inSicilyeven in excellent conditions, a rushed effort to develop an even smaller and lighter Rebecca III system started. Cossor was again selected for the development, using a super-regenerative receiver and batteries for only 30 minutes of operation. The Rebecca IIIN version was used for strike aircraft in the Pacific theatre. These versions used capacitors in all five positions of the rotary switch.
The introduction of the miniature B7G tubes in 1944 led to a new round of development of the Rebecca/Eureka. Dozens of different variations were eventually developed.
The airborne Rebecca interrogator transmitted a 4-5 μs (microsecond) long pulse at a rate of 300 pulses per second on a frequency between 170 and 234 MHz. Upon receiving this signal, the Eureka rebroadcast the pulses on a different frequency. The Eureka unit also included a keying system that periodically lengthened the pulses over a period of seconds, allowing amorse codesignal to be sent for station identification.
This rebroadcast signal was received by two directionalyagi antennason the aircraft carrying the Rebecca unit, the usual location for the aerials being on either side of the aircraft cockpit. The signal was then sent to a conventional ASV radar display, with the vertical axis measuring time (and thus distance) and the horizontal showing the strength of the signal. If the aircraft was approaching the Eureka from the side, the horizontal pulse would extend further on one side of the display than the other, indicating the need for the aircraft to turn toward the shorter blip in order to fly directly toward the Eureka.
There was a slight delay in the Eureka between signal reception and the return pulse. As the Rebecca units approached the Eureka the return signal would eventually overlap the interrogation pulse, and render the system ineffective. This occurred at a range of about two miles. At this time the crew had to switch to visual means of locating the drop zone. Reliance on Eureka without visual confirmation invariably resulted in premature drops, as during theAmerican airborne landings in Normandy.
There were many versions of the system. Early models were limited to a single frequency; later ones could switch between five frequencies.
Eureka Mk VII was a rack-mounted, non-mobile transponder used at RAF bases for aircraft to home onto.
A Mark X version of both Rebecca and Eureka that worked in the 1000 MHz range. This was developed for use duringin-flight refueling, enabling the receiving aircraft to locate the tanker while maintaining radio silence. The tanker aircraft carried the Eureka and the receiving aircraft carried the Rebecca. This equipment was trialled by 214 Squadron in the early 1960s.
TheRebeccacode namewas derived from the phrase "recognition of beacons".
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https://en.wikipedia.org/wiki/Rebecca/Eureka_transponding_radar
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Incomputational complexity theory, theelement distinctness problemorelement uniqueness problemis the problem of determining whether all the elements of a list are distinct.
It is a well studied problem in many different models of computation. The problem may be solved bysortingthe list and then checking if there are any consecutive equal elements; it may also be solved in linear expected time by arandomized algorithmthat inserts each item into ahash tableand compares only those elements that are placed in the same hash table cell.[1]
Several lower bounds in computational complexity are proved by reducing the element distinctness problem to the problem in question, i.e., by demonstrating that the solution of the element uniqueness problem may be quickly found after solving the problem in question.
The number of comparisons needed to solve the problem of sizen{\displaystyle n}, in a comparison-based model of computation such as adecision treeoralgebraic decision tree, isΘ(nlogn){\displaystyle \Theta (n\log n)}. Here,Θ{\displaystyle \Theta }invokesbig theta notation, meaning that the problem can be solved in a number of comparisons proportional tonlogn{\displaystyle n\log n}(alinearithmic function) and that all solutions require this many comparisons.[2]In these models of computation, the input numbers may not be used to index the computer's memory (as in the hash table solution) but may only be accessed by computing and comparing simple algebraic functions of their values. For these models, an algorithm based oncomparison sortsolves the problem within a constant factor of the best possible number of comparisons. The same lower bound applies as well to theexpected numberof comparisons in therandomizedalgebraic decision treemodel.[3][4]
If the elements in the problem arereal numbers, the decision-tree lower bound extends to thereal random-access machinemodel with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor").[5]It follows that the problem's complexity in this model is alsoΘ(nlogn){\displaystyle \Theta (n\log n)}. This RAM model covers more algorithms than the algebraic decision-tree model, as it encompasses algorithms that use indexing into tables. However, in this model all program steps are counted, not just decisions.
A single-tape deterministicTuring machinecan solve the problem, fornelements ofm≥ lognbits each, in timeO(n2m(m+2–logn)),
while on a nondeterministic machine the time complexity isO(nm(n+ logm)).[6]
Quantum algorithmscan solve this problem faster, inΘ(n2/3){\textstyle \Theta (n^{2/3})}queries. The optimal algorithm is byAndris Ambainis.[7]Yaoyun Shifirst proved a tight lower bound when the size of the range is sufficiently large.[8]Ambainis[9]and Kutin[10]independently (and via different proofs) extended his work to obtain the lower bound for all functions.
Elements that occur more thann/k{\displaystyle n/k}times in a multiset of sizen{\displaystyle n}may be found by a comparison-based algorithm, theMisra–Gries heavy hitters algorithm, in timeO(nlogk){\displaystyle O(n\log k)}. The element distinctness problem is a special case of this problem wherek=n{\displaystyle k=n}. This time is optimal under thedecision tree modelof computation.[11]
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https://en.wikipedia.org/wiki/Element_distinctness_problem
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Inquantum computing,Grover's algorithm, also known as thequantum search algorithm, is aquantum algorithmfor unstructured search that findswith high probabilitythe unique input to ablack boxfunction that produces a particular output value, using justO(N){\displaystyle O({\sqrt {N}})}evaluations of the function, whereN{\displaystyle N}is the size of the function'sdomain. It was devised byLov Groverin 1996.[1]
The analogous problem in classical computation would have aquery complexityO(N){\displaystyle O(N)}(i.e., the function would have to be evaluatedO(N){\displaystyle O(N)}times: there is no better approach than trying out all input values one after the other, which, on average, takesN/2{\displaystyle N/2}steps).[1]
Charles H. Bennett, Ethan Bernstein,Gilles Brassard, andUmesh Vaziraniproved that any quantum solution to the problem needs to evaluate the functionΩ(N){\displaystyle \Omega ({\sqrt {N}})}times, so Grover's algorithm isasymptotically optimal.[2]Since classical algorithms forNP-complete problemsrequire exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution for unstructured search, this suggests that Grover's algorithm by itself will not providepolynomial-timesolutions for NP-complete problems (as the square root of an exponential function is still an exponential, not a polynomial function).[3]
Unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts, Grover's algorithm provides only a quadratic speedup. However, even quadratic speedup is considerable whenN{\displaystyle N}is large, and Grover's algorithm can be applied to speed up broad classes of algorithms.[3]Grover's algorithm couldbrute-forcea 128-bit symmetric cryptographic key in roughly 264iterations, or a 256-bit key in roughly 2128iterations. It may not be the case that Grover's algorithm poses a significantly increased risk to encryption over existing classical algorithms, however.[4]
Grover's algorithm, along with variants likeamplitude amplification, can be used to speed up a broad range of algorithms.[5][6][7]In particular, algorithms for NP-complete problems which contain exhaustive search as a subroutine can be sped up by Grover's algorithm.[6]The current theoretical best algorithm, in terms of worst-case complexity, for3SATis one such example. Genericconstraint satisfaction problemsalso see quadratic speedups with Grover.[8]These algorithms do not require that the input be given in the form of an oracle, since Grover's algorithm is being applied with an explicit function, e.g. the function checking that a set of bits satisfies a 3SAT instance. However, it is unclear whether Grover's algorithm could speed up best practical algorithms for these problems.
Grover's algorithm can also give provable speedups for black-box problems inquantum query complexity, including element distinctness[9]and thecollision problem[10](solved with theBrassard–Høyer–Tapp algorithm). In these types of problems, one treats the oracle functionfas a database, and the goal is to use the quantum query to this function as few times as possible.
Grover's algorithm essentially solves the task offunction inversion. Roughly speaking, if we have a functiony=f(x){\displaystyle y=f(x)}that can be evaluated on a quantum computer, Grover's algorithm allows us to calculatex{\displaystyle x}when giveny{\displaystyle y}. Consequently, Grover's algorithm gives broad asymptotic speed-ups to many kinds ofbrute-force attacksonsymmetric-key cryptography, includingcollision attacksandpre-image attacks.[11]However, this may not necessarily be the most efficient algorithm since, for example, thePollard's rho algorithmis able to find a collision inSHA-2more efficiently than Grover's algorithm.[12]
Grover's original paper described the algorithm as a database search algorithm, and this description is still common. The database in this analogy is a table of all of the function's outputs, indexed by the corresponding input. However, this database is not represented explicitly. Instead, an oracle is invoked to evaluate an item by its index. Reading a full database item by item and converting it into such a representation may take a lot longer than Grover's search. To account for such effects, Grover's algorithm can be viewed as solving an equation orsatisfying a constraint. In such applications, the oracle is a way to check the constraint and is not related to the search algorithm. This separation usually prevents algorithmic optimizations, whereas conventional search algorithms often rely on such optimizations and avoid exhaustive search.[13]Fortunately, fast Grover's oracle implementation is possible for many constraint satisfaction and optimization problems.[14]
The major barrier to instantiating a speedup from Grover's algorithm is that the quadratic speedup achieved is too modest to overcome the large overhead of near-term quantum computers.[15]However, later generations offault-tolerantquantum computers with better hardware performance may be able to realize these speedups for practical instances of data.
As input for Grover's algorithm, suppose we have a functionf:{0,1,…,N−1}→{0,1}{\displaystyle f\colon \{0,1,\ldots ,N-1\}\to \{0,1\}}. In the "unstructured database" analogy, the domain represent indices to a database, andf(x) = 1if and only if the data thatxpoints to satisfies the search criterion. We additionally assume that only one index satisfiesf(x) = 1, and we call this indexω. Our goal is to identifyω.
We can accessfwith asubroutine(sometimes called anoracle) in the form of aunitary operatorUωthat acts as follows:
{Uω|x⟩=−|x⟩forx=ω, that is,f(x)=1,Uω|x⟩=|x⟩forx≠ω, that is,f(x)=0.{\displaystyle {\begin{cases}U_{\omega }|x\rangle =-|x\rangle &{\text{for }}x=\omega {\text{, that is, }}f(x)=1,\\U_{\omega }|x\rangle =|x\rangle &{\text{for }}x\neq \omega {\text{, that is, }}f(x)=0.\end{cases}}}
This uses theN{\displaystyle N}-dimensionalstate spaceH{\displaystyle {\mathcal {H}}}, which is supplied by aregisterwithn=⌈log2N⌉{\displaystyle n=\lceil \log _{2}N\rceil }qubits.
This is often written as
Uω|x⟩=(−1)f(x)|x⟩.{\displaystyle U_{\omega }|x\rangle =(-1)^{f(x)}|x\rangle .}
Grover's algorithm outputsωwith probability at least1/2usingO(N){\displaystyle O({\sqrt {N}})}applications ofUω. This probability can be made arbitrarily large by running Grover's algorithm multiple times. If one runs Grover's algorithm untilωis found, theexpectednumber of applications is stillO(N){\displaystyle O({\sqrt {N}})}, since it will only be run twice on average.
This section compares the above oracleUω{\displaystyle U_{\omega }}with an oracleUf{\displaystyle U_{f}}.
Uωis different from the standardquantum oraclefor a functionf. This standard oracle, denoted here asUf, uses anancillary qubitsystem. The operation then represents an inversion (NOT gate) on the main system conditioned by the value off(x) from the ancillary system:
{Uf|x⟩|y⟩=|x⟩|¬y⟩forx=ω, that is,f(x)=1,Uf|x⟩|y⟩=|x⟩|y⟩forx≠ω, that is,f(x)=0,{\displaystyle {\begin{cases}U_{f}|x\rangle |y\rangle =|x\rangle |\neg y\rangle &{\text{for }}x=\omega {\text{, that is, }}f(x)=1,\\U_{f}|x\rangle |y\rangle =|x\rangle |y\rangle &{\text{for }}x\neq \omega {\text{, that is, }}f(x)=0,\end{cases}}}
or briefly,
Uf|x⟩|y⟩=|x⟩|y⊕f(x)⟩.{\displaystyle U_{f}|x\rangle |y\rangle =|x\rangle |y\oplus f(x)\rangle .}
These oracles are typically realized usinguncomputation.
If we are givenUfas our oracle, then we can also implementUω, sinceUωisUfwhen the ancillary qubit is in the state|−⟩=12(|0⟩−|1⟩)=H|1⟩{\displaystyle |-\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle -|1\rangle {\big )}=H|1\rangle }:
Uf(|x⟩⊗|−⟩)=12(Uf|x⟩|0⟩−Uf|x⟩|1⟩)=12(|x⟩|0⊕f(x)⟩−|x⟩|1⊕f(x)⟩)={12(−|x⟩|0⟩+|x⟩|1⟩)iff(x)=1,12(|x⟩|0⟩−|x⟩|1⟩)iff(x)=0=(Uω|x⟩)⊗|−⟩{\displaystyle {\begin{aligned}U_{f}{\big (}|x\rangle \otimes |-\rangle {\big )}&={\frac {1}{\sqrt {2}}}\left(U_{f}|x\rangle |0\rangle -U_{f}|x\rangle |1\rangle \right)\\&={\frac {1}{\sqrt {2}}}\left(|x\rangle |0\oplus f(x)\rangle -|x\rangle |1\oplus f(x)\rangle \right)\\&={\begin{cases}{\frac {1}{\sqrt {2}}}\left(-|x\rangle |0\rangle +|x\rangle |1\rangle \right)&{\text{if }}f(x)=1,\\{\frac {1}{\sqrt {2}}}\left(|x\rangle |0\rangle -|x\rangle |1\rangle \right)&{\text{if }}f(x)=0\end{cases}}\\&=(U_{\omega }|x\rangle )\otimes |-\rangle \end{aligned}}}
So, Grover's algorithm can be run regardless of which oracle is given.[3]IfUfis given, then we must maintain an additional qubit in the state|−⟩{\displaystyle |-\rangle }and applyUfin place ofUω.
The steps of Grover's algorithm are given as follows:
For the correctly chosen value ofr{\displaystyle r}, the output will be|ω⟩{\displaystyle |\omega \rangle }with probability approaching 1 forN≫ 1. Analysis shows that this eventual value forr(N){\displaystyle r(N)}satisfiesr(N)≤⌈π4N⌉{\displaystyle r(N)\leq {\Big \lceil }{\frac {\pi }{4}}{\sqrt {N}}{\Big \rceil }}.
Implementing the steps for this algorithm can be done using a number of gates linear in the number of qubits.[3]Thus, the gate complexity of this algorithm isO(log(N)r(N)){\displaystyle O(\log(N)r(N))}, orO(log(N)){\displaystyle O(\log(N))}per iteration.
There is a geometric interpretation of Grover's algorithm, following from the observation that the quantum state of Grover's algorithm stays in a two-dimensional subspace after each step. Consider the plane spanned by|s⟩{\displaystyle |s\rangle }and|ω⟩{\displaystyle |\omega \rangle }; equivalently, the plane spanned by|ω⟩{\displaystyle |\omega \rangle }and the perpendicularket|s′⟩=1N−1∑x≠ω|x⟩{\displaystyle \textstyle |s'\rangle ={\frac {1}{\sqrt {N-1}}}\sum _{x\neq \omega }|x\rangle }.
Grover's algorithm begins with the initial ket|s⟩{\displaystyle |s\rangle }, which lies in the subspace. The operatorUω{\displaystyle U_{\omega }}is a reflection at the hyperplane orthogonal to|ω⟩{\displaystyle |\omega \rangle }for vectors in the plane spanned by|s′⟩{\displaystyle |s'\rangle }and|ω⟩{\displaystyle |\omega \rangle }, i.e. it acts as a reflection across|s′⟩{\displaystyle |s'\rangle }. This can be seen by writingUω{\displaystyle U_{\omega }}in the form of aHouseholder reflection:
Uω=I−2|ω⟩⟨ω|.{\displaystyle U_{\omega }=I-2|\omega \rangle \langle \omega |.}
The operatorUs=2|s⟩⟨s|−I{\displaystyle U_{s}=2|s\rangle \langle s|-I}is a reflection through|s⟩{\displaystyle |s\rangle }. Both operatorsUs{\displaystyle U_{s}}andUω{\displaystyle U_{\omega }}take states in the plane spanned by|s′⟩{\displaystyle |s'\rangle }and|ω⟩{\displaystyle |\omega \rangle }to states in the plane. Therefore, Grover's algorithm stays in this plane for the entire algorithm.
It is straightforward to check that the operatorUsUω{\displaystyle U_{s}U_{\omega }}of each Grover iteration step rotates the state vector by an angle ofθ=2arcsin1N{\displaystyle \theta =2\arcsin {\tfrac {1}{\sqrt {N}}}}. So, with enough iterations, one can rotate from the initial state|s⟩{\displaystyle |s\rangle }to the desired output state|ω⟩{\displaystyle |\omega \rangle }. The initial ket is close to the state orthogonal to|ω⟩{\displaystyle |\omega \rangle }:
⟨s′|s⟩=N−1N.{\displaystyle \langle s'|s\rangle ={\sqrt {\frac {N-1}{N}}}.}
In geometric terms, the angleθ/2{\displaystyle \theta /2}between|s⟩{\displaystyle |s\rangle }and|s′⟩{\displaystyle |s'\rangle }is given by
sinθ2=1N.{\displaystyle \sin {\frac {\theta }{2}}={\frac {1}{\sqrt {N}}}.}
We need to stop when the state vector passes close to|ω⟩{\displaystyle |\omega \rangle }; after this, subsequent iterations rotate the state vectorawayfrom|ω⟩{\displaystyle |\omega \rangle }, reducing the probability of obtaining the correct answer. The exact probability of measuring the correct answer is
sin2((r+12)θ),{\displaystyle \sin ^{2}\left({\Big (}r+{\frac {1}{2}}{\Big )}\theta \right),}
whereris the (integer) number of Grover iterations. The earliest time that we get a near-optimal measurement is thereforer≈πN/4{\displaystyle r\approx \pi {\sqrt {N}}/4}.
To complete the algebraic analysis, we need to find out what happens when we repeatedly applyUsUω{\displaystyle U_{s}U_{\omega }}. A natural way to do this is by eigenvalue analysis of a matrix. Notice that during the entire computation, the state of the algorithm is a linear combination ofs{\displaystyle s}andω{\displaystyle \omega }. We can write the action ofUs{\displaystyle U_{s}}andUω{\displaystyle U_{\omega }}in the space spanned by{|s⟩,|ω⟩}{\displaystyle \{|s\rangle ,|\omega \rangle \}}as:
Us:a|ω⟩+b|s⟩↦[|ω⟩|s⟩][−102/N1][ab].Uω:a|ω⟩+b|s⟩↦[|ω⟩|s⟩][−1−2/N01][ab].{\displaystyle {\begin{aligned}U_{s}:a|\omega \rangle +b|s\rangle &\mapsto [|\omega \rangle \,|s\rangle ]{\begin{bmatrix}-1&0\\2/{\sqrt {N}}&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}.\\U_{\omega }:a|\omega \rangle +b|s\rangle &\mapsto [|\omega \rangle \,|s\rangle ]{\begin{bmatrix}-1&-2/{\sqrt {N}}\\0&1\end{bmatrix}}{\begin{bmatrix}a\\b\end{bmatrix}}.\end{aligned}}}
So in the basis{|ω⟩,|s⟩}{\displaystyle \{|\omega \rangle ,|s\rangle \}}(which is neither orthogonal nor a basis of the whole space) the actionUsUω{\displaystyle U_{s}U_{\omega }}of applyingUω{\displaystyle U_{\omega }}followed byUs{\displaystyle U_{s}}is given by the matrix
UsUω=[−102/N1][−1−2/N01]=[12/N−2/N1−4/N].{\displaystyle U_{s}U_{\omega }={\begin{bmatrix}-1&0\\2/{\sqrt {N}}&1\end{bmatrix}}{\begin{bmatrix}-1&-2/{\sqrt {N}}\\0&1\end{bmatrix}}={\begin{bmatrix}1&2/{\sqrt {N}}\\-2/{\sqrt {N}}&1-4/N\end{bmatrix}}.}
This matrix happens to have a very convenientJordan form. If we definet=arcsin(1/N){\displaystyle t=\arcsin(1/{\sqrt {N}})}, it is
UsUω=M[e2it00e−2it]M−1{\displaystyle U_{s}U_{\omega }=M{\begin{bmatrix}e^{2it}&0\\0&e^{-2it}\end{bmatrix}}M^{-1}}
whereM=[−iieite−it].{\displaystyle M={\begin{bmatrix}-i&i\\e^{it}&e^{-it}\end{bmatrix}}.}
It follows thatr-th power of the matrix (corresponding toriterations) is
(UsUω)r=M[e2rit00e−2rit]M−1.{\displaystyle (U_{s}U_{\omega })^{r}=M{\begin{bmatrix}e^{2rit}&0\\0&e^{-2rit}\end{bmatrix}}M^{-1}.}
Using this form, we can use trigonometric identities to compute the probability of observingωafterriterations mentioned in the previous section,
|[⟨ω|ω⟩⟨ω|s⟩](UsUω)r[01]|2=sin2((2r+1)t).{\displaystyle \left|{\begin{bmatrix}\langle \omega |\omega \rangle &\langle \omega |s\rangle \end{bmatrix}}(U_{s}U_{\omega })^{r}{\begin{bmatrix}0\\1\end{bmatrix}}\right|^{2}=\sin ^{2}\left((2r+1)t\right).}
Alternatively, one might reasonably imagine that a near-optimal time to distinguish would be when the angles 2rtand −2rtare as far apart as possible, which corresponds to2rt≈π/2{\displaystyle 2rt\approx \pi /2}, orr=π/4t=π/4arcsin(1/N)≈πN/4{\displaystyle r=\pi /4t=\pi /4\arcsin(1/{\sqrt {N}})\approx \pi {\sqrt {N}}/4}. Then the system is in state
[|ω⟩|s⟩](UsUω)r[01]≈[|ω⟩|s⟩]M[i00−i]M−1[01]=|ω⟩1cos(t)−|s⟩sin(t)cos(t).{\displaystyle [|\omega \rangle \,|s\rangle ](U_{s}U_{\omega })^{r}{\begin{bmatrix}0\\1\end{bmatrix}}\approx [|\omega \rangle \,|s\rangle ]M{\begin{bmatrix}i&0\\0&-i\end{bmatrix}}M^{-1}{\begin{bmatrix}0\\1\end{bmatrix}}=|\omega \rangle {\frac {1}{\cos(t)}}-|s\rangle {\frac {\sin(t)}{\cos(t)}}.}
A short calculation now shows that the observation yields the correct answerωwith errorO(1N){\displaystyle O\left({\frac {1}{N}}\right)}.
If, instead of 1 matching entry, there arekmatching entries, the same algorithm works, but the number of iterations must beπ4(Nk)1/2{\textstyle {\frac {\pi }{4}}{\left({\frac {N}{k}}\right)^{1/2}}}instead ofπ4N1/2.{\textstyle {\frac {\pi }{4}}{N^{1/2}}.}
There are several ways to handle the case ifkis unknown.[16]A simple solution performs optimally up to a constant factor: run Grover's algorithm repeatedly for increasingly small values ofk, e.g., takingk=N,N/2,N/4, ..., and so on, takingk=N/2t{\displaystyle k=N/2^{t}}for iterationtuntil a matching entry is found.
With sufficiently high probability, a marked entry will be found by iterationt=log2(N/k)+c{\displaystyle t=\log _{2}(N/k)+c}for some constantc. Thus, the total number of iterations taken is at most
π4(1+2+4+⋯+Nk2c)=O(N/k).{\displaystyle {\frac {\pi }{4}}{\Big (}1+{\sqrt {2}}+{\sqrt {4}}+\cdots +{\sqrt {\frac {N}{k2^{c}}}}{\Big )}=O{\big (}{\sqrt {N/k}}{\big )}.}
Another approach ifkis unknown is to derive it via thequantum counting algorithmprior.
Ifk=N/2{\displaystyle k=N/2}(or the traditional one marked state Grover's Algorithm if run withN=2{\displaystyle N=2}), the algorithm will provide no amplification. Ifk>N/2{\displaystyle k>N/2}, increasingkwill begin to increase the number of iterations necessary to obtain a solution.[17]On the other hand, ifk≥N/2{\displaystyle k\geq N/2}, a classical running of the checking oracle on a single random choice of input will more likely than not give a correct solution.
A version of this algorithm is used in order to solve thecollision problem.[18][19]
A modification of Grover's algorithm called quantum partial search was described by Grover and Radhakrishnan in 2004.[20]In partial search, one is not interested in finding the exact address of the target item, only the first few digits of the address. Equivalently, we can think of "chunking" the search space into blocks, and then asking "in which block is the target item?". In many applications, such a search yields enough information if the target address contains the information wanted. For instance, to use the example given by L. K. Grover, if one has a list of students organized by class rank, we may only be interested in whether a student is in the lower 25%, 25–50%, 50–75% or 75–100% percentile.
To describe partial search, we consider a database separated intoK{\displaystyle K}blocks, each of sizeb=N/K{\displaystyle b=N/K}. The partial search problem is easier. Consider the approach we would take classically – we pick one block at random, and then perform a normal search through the rest of the blocks (in set theory language, the complement). If we do not find the target, then we know it is in the block we did not search. The average number of iterations drops fromN/2{\displaystyle N/2}to(N−b)/2{\displaystyle (N-b)/2}.
Grover's algorithm requiresπ4N{\textstyle {\frac {\pi }{4}}{\sqrt {N}}}iterations. Partial search will be faster by a numerical factor that depends on the number of blocksK{\displaystyle K}. Partial search usesn1{\displaystyle n_{1}}global iterations andn2{\displaystyle n_{2}}local iterations. The global Grover operator is designatedG1{\displaystyle G_{1}}and the local Grover operator is designatedG2{\displaystyle G_{2}}.
The global Grover operator acts on the blocks. Essentially, it is given as follows:
The optimal values ofj1{\displaystyle j_{1}}andj2{\displaystyle j_{2}}are discussed in the paper by Grover and Radhakrishnan. One might also wonder what happens if one applies successive partial searches at different levels of "resolution". This idea was studied in detail byVladimir Korepinand Xu, who called it binary quantum search. They proved that it is not in fact any faster than performing a single partial search.
Grover's algorithm is optimal up to sub-constant factors. That is, any algorithm that accesses the database only by using the operatorUωmust applyUωat least a1−o(1){\displaystyle 1-o(1)}fraction as many times as Grover's algorithm.[21]The extension of Grover's algorithm tokmatching entries,π(N/k)1/2/4, is also optimal.[18]This result is important in understanding the limits of quantum computation.
If the Grover's search problem was solvable withlogcNapplications ofUω, that would imply thatNPis contained inBQP, by transforming problems in NP into Grover-type search problems. The optimality of Grover's algorithm suggests that quantum computers cannot solveNP-Completeproblems in polynomial time, and thus NP is not contained in BQP.
It has been shown that a class of non-localhidden variablequantum computers could implement a search of anN{\displaystyle N}-item database in at mostO(N3){\displaystyle O({\sqrt[{3}]{N}})}steps. This is faster than theO(N){\displaystyle O({\sqrt {N}})}steps taken by Grover's algorithm.[22]
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https://en.wikipedia.org/wiki/Grover%27s_algorithm
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Sudoku(/suːˈdoʊkuː,-ˈdɒk-,sə-/;Japanese:数独,romanized:sūdoku,lit.'digit-single'; originally calledNumber Place)[1]is alogic-based,[2][3]combinatorial[4]number-placementpuzzle. In classic Sudoku, the objective is to fill a 9 × 9 grid with digits so that each column, each row, and each of the nine 3 × 3 subgrids that compose the grid (also called "boxes", "blocks", or "regions") contains all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which for awell-posedpuzzle has a single solution.
French newspapers featured similar puzzles in the 19th century, and the modern form of the puzzle first appeared in 1979puzzle booksbyDell Magazinesunder the name Number Place.[5]However, the puzzle type only began to gain widespread popularity in 1986 when it was published by the Japanese puzzle companyNikoliunder the name Sudoku, meaning "single number".[6]In newspapers outside of Japan, it first appeared inThe Conway Daily Sun(New Hampshire) in September 2004, and thenThe Times(London) in November 2004, both of which were thanks to the efforts of the Hong Kong judgeWayne Gould, who devised acomputer programto rapidly produce unique puzzles.
Number puzzles appeared in newspapers in the late 19th century, when French puzzle setters began experimenting with removing numbers frommagic squares.Le Siècle, a Paris daily, published a partially completed 9×9 magic square with 3×3 subsquares on November 19, 1892.[7]It was not a Sudoku because it contained double-digit numbers and required arithmetic rather than logic to solve, but it shared key characteristics: each row, column, and subsquare added up to the same number.
On July 6, 1895,Le Siècle'srival,La France, refined the puzzle so that it was almost a modern Sudoku and named itcarré magique diabolique('diabolical magic square'). It simplified the 9×9 magic square puzzle so that each row, column, andbroken diagonalscontained only the numbers 1–9, but did not mark the subsquares. Although they were unmarked, each 3×3 subsquare did indeed comprise the numbers 1–9, and the additional constraint on the broken diagonals led to only one solution.[8]
These weekly puzzles were a feature of French newspapers such asL'Écho de Parisfor about a decade, but disappeared about the time ofWorld War I.[9]
The modern Sudoku was most likely designed anonymously byHoward Garns, a 74-year-old retired architect and freelance puzzle constructor fromConnersville, Indiana, and first published in 1979 byDell Magazinesas Number Place (the earliest known examples of modern Sudoku).[1]Garns' name was always present on the list of contributors in issues ofDell Pencil Puzzles and Word Gamesthat included Number Place and was always absent from issues that did not.[10]He died in 1989 before getting a chance to see his creation as a worldwide phenomenon.[10]Whether or not Garns was familiar with any of the French newspapers listed above is unclear.
The puzzle was introduced in Japan byMaki Kaji(鍜治 真起,Kaji Maki), president of theNikolipuzzle company, in the paperMonthly Nikolistin April 1984[10]asSūji wa dokushin ni kagiru(数字は独身に限る), which can be translated as "the digits must be single", or as "the digits are limited to one occurrence" (In Japanese,dokushinmeans an "unmarried person"). The name was later abbreviated toSudoku(数独), taking only the firstkanjiof compound words to form a shorter version.[10]"Sudoku" is a registered trademark in Japan[11]and the puzzle is generally referred to as Number Place(ナンバープレース,Nanbāpurēsu)or, more informally, a shortening of the two words, Num(ber) Pla(ce)(ナンプレ,Nanpure). In 1986, Nikoli introduced two innovations: the number of givens was restricted to no more than 32, and puzzles became "symmetrical" (meaning the givens were distributed inrotationally symmetric cells). It is now published in mainstream Japanese periodicals, such as theAsahi Shimbun.
In 1997, Hong Kong judgeWayne Gouldsaw a partly completed puzzle in a Japanese bookshop. Over six years, he developed a computer program to produce unique puzzles rapidly.[5]
The first newspaper outside of Japan to publish a Sudoku puzzle wasThe Conway Daily Sun(New Hampshire), which published a puzzle by Gould in September 2004.[12][13]
Gould pitched the idea of publishing Sudoku puzzles to newspapers, offering the puzzles for free in exchange for the newspapers' attributing them to him and linking to his website for solutions and other puzzles.
Knowing that British newspapers have a long history of publishingcrosswordsand other puzzles, he promoted Sudoku toThe Timesin Britain, which launched it on November 12, 2004 (calling it Su Doku). The first letter toThe Timesregarding Su Doku was published the following day on November 13 from Ian Payn ofBrentford, complaining that the puzzle had caused him to miss his stop on thetube.[14]Sudoku puzzles rapidly spread to other newspapers as a regular feature.[5][15]
The rapid rise of Sudoku in Britain from relative obscurity to a front-page feature in national newspapers attracted commentary in the media and parody (such as whenThe Guardian'sG2section advertised itself as the first newspaper supplement with a Sudoku grid on every page).[16]Recognizing the different psychological appeals of easy and difficult puzzles,The Timesintroduced both, side by side, on June 20, 2005. From July 2005,Channel 4included a daily Sudoku game in theirteletextservice. On August 2, the BBC's program guideRadio Timesfeatured a weekly Super Sudoku with a 16×16 grid.
The world's first live TV Sudoku show,Sudoku Live, was apuzzle contestfirst broadcast on July 1, 2005, on the British pay-television channelSky One. It was presented byCarol Vorderman. Nine teams of nine players (with one celebrity in each team) representing geographical regions competed to solve a puzzle. Each player had a hand-held device for entering numbers corresponding to answers for four cells. Phil Kollin ofWinchelsea, England, was the series grand prize winner, taking home over £23,000 over a series of games. The audience at home was in a separate interactive competition, which was won by Hannah Withey ofCheshire.
Later in 2005, theBBClaunchedSUDO-Q, agame showthat combined Sudoku with general knowledge. However, it used only 4×4 and 6×6 puzzles. Four seasons were produced before the show ended in 2007.
An annualWorld Sudoku Championshipseries has been organized by theWorld Puzzle Federationsince 2006, except in 2020 and 2021 during theCOVID-19 pandemic.
In 2006, a Sudoku website published a tribute song by Australian songwriter Peter Levy, but the song download was later removed due to heavy traffic. The Japanese Embassy nominated the song for an award, and Levy claimed he was in discussions withSonyin Japan to release the song as a single.[17]
Sudoku software is very popular on PCs, websites, and mobile phones. It comes with many distributions ofLinux. The software has also been released on video game consoles, such as theNintendo DS,PlayStation Portable, theGame Boy Advance,Xbox Live Arcade, theNooke-book reader, Kindle Fire tablet, severaliPodmodels, and theiPhone. ManyNokiaphones also had Sudoku. In fact, just two weeks afterApple Inc.debuted the onlineApp Storewithin itsiTunes Storeon July 11, 2008, nearly 30 different Sudoku games were already in it, created by varioussoftware developers, specifically for the iPhone and iPod Touch. Sudoku games also rapidly became available forweb browserusers and for basically all gaming, cellphone, and computer platforms.
In June 2008, an Australian drugs-related jury trial costing overA$1 million was aborted when it was discovered that four or five of the twelve jurors had been playing Sudoku instead of listening to the evidence.[18]
Although the 9×9 grid with 3×3 regions is by far the most common, many other variations exist. Sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids withpentominoregions have been published under the name Logi-5; theWorld Puzzle Championshiphas featured a 6×6 grid with 2×3 regions and a 7×7 grid with sixheptominoregions and a disjoint region. Larger grids are also possible, or different irregular shapes (under various names such asSuguru,Tectonic,Jigsaw Sudokuetc.).The Timesoffers a 12×12-grid "Dodeka Sudoku" with 12 regions of 4×3 squares. Dell Magazines regularly publishes 16×16 "Number Place Challenger" puzzles (using the numbers 1–16 or the letters A-P). Nikoli offers 25×25 "Sudoku the Giant" behemoths. A 100×100-grid puzzle dubbed Sudoku-zilla was published in 2010.[19]
Under the name "Mini Sudoku", a 6×6 variant with 3×2 regions appears in the American newspaperUSA Todayand elsewhere. The object is the same as that of standard Sudoku, but the puzzle only uses the numbers 1 through 6. A similar form, for younger solvers of puzzles, called "The Junior Sudoku", has appeared in some newspapers, such as some editions ofThe Daily Mail.
Another common variant is to add limits on the placement of numbers beyond the usual row, column, and box requirements. Often, the limit takes the form of an extra "dimension"; the most common is to require the numbers in the main diagonals of the grid to also be unique. The aforementioned "Number Place Challenger" puzzles are all of this variant, as are the Sudoku X puzzles inThe Daily Mail, which use 6×6 grids.
The killer sudoku variant combines elements of sudoku andkakuro. A killer sudoku puzzle is made up of 'cages', typically depicted by boxes outlined with dashes or colours. The sum of the numbers in a cage is written in the top left corner of the cage, and numbers cannot be repeated in a cage.
Puzzles constructed from more than two grids are also common. Five 9×9 grids that overlap at the corner regions in the shape of aquincunxis known in Japan asGattai5 (five merged) Sudoku. InThe Times,The Age, andThe Sydney Morning Herald, this form of puzzle is known as Samurai Sudoku.The Baltimore Sunand theToronto Starpublish a puzzle of this variant (titled High Five) in their Sunday edition. Often, no givens are placed in the overlapping regions. Sequential grids, as opposed to overlapping, are also published, with values in specific locations in grids needing to be transferred to others.
A tabletop version of Sudoku can be played with a standard 81-card Set deck (seeSet game). A three-dimensional Sudoku puzzle was published inThe Daily Telegraphin May 2005.The Timesalso publishes a three-dimensional version under the name Tredoku. Also, a Sudoku version of theRubik's Cubeis namedSudoku Cube.
Many other variants have been developed.[20][21][22]Some are different shapes in the arrangement of overlapping 9×9 grids, such as butterfly, windmill, or flower.[23]Others vary the logic for solving the grid. One of these is "Greater Than Sudoku". In this, a 3×3 grid of the Sudoku is given with 12 symbols of Greater Than (>) or Less Than (<) on the common line of the two adjacent numbers.[10]Another variant on the logic of the solution is "Clueless Sudoku", in which nine 9×9 Sudoku grids are each placed in a 3×3 array. The center cell in each 3×3 grid of all nine puzzles is left blank and forms a tenth Sudoku puzzle without any cell completed; hence, "clueless".[23]Examples and other variants can be found in theGlossary of Sudoku.
This section refers to classic Sudoku, disregarding jigsaw, hyper, and other variants. A completed Sudoku grid is a special type ofLatin squarewith the additional property of no repeated values in any of the nine blocks (orboxesof 3×3 cells).[24]
The general problem of solving Sudoku puzzles onn2×n2grids ofn×nblocks is known to beNP-complete.[25]ManySudoku solving algorithms, such asbrute force-backtracking anddancing linkscan solve most 9×9 puzzles efficiently, butcombinatorial explosionoccurs asnincreases, creating practical limits to the properties of Sudokus that can be constructed, analyzed, and solved asnincreases. A Sudoku puzzle can be expressed as agraph coloringproblem.[26]The aim is to construct a 9-coloring of a particular graph, given a partial 9-coloring.
The fewest clues possible for a proper Sudoku is 17.[27]Tens of thousands of distinct Sudoku puzzles have only 17 clues.[28]
The number of classic 9×9 Sudoku solution grids is 6,670,903,752,021,072,936,960, or around6.67×1021.[29]The number of essentially different solutions, whensymmetriessuch as rotation, reflection, permutation, and relabelling are taken into account, is much smaller, 5,472,730,538.[30]
Unlike the number of complete Sudoku grids, the number of minimal 9×9 Sudoku puzzles is not precisely known. (A minimal puzzle is one in which no clue can be deleted without losing the uniqueness of the solution.) However, statistical techniques combined with a puzzle generator show that about (with 0.065% relative error) 3.10 × 1037minimal puzzles and 2.55 × 1025nonessentially equivalent minimal puzzles exist.[31]
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Inmathematics, acombinatorial explosionis the rapid growth of the complexity of a problem due to the way itscombinatoricsdepends on input, constraints and bounds. Combinatorial explosion is sometimes used to justify the intractability of certain problems.[1][2]Examples of such problems include certain mathematicalfunctions, the analysis of some puzzles and games, and some pathological examples which can be modelled as theAckermann function.
ALatin squareof ordernis ann×narray with entries from a set ofnelements with the property that each element of the set occurs exactly once in each row and each column of the array. An example of a Latin square of order three is given by,
A common example of a Latin square would be a completedSudokupuzzle.[3]A Latin square is a combinatorial object (as opposed to an algebraic object) since only the arrangement of entries matters and not what the entries actually are. The number of Latin squares as a function of the order (independent of the set from which the entries are drawn) (sequenceA002860in theOEIS) provides an example of combinatorial explosion as illustrated by the following table.
A combinatorial explosion can also occur in some puzzles played on a grid, such as Sudoku.[2]A Sudoku is a type of Latin square with the additional property that each element occurs exactly once in sub-sections of size√n×√n(calledboxes). Combinatorial explosion occurs asnincreases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved, as illustrated in the following table.
One example in a game where combinatorial complexity leads to a solvability limit is insolving chess(a game with 64 squares and 32 pieces). Chess is not asolved game. In 2005 all chess game endings with six pieces or fewer were solved, showing the result of each position if played perfectly. It took ten more years to complete the tablebase with one more chess piece added, thus completing a 7-piece tablebase. Adding one more piece to a chess ending (thus making an 8-piece tablebase) is considered intractable due to the added combinatorial complexity.[6][7]
Furthermore, the prospect of solving larger chess-like games becomes more difficult as the board-size is increased, such as in largechess variants, andinfinite chess.[8]
Combinatorial explosion can occur in computing environments in a way analogous to communications andmulti-dimensional space. Imagine a simple system with only one variable, abooleancalledA. The system has two possible states,A= true orA= false. Adding another boolean variableBwill give the system four possible states,A= true andB= true,A= true andB= false,A= false andB= true,A= false andB= false. A system withnbooleans has 2npossible states, while a system ofnvariables each withZallowed values (rather than just the 2 (true and false) of booleans) will haveZnpossible states.
The possible states can be thought of as the leaf nodes of atreeof heightn, where each node hasZchildren. This rapid increase of leaf nodes can be useful in areas likesearching, since many results can be accessed without having to descend very far. It can also be a hindrance when manipulating such structures.
Aclass hierarchyin anobject-oriented languagecan be thought of as a tree, with different types of object inheriting from their parents. If different classes need to be combined, such as in a comparison (likeA<B) then the number of possible combinations which may occur explodes. If each type of comparison needs to be programmed then this soon becomes intractable for even small numbers of classes.Multiple inheritancecan solve this, by allowing subclasses to have multiple parents, and thus a few parent classes can be considered rather than every child, without disrupting any existing hierarchy.
An example is a taxonomy where different vegetables inherit from their ancestor species. Attempting to compare the tastiness of each vegetable with the others becomes intractable since the hierarchy only contains information about genetics and makes no mention of tastiness. However, instead of having to write comparisons for carrot/carrot, carrot/potato, carrot/sprout, potato/potato, potato/sprout, sprout/sprout, they can allmultiply inheritfrom a separate class of tasty whilst keeping their current ancestor-based hierarchy, then all of the above can be implemented with only a tasty/tasty comparison.
Suppose we take thefactorialofn:
Then 1! = 1, 2! = 2, 3! = 6, and 4! = 24. However, we quickly get to extremely large numbers, even for relatively smalln. For example, 100! ≈9.33262154×10157, a number so large that it cannot be displayed on most calculators, and vastly larger than the estimated number of fundamental particles in the observable universe.[9]
In administration andcomputing, acombinatorial explosionis the rapidly accelerating increase in communication lines as organizations are added in a process. (This growth is often casually described as "exponential" but is actuallypolynomial.)
If two organizations need to communicate about a particular topic, it may be easiest to communicate directly in an ad hoc manner—only onechannel of communicationis required. However, if a third organization is added, three separate channels are required. Adding a fourth organization requires six channels; five, ten; six, fifteen; etc.
In general, it will takel=n(n−1)2=(n2){\displaystyle l={\frac {n(n-1)}{2}}={n \choose 2}}communication lines fornorganizations, which is just the number of 2-combinationsofnelements (see alsoBinomial coefficient).[10]
The alternative approach is to realize when this communication will not be a one-off requirement, and produce a generic or intermediate way of passing information. The drawback is that this requires more work for the first pair, since each must convert its internal approach to the common one, rather than the superficially easier approach of just understanding the other.
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This is a glossary ofSudokuterms and jargon. Sudoku with a 9×9 grid is assumed, unless otherwise noted.
ASudoku(i.e. thepuzzle) is a partially completedgrid. A grid has 9rows, 9columnsand 9boxes, each having 9cells(81 total). Boxes can also be calledblocksorregions.[1]Three horizontally adjacent blocks are aband, and three vertically adjacent blocks are astack.[2]The initially defined values arecluesorgivens. An ordinary Sudoku (i.e. a proper Sudoku) has one solution. Rows, columns and regions can be collectively referred to asgroups, of which the grid has 27. TheOne Ruleencapsulates the three prime rules, i.e. eachdigit(or number) can occur only once in each row, column, and box; and can be compactly stated as: "Each digit appears once in each group."
The classic 9×9 Sudoku format can be generalized to an
This accommodates variants by region size and shape, e.g. 6-cell rectangular regions. (N×NSudoku is square). ForprimeN,polyomino-shaped regions can be used and the requirement to use equal-sized regions, or have the regions entirely cover the grid can be relaxed.
Other variations include additional value placement constraints, alternate symbols (e.g. letters), alternate mechanism for expressing the clues, and compositions withoverlapping grids. SeeSudoku – Variantsfor details and additional variants.
Sudokus variants can also have additional constraints on the placement of digits, such as "< >" relations, sums, linked cells, etc.
The meanings of most of these terms can be extended to region shapes other than boxes (square-shaped). To simplify reading, definitions are given only in terms of boxes.
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Inmathematics, afractalis ageometric shapecontaining detailed structure at arbitrarily small scales, usually having afractal dimensionstrictly exceeding thetopological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of theMandelbrot set.[1][2][3][4]This exhibition of similar patterns at increasingly smaller scales is calledself-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in theMenger sponge, the shape is calledaffineself-similar.[5]Fractal geometry lies within the mathematical branch ofmeasure theory.
One way that fractals are different from finitegeometric figuresis how theyscale. Doubling the edge lengths of a filledpolygonmultiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if theradiusof a filled sphere is doubled, itsvolumescales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily anintegerand is in general greater than its conventional dimension.[1]This power is called thefractal dimensionof the geometric object, to distinguish it from the conventional dimension (which is formally called thetopological dimension).[6]
Analytically, many fractals are nowheredifferentiable.[1][4]An infinitefractal curvecan be conceived of as winding through space differently from an ordinary line – although it is stilltopologically 1-dimensional, its fractal dimension indicates that it locally fills space more efficiently than an ordinary line.[1][6]
Starting in the 17th century with notions ofrecursion, fractals have moved through increasingly rigorous mathematical treatment to the study ofcontinuousbut notdifferentiablefunctions in the 19th century by the seminal work ofBernard Bolzano,Bernhard Riemann, andKarl Weierstrass,[7]and on to the coining of the wordfractalin the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.[8][9]
There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."[10]More formally, in 1982 Mandelbrot definedfractalas follows: "A fractal is by definition a set for which theHausdorff–Besicovitch dimensionstrictly exceeds thetopological dimension."[11]Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or fragmentedgeometric shapethat can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole."[1]Still later, Mandelbrot proposed "to usefractalwithout a pedantic definition, to usefractal dimensionas a generic term applicable toallthe variants".[12]
The consensus among mathematicians is that theoretical fractals are infinitely self-similariteratedand detailed mathematical constructs, of which manyexampleshave been formulated and studied.[1][2][3]Fractals are not limited to geometric patterns, but can also describe processes in time.[5][4][13]Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media[14]and found innature,[15][16][17][18]technology,[19][20][21][22]art,[23][24]andarchitecture.[25]Fractals are of particular relevance in the field ofchaos theorybecause they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).[26]
The term "fractal" was coined by the mathematicianBenoît Mandelbrotin 1975.[27]Mandelbrot based it on the Latinfrāctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractionaldimensionsto geometricpatterns in nature.[1][28][29]
The word "fractal" often has different connotations for mathematicians and the general public, where the public is more likely to be familiar withfractal artthan the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background.
The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in theinfinite regressin parallel mirrors or thehomunculus, the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed.[1]: 166, 18[2][28]
This idea of being detailed relates to another feature that can be understood without much mathematical background: Having afractal dimensiongreater than its topological dimension, for instance, refers to how a fractal scales compared to how geometricshapesare usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure isrep-tiledinto pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 32= 9 pieces.
We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total ofrnpieces. Now, consider theKoch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real numberDthat satisfies 3D= 4. This number is called thefractal dimensionof the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from theconventionally understooddimension (formally called the topological dimension).
This also leads to understanding a third feature, that fractals as mathematical equations are "nowheredifferentiable". In a concrete sense, this means fractals cannot be measured in traditional ways.[1][4][30]To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner ofmeasuringwith a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter.[1]
The history of fractals traces a path from chiefly theoretical studies to modern applications incomputer graphics, with several notable people contributing canonical fractal forms along the way.[8][9]A common theme in traditionalAfrican architectureis the use of fractal scaling, whereby small parts of the structure tend to look similar to larger parts, such as a circular village made of circular houses.[31]According toPickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopherGottfried Leibnizponderedrecursiveself-similarity(although he made the mistake of thinking that only thestraight linewas self-similar in this sense).[32]
In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.[1]: 405Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters".[30][8][9]Thus, it was not until two centuries had passed that on July 18, 1872Karl Weierstrasspresented the first definition of afunctionwith agraphthat would today be considered a fractal, having the non-intuitiveproperty of being everywherecontinuousbutnowhere differentiableat the Royal Prussian Academy of Sciences.[8]: 7[9]
In addition, the quotient difference becomes arbitrarily large as the summation index increases.[33]Not long after that, in 1883,Georg Cantor, who attended lectures by Weierstrass,[9]published examples ofsubsetsof the real line known asCantor sets, which had unusual properties and are now recognized as fractals.[8]: 11–24Also in the last part of that century,Felix KleinandHenri Poincaréintroduced a category of fractal that has come to be called "self-inverse" fractals.[1]: 166
One of the next milestones came in 1904, whenHelge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called theKoch snowflake.[8]: 25[9]Another milestone came a decade later in 1915, whenWacław Sierpińskiconstructed his famoustrianglethen, one year later, hiscarpet. By 1918, two French mathematicians,Pierre FatouandGaston Julia, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mappingcomplex numbersand iterative functions and leading to further ideas aboutattractors and repellors(i.e., points that attract or repel other points), which have become very important in the study of fractals.[4][8][9]
Very shortly after that work was submitted, by March 1918,Felix Hausdorffexpanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions.[9]The idea of self-similar curves was taken further byPaul Lévy, who, in his 1938 paperPlane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, theLévy C curve.[notes 1]
Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings).[1]: 179[30][9]That changed, however, in the 1960s, whenBenoit Mandelbrotstarted writing about self-similarity in papers such asHow Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,[34][35]which built on earlier work byLewis Fry Richardson.
In 1975,[28]Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonicalMandelbrot set, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".[36][30][8][32]
In 1980,Loren Carpentergave a presentation at theSIGGRAPHwhere he introduced his software for generating and rendering fractally generated landscapes.[37]
One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmentedgeometric shapethat can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole";[1]this is generally helpful but limited. Authors disagree on the exact definition offractal, but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in.[1][5][2][4][38]
One point agreed on is that fractal patterns are characterized byfractal dimensions, but whereas these numbers quantifycomplexity(i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.[39]In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whoseHausdorff–Besicovitch dimensionis greater than itstopological dimension.[28]However, this requirement is not met byspace-filling curvessuch as theHilbert curve.[notes 2]
Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According toFalconer, fractals should be only generally characterized by agestaltof the following features;[2]
As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion.[1][4]
Images of fractals can be created byfractal generating programs. Because of thebutterfly effect, a small change in a single variable can have anunpredictableoutcome.
Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals ornatural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks forfractal analysis. Some specific applications of fractals to technology are listedelsewhere. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or displayartifactswhich are not characteristics of true fractals.
Modeled fractals may be sounds,[14]digital images, electrochemical patterns,circadian rhythms,[46]etc.
Fractal patterns have been reconstructed in physical 3-dimensional space[21]: 10and virtually, often called "in silico" modeling.[43]Models of fractals are generally created usingfractal-generating softwarethat implements techniques such as those outlined above.[4][13][21]As one illustration, trees, ferns, cells of the nervous system,[18]blood and lung vasculature,[43]and other branchingpatterns in naturecan be modeled on a computer by using recursivealgorithmsandL-systemstechniques.[18]
The recursive nature of some patterns is obvious in certain examples—a branch from a tree or afrondfrom afernis a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects, such as coastlines and mountains. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.
Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees.[47]Phenomena known to have fractal features include:
Fractals often appear in the realm of living organisms where they arise through branching processes and other complex pattern formation. Ian Wong and co-workers have shown that migrating cells can form fractals by clustering andbranching.[70]Nerve cellsfunction through processes at the cell surface, with phenomena that are enhanced by largely increasing the surface to volume ratio. As a consequence nerve cells often are found to form into fractal patterns.[71]These processes are crucial in cellphysiologyand differentpathologies.[72]
Multiple subcellular structures also are found to assemble into fractals.Diego Krapfhas shown that through branching processes theactinfilaments in human cells assemble into fractal patterns.[73]Similarly Matthias Weiss showed that theendoplasmic reticulumdisplays fractal features.[74]The current understanding is that fractals are ubiquitous in cell biology, fromproteins, toorganelles, to whole cells.
Since 1999 numerous scientific groups have performed fractal analysis on over 50 paintings created byJackson Pollockby pouring paint directly onto horizontal canvasses.[75][76][77]
Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.[78]Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.[79]
Decalcomania, a technique used by artists such asMax Ernst, can produce fractal-like patterns.[80]It involves pressing paint between two surfaces and pulling them apart.
CyberneticistRon Eglashhas suggested that fractal geometry and mathematics are prevalent inAfrican art, games,divination, trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.[24][81]Hokky Situngkiralso suggested the similar properties in Indonesian traditional art,batik, andornamentsfound in traditional houses.[82][83]
Ethnomathematician Ron Eglash has discussed the planned layout ofBenin cityusing fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet."[84]
In a 1996 interview withMichael Silverblatt,David Foster Wallaceexplained that the structure of the first draft ofInfinite Jesthe gave to his editor Michael Pietsch was inspired by fractals, specifically theSierpinski triangle(a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".[23]
Some works by the Dutch artistM. C. Escher, such asCircle Limit III, contain shapes repeated to infinity that become smaller and smaller as they get near to the edges, in a pattern that would always look the same if zoomed in.
Aesthetics and Psychological Effects of Fractal Based Design:[85]Highly prevalent in nature, fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on the impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant well-being. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create a ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay the same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant well-being.
Humans appear to be especially well-adapted to processing fractal patterns withfractal dimensionbetween 1.3 and 1.5.[86]When humans view fractal patterns with fractal dimension between 1.3 and 1.5, this tends to reduce physiological stress.[87][88]
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Inmathematics, afundamental theoremis atheoremwhich is considered to be central and conceptually important for some topic. For example, thefundamental theorem of calculusgives the relationship betweendifferential calculusandintegral calculus.[1]The names are mostly traditional, so that for example thefundamental theorem of arithmeticis basic to what would now be callednumber theory.[2]Some of these areclassification theoremsof objects which are mainly dealt with in the field. For instance, thefundamental theorem of curvesdescribes classification ofregular curvesin space up totranslationandrotation.
Likewise, the mathematical literature sometimes refers to thefundamental lemmaof a field. The termlemmais conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself.
Carl Friedrich Gaussreferred to the law ofquadratic reciprocityas the "fundamental theorem" ofquadratic residues.[3]
There are also a number of "fundamental theorems" that are not directly related to mathematics:
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Inmathematics, theprime signatureof a number is themultisetof (nonzero) exponents of itsprime factorization. The prime signature of a number having prime factorizationp1m1p2m2…pnmn{\displaystyle p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{n}^{m_{n}}}is the multiset{m1,m2,…,mn}{\displaystyle \left\{m_{1},m_{2},\dots ,m_{n}\right\}}.
For example, allprime numbershave a prime signature of {1}, thesquaresof primes have a prime signature of {2}, the products of 2 distinct primes have a prime signature of{1, 1} and the products of a square of a prime and a different prime (e.g. 12, 18, 20, ...) have a prime signature of{2, 1}.
Thedivisor functionτ(n), theMöbius functionμ(n), the number of distinct prime divisors ω(n) ofn, the number of prime divisors Ω(n) ofn, theindicator functionof thesquarefree integers, and many other important functions in number theory, are functions of the prime signature ofn.
In particular, τ(n) equals the product of the incremented by 1 exponents from the prime signature ofn.
For example, 20 has prime signature {2,1} and so the number of divisors is (2+1) × (1+1) = 6. Indeed, there are six divisors: 1, 2, 4, 5, 10 and 20.
The smallest number of each prime signature is a product ofprimorials. The first few are:
A number cannot divide another unless its prime signature is included in the other numbers prime signature in theYoung's lattice.
Given a number with prime signatureS, it is
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Euler's factorization methodis a technique forfactoringa number by writing it as a sum of two squares in two different ways. For example the number1000009{\displaystyle 1000009}can be written as10002+32{\displaystyle 1000^{2}+3^{2}}or as9722+2352{\displaystyle 972^{2}+235^{2}}and Euler's method gives the factorization1000009=293⋅3413{\displaystyle 1000009=293\cdot 3413}.
The idea that two distinct representations of an odd positive integer may lead to a factorization was apparently first proposed byMarin Mersenne. However, it was not put to use extensively until one hundred years later by Euler. His most celebrated use of the method that now bears his name was to factor the number1000009{\displaystyle 1000009}, which apparently was previously thought to be prime even though it is not apseudoprimeby any major primality test.
Euler's factorization method is more effective than Fermat's for integers whose factors are not close together and potentially much more efficient than trial division if one can find representations of numbers as sums of two squares reasonably easily. The methods used to find representations of numbers as sums of two squares are essentially the same as with finding differences of squares inFermat's factorization method.
The great disadvantage of Euler's factorization method is that it cannot be applied to factoring an integer with any prime factor of the form 4k+ 3 occurring to an odd power in its prime factorization, as such a number can never be the sum of two squares. Even oddcomposite numbersof the form 4k+ 1 are often the product of two primes of the form 4k+ 3 (e.g. 3053 = 43 × 71) and again cannot be factored by Euler's method.
This restricted applicability has made Euler's factorization method disfavoured forcomputerfactoringalgorithms, since any user attempting to factor a random integer is unlikely to know whether Euler's method can actually be applied to the integer in question. It is only relatively recently that there have been attempts to develop Euler's method into computer algorithms for use on specialised numbers where it is known Euler's method can be applied.
TheBrahmagupta–Fibonacci identitystates that the product of two sums of two squares is a sum of two squares. Euler's method relies on this theorem but it can be viewed as the converse, givenn=a2+b2=c2+d2{\displaystyle n=a^{2}+b^{2}=c^{2}+d^{2}}we findn{\displaystyle n}as a product of sums of two squares.
First deduce that
and factor both sides to get
Now letk=gcd(a−c,d−b){\displaystyle k=\operatorname {gcd} (a-c,d-b)}andh=gcd(a+c,d+b){\displaystyle h=\operatorname {gcd} (a+c,d+b)}so that there exists some constantsl,m,l′,m′{\displaystyle l,m,l',m'}satisfying
gcd(l,m)=1{\displaystyle \operatorname {gcd} (l,m)=1}
gcd(l′,m′)=1{\displaystyle \operatorname {gcd} (l',m')=1}
Substituting these into equation (1) gives
Canceling common factors yields
Now using the fact that(l,m){\displaystyle (l,m)}and(l′,m′){\displaystyle \left(l',m'\right)}are pairs of relatively prime numbers, we find that
So
We now see thatm=gcd(a+c,d−b){\displaystyle m=\operatorname {gcd} (a+c,d-b)}andl=gcd(a−c,d+b){\displaystyle l=\operatorname {gcd} (a-c,d+b)}
Applying theBrahmagupta–Fibonacci identitywe get
As each factor is a sum of two squares, one of these must contain both even numbers: either(k,h){\displaystyle (k,h)}or(l,m){\displaystyle (l,m)}. Without loss of generality, assume that pair(k,h){\displaystyle (k,h)}is even. The factorization then becomes
Since:1000009=10002+32=9722+2352{\displaystyle \ 1000009=1000^{2}+3^{2}=972^{2}+235^{2}}
we have from the formula above:
Thus,
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Fermat's factorization method, named afterPierre de Fermat, is based on the representation of anoddintegeras thedifference of two squares:
That difference isalgebraicallyfactorable as(a+b)(a−b){\displaystyle (a+b)(a-b)}; if neither factor equals one, it is a proper factorization ofN.
Each odd number has such a representation. Indeed, ifN=cd{\displaystyle N=cd}is a factorization ofN, then
SinceNis odd, thencanddare also odd, so those halves are integers. (A multiple of four is also a difference of squares: letcanddbe even.)
In its simplest form, Fermat's method might be even slower than trial division (worst case). Nonetheless, the combination of trial division and Fermat's is more effective than either by itself.
One tries various values ofa, hoping thata2−N=b2{\displaystyle a^{2}-N=b^{2}}, a square.
For example, to factorN=5959{\displaystyle N=5959}, the first try forais the square root of5959rounded up to the next integer, which is78. Thenb2=782−5959=125{\displaystyle b^{2}=78^{2}-5959=125}. Since 125 is not a square, a second try is made by increasing the value ofaby 1. The second attempt also fails, because 282 is again not a square.
The third try produces the perfect square of 441. Thus,a=80{\displaystyle a=80},b=21{\displaystyle b=21}, and the factors of5959area−b=59{\displaystyle a-b=59}anda+b=101{\displaystyle a+b=101}.
Suppose N has more than two prime factors. That procedure first finds the factorization with the least values ofaandb. That is,a+b{\displaystyle a+b}is the smallest factor ≥ the square-root ofN, and soa−b=N/(a+b){\displaystyle a-b=N/(a+b)}is the largest factor ≤ root-N. If the procedure findsN=1⋅N{\displaystyle N=1\cdot N}, that shows thatNis prime.
ForN=cd{\displaystyle N=cd}, letcbe the largest subroot factor.a=(c+d)/2{\displaystyle a=(c+d)/2}, so the number of steps is approximately(c+d)/2−N=(d−c)2/2=(N−c)2/2c{\displaystyle (c+d)/2-{\sqrt {N}}=({\sqrt {d}}-{\sqrt {c}})^{2}/2=({\sqrt {N}}-c)^{2}/2c}.
IfNis prime (so thatc=1{\displaystyle c=1}), one needsO(N){\displaystyle O(N)}steps. This is a bad way to prove primality. But ifNhas a factor close to its square root, the method works quickly. More precisely, ifcdiffers less than(4N)1/4{\displaystyle {\left(4N\right)}^{1/4}}fromN{\displaystyle {\sqrt {N}}}, the method requires only one step; this is independent of the size ofN.[citation needed]
Consider trying to factor the prime numberN= 2,345,678,917, but also computebanda−bthroughout. Going up fromN{\displaystyle {\sqrt {N}}}rounded up to the next integer, which is 48,433, we can tabulate:
In practice, one wouldn't bother with that last row untilbis an integer. But observe that ifNhad a subroot factor abovea−b=47830.1{\displaystyle a-b=47830.1}, Fermat's method would have found it already.
Trial division would normally try up to 48,432; but after only four Fermat steps, we need only divide up to 47830, to find a factor or prove primality.
This all suggests a combined factoring method. Choose some boundamax>N{\displaystyle a_{\mathrm {max} }>{\sqrt {N}}}; use Fermat's method for factors betweenN{\displaystyle {\sqrt {N}}}andamax{\displaystyle a_{\mathrm {max} }}. This gives a bound for trial division which isamax−amax2−N{\displaystyle a_{\mathrm {max} }-{\sqrt {a_{\mathrm {max} }^{2}-N}}}. In the above example, withamax=48436{\displaystyle a_{\mathrm {max} }=48436}the bound for trial division is 47830. A reasonable choice could beamax=55000{\displaystyle a_{\mathrm {max} }=55000}giving a bound of 28937.
In this regard, Fermat's method gives diminishing returns. One would surely stop before this point:
When considering the table forN=2345678917{\displaystyle N=2345678917}, one can quickly tell that none of the values ofb2{\displaystyle b^{2}}are squares:
It is not necessary to compute all the square-roots ofa2−N{\displaystyle a^{2}-N}, nor even examine all the values fora. Squares are always congruent to 0, 1, 4, 5, 9, 16modulo20. The values repeat with each increase ofaby 10. In this example, N is 17 mod 20, so subtracting 17 mod 20 (or adding 3),a2−N{\displaystyle a^{2}-N}produces 3, 4, 7, 8, 12, and 19 modulo 20 for these values. It is apparent that only the 4 from this list can be a square. Thus,a2{\displaystyle a^{2}}must be 1 mod 20, which means thatais 1, 9, 11 or 19 mod 20; it will produce ab2{\displaystyle b^{2}}which ends in 4 mod 20 and, if square,bwill end in 2 or 8 mod 10.
This can be performed with any modulus. Using the sameN=2345678917{\displaystyle N=2345678917},
One generally chooses a power of a different prime for each modulus.
Given a sequence ofa-values (start, end, and step) and a modulus, one can proceed thus:
But therecursionis stopped when fewa-values remain; that is, when (aend-astart)/astep is small. Also, becausea's step-size is constant, one can compute successive b2's with additions.
Fermat's method works best when there is a factor near the square-root ofN.
If the approximate ratio of two factors (d/c{\displaystyle d/c}) is known, then arational numberv/u{\displaystyle v/u}can be picked near that value.Nuv=cv⋅du{\displaystyle Nuv=cv\cdot du}, and Fermat's method, applied toNuv, will find the factorscv{\displaystyle cv}anddu{\displaystyle du}quickly. Thengcd(N,cv)=c{\displaystyle \gcd(N,cv)=c}andgcd(N,du)=d{\displaystyle \gcd(N,du)=d}. (Unlesscdividesuorddividesv.)
Generally, if the ratio is not known, variousu/v{\displaystyle u/v}values can be tried, and try to factor each resultingNuv. R. Lehman devised a systematic way to do this, so that Fermat's plus trial division can factor N inO(N1/3){\displaystyle O(N^{1/3})}time.[1]
The fundamental ideas of Fermat's factorization method are the basis of thequadratic sieveandgeneral number field sieve, the best-known algorithms for factoring largesemiprimes, which are the "worst-case". The primary improvement that quadratic sieve makes over Fermat's factorization method is that instead of simply finding a square in the sequence ofa2−n{\displaystyle a^{2}-n}, it finds a subset of elements of this sequence whoseproductis a square, and it does this in a highly efficient manner. The end result is the same: a difference of squares modnthat, if nontrivial, can be used to factorn.
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Inmathematics, afactorisationof afree monoidis a sequence ofsubsetsof words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets. The Chen–Fox–Lyndontheorem states that theLyndon wordsfurnish a factorisation. TheSchützenbergertheorem relates the definition in terms of a multiplicative property to an additive property.[clarification needed]
LetA∗be thefree monoidon an alphabetA. LetXibe a sequence of subsets ofA∗indexed by atotally orderedindex setI. A factorisation of a wordwinA∗is an expression
withxij∈Xij{\displaystyle x_{i_{j}}\in X_{i_{j}}}andi1≥i2≥…≥in{\displaystyle i_{1}\geq i_{2}\geq \ldots \geq i_{n}}. Some authors reverse the order of the inequalities.
ALyndon wordover a totally ordered alphabetAis a word that islexicographicallyless than all its rotations.[1]TheChen–Fox–Lyndon theoremstates that every string may be formed in a unique way by concatenating a lexicographically non-increasing sequence of Lyndon words. Hence takingXlto be thesingleton set{l}for each Lyndon wordl, with the index setLof Lyndon words ordered lexicographically, we obtain a factorisation ofA∗.[2]Such a factorisation can be found inlinear timeand constant space by Duval's algorithm.[3]The algorithm[4]inPythoncode is:
TheHall setprovides a factorization.[5]Indeed, Lyndon words are a special case of Hall words. The article onHall wordsprovides a sketch of all of the mechanisms needed to establish a proof of this factorization.
Abisectionof a free monoid is a factorisation with just two classesX0,X1.[6]
Examples:
IfX,Yaredisjoint setsof non-empty words, then (X,Y) is a bisection ofA∗if and only if[7]
As a consequence, for any partitionP,QofA+there is a unique bisection (X,Y) withXa subset ofPandYa subset ofQ.[8]
This theorem states that a sequenceXiof subsets ofA∗forms a factorisation if and only if two of the following three statements hold:
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AGaussian integeris either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table ofGaussian Integersx+iyfollowed either by an explicit factorization or followed by the label (p) if the integer is aGaussian prime. The factorizations take the form of an optionalunitmultiplied by integer powers of Gaussian primes.
Note that there arerational primeswhich are not Gaussian primes. A simple example is the rational prime 5, which is factored as5=(2+i)(2−i)in the table, and therefore not a Gaussian prime.
The second column of the table contains only integers in the first quadrant, which means the real partxis positive and the imaginary partyis non-negative. The table might have been further reduced to the integers in the firstoctantof thecomplex planeusing the symmetryy+ix=i(x−iy).
The factorizations are often not unique in the sense that the unit could be absorbed into any other factor with exponent equal to one. The entry4+2i = −i(1+i)2(2+i), for example, could also be written as4+2i= (1+i)2(1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right complex half plane withabsolute valueof the real part larger than or equal to the absolute value of the imaginary part.
The entries are sorted according to increasing normx2+y2(sequenceA001481in theOEIS). The table is complete up to the maximum norm at the end of the table in the sense that
each composite or prime in the first quadrant appears in the second column.
Gaussian primes occur only for a subset of norms, detailed in sequenceOEIS:A055025. This here is a composition of sequencesOEIS:A103431andOEIS:A103432.
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Innumber theoryandcombinatorics, apartitionof a non-negativeintegern, also called aninteger partition, is a way of writingnas asumofpositive integers. Two sums that differ only in the order of theirsummandsare considered the same partition. (If order matters, the sum becomes acomposition.) For example,4can be partitioned in five distinct ways:
The only partition of zero is the empty sum, having no parts.
The order-dependent composition1 + 3is the same partition as3 + 1, and the two distinct compositions1 + 2 + 1and1 + 1 + 2represent the same partition as2 + 1 + 1.
An individual summand in a partition is called apart. The number of partitions ofnis given by thepartition functionp(n). Sop(4) = 5. The notationλ⊢nmeans thatλis a partition ofn.
Partitions can be graphically visualized withYoung diagramsorFerrers diagrams. They occur in a number of branches ofmathematicsandphysics, including the study ofsymmetric polynomialsand of thesymmetric groupand ingroup representation theoryin general.
The seven partitions of 5 are
Some authors treat a partition as a non-increasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as thetuple(2, 2, 1)or in the even more compact form(22, 1)where the superscript indicates the number of repetitions of a part.
This multiplicity notation for a partition can be written alternatively as1m12m23m3⋯{\displaystyle 1^{m_{1}}2^{m_{2}}3^{m_{3}}\cdots }, wherem1is the number of 1's,m2is the number of 2's, etc. (Components withmi= 0may be omitted.) For example, in this notation, the partitions of 5 are written51,1141,2131,1231,1122,1321{\displaystyle 5^{1},1^{1}4^{1},2^{1}3^{1},1^{2}3^{1},1^{1}2^{2},1^{3}2^{1}}, and15{\displaystyle 1^{5}}.
There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named afterNorman Macleod Ferrers, and as Young diagrams, named afterAlfred Young. Both have several possible conventions; here, we useEnglish notation, with diagrams aligned in the upper-left corner.
The partition 6 + 4 + 3 + 1 of the number 14 can be represented by the following diagram:
The 14 circles are lined up in 4 rows, each having the size of a part of the partition.
The diagrams for the 5 partitions of the number 4 are shown below:
An alternative visual representation of an integer partition is itsYoung diagram(often also called a Ferrers diagram). Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for the partition 5 + 4 + 1 is
while the Ferrers diagram for the same partition is
While this seemingly trivial variation does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study ofsymmetric functionsandgroup representation theory: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects calledYoung tableaux, and these tableaux have combinatorial and representation-theoretic significance.[1]As a type of shape made by adjacent squares joined together, Young diagrams are a special kind ofpolyomino.[2]
Thepartition functionp(n){\displaystyle p(n)}counts the partitions of a non-negative integern{\displaystyle n}. For instance,p(4)=5{\displaystyle p(4)=5}because the integer4{\displaystyle 4}has the five partitions1+1+1+1{\displaystyle 1+1+1+1},1+1+2{\displaystyle 1+1+2},1+3{\displaystyle 1+3},2+2{\displaystyle 2+2}, and4{\displaystyle 4}.
The values of this function forn=0,1,2,…{\displaystyle n=0,1,2,\dots }are:
Thegenerating functionofp{\displaystyle p}is
Noclosed-form expressionfor the partition function is known, but it has bothasymptotic expansionsthat accurately approximate it andrecurrence relationsby which it can be calculated exactly. It grows as anexponential functionof thesquare rootof its argument.,[3]as follows:
In 1937,Hans Rademacherfound a way to represent the partition functionp(n){\displaystyle p(n)}by theconvergent series
p(n)=1π2∑k=1∞Ak(n)k⋅ddn(1n−124sinh[πk23(n−124)]){\displaystyle p(n)={\frac {1}{\pi {\sqrt {2}}}}\sum _{k=1}^{\infty }A_{k}(n){\sqrt {k}}\cdot {\frac {d}{dn}}\left({{\frac {1}{\sqrt {n-{\frac {1}{24}}}}}\sinh \left[{{\frac {\pi }{k}}{\sqrt {{\frac {2}{3}}\left(n-{\frac {1}{24}}\right)}}}\,\,\,\right]}\right)}where
Ak(n)=∑0≤m<k,(m,k)=1eπi(s(m,k)−2nm/k).{\displaystyle A_{k}(n)=\sum _{0\leq m<k,\;(m,k)=1}e^{\pi i\left(s(m,k)-2nm/k\right)}.}ands(m,k){\displaystyle s(m,k)}is theDedekind sum.
Themultiplicative inverseof its generating function is theEuler function; by Euler'spentagonal number theoremthis function is an alternating sum ofpentagonal numberpowers of its argument.
Srinivasa Ramanujandiscovered that the partition function has nontrivial patterns inmodular arithmetic, now known asRamanujan's congruences. For instance, whenever the decimal representation ofn{\displaystyle n}ends in the digit 4 or 9, the number of partitions ofn{\displaystyle n}will be divisible by 5.[4]
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied.[5]This section surveys a few such restrictions.
If we flip the diagram of the partition 6 + 4 + 3 + 1 along itsmain diagonal, we obtain another partition of 14:
By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions are said to beconjugateof one another.[6]In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest are partitions, such as 2 + 2, which have themselves as conjugate. Such partitions are said to beself-conjugate.[7]
Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
Proof (outline): The crucial observation is that every odd part can be "folded" in the middle to form a self-conjugate diagram:
One can then obtain abijectionbetween the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example:
Among the 22 partitions of the number 8, there are 6 that contain onlyodd parts:
Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called apartition with distinct parts. If we count the partitions of 8 with distinct parts, we also obtain 6:
This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted byq(n).[8][9]This result was proved byLeonhard Eulerin 1748[10]and later was generalized asGlaisher's theorem.
For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example isq(n) (partitions into distinct parts). The first few values ofq(n) are (starting withq(0)=1):
Thegenerating functionforq(n) is given by[11]
Thepentagonal number theoremgives a recurrence forq:[12]
whereakis (−1)mifk= 3m2−mfor some integermand is 0 otherwise.
By taking conjugates, the numberpk(n)of partitions ofninto exactlykparts is equal to the number of partitions ofnin which the largest part has sizek. The functionpk(n)satisfies the recurrence
with initial valuesp0(0) = 1andpk(n) = 0ifn≤ 0 ork≤ 0andnandkare not both zero.[13]
One recovers the functionp(n) by
One possible generating function for such partitions, takingkfixed andnvariable, is
More generally, ifTis a set of positive integers then the number of partitions ofn, all of whose parts belong toT, hasgenerating function
This can be used to solvechange-making problems(where the setTspecifies the available coins). As two particular cases, one has that the number of partitions ofnin which all parts are 1 or 2 (or, equivalently, the number of partitions ofninto 1 or 2 parts) is
and the number of partitions ofnin which all parts are 1, 2 or 3 (or, equivalently, the number of partitions ofninto at most three parts) is the nearest integer to (n+ 3)2/ 12.[14]
One may also simultaneously limit the number and size of the parts. Letp(N,M;n)denote the number of partitions ofnwith at mostMparts, each of size at mostN. Equivalently, these are the partitions whose Young diagram fits inside anM×Nrectangle. There is a recurrence relationp(N,M;n)=p(N,M−1;n)+p(N−1,M;n−M){\displaystyle p(N,M;n)=p(N,M-1;n)+p(N-1,M;n-M)}obtained by observing thatp(N,M;n)−p(N,M−1;n){\displaystyle p(N,M;n)-p(N,M-1;n)}counts the partitions ofninto exactlyMparts of size at mostN, and subtracting 1 from each part of such a partition yields a partition ofn−Minto at mostMparts.[15]
The Gaussian binomial coefficient is defined as:(k+ℓℓ)q=(k+ℓk)q=∏j=1k+ℓ(1−qj)∏j=1k(1−qj)∏j=1ℓ(1−qj).{\displaystyle {k+\ell \choose \ell }_{q}={k+\ell \choose k}_{q}={\frac {\prod _{j=1}^{k+\ell }(1-q^{j})}{\prod _{j=1}^{k}(1-q^{j})\prod _{j=1}^{\ell }(1-q^{j})}}.}The Gaussian binomial coefficient is related to thegenerating functionofp(N,M;n)by the equality∑n=0MNp(N,M;n)qn=(M+NM)q.{\displaystyle \sum _{n=0}^{MN}p(N,M;n)q^{n}={M+N \choose M}_{q}.}
Therankof a partition is the largest numberksuch that the partition contains at leastkparts of size at leastk. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rankr, ther×rsquare of entries in the upper-left is known as theDurfee square:
The Durfee square has applications within combinatorics in the proofs of various partition identities.[16]It also has some practical significance in the form of theh-index.
A different statistic is also sometimes called therank of a partition(or Dyson rank), namely, the differenceλk−k{\displaystyle \lambda _{k}-k}for a partition ofkparts with largest partλk{\displaystyle \lambda _{k}}. This statistic (which is unrelated to the one described above) appears in the study ofRamanujan congruences.
There is a naturalpartial orderon partitions given by inclusion of Young diagrams. This partially ordered set is known asYoung's lattice. The lattice was originally defined in the context ofrepresentation theory, where it is used to describe theirreducible representationsofsymmetric groupsSnfor alln, together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of adifferential poset.
There is a deep theory of random partitions chosen according to the uniform probability distribution on thesymmetric groupvia theRobinson–Schensted correspondence. In 1977, Logan and Shepp, as well as Vershik and Kerov, showed that the Young diagram of a typical large partition becomes asymptotically close to the graph of a certain analytic function minimizing a certain functional. In 1988, Baik, Deift and Johansson extended these results to determine the distribution of the longest increasing subsequence of a random permutation in terms of theTracy–Widom distribution.[17]Okounkovrelated these results to the combinatorics ofRiemann surfacesand representation theory.[18][19]
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Inmathematics, adivisorof an integern,{\displaystyle n,}also called afactorofn,{\displaystyle n,}is anintegerm{\displaystyle m}that may be multiplied by some integer to producen.{\displaystyle n.}[1]In this case, one also says thatn{\displaystyle n}is amultipleofm.{\displaystyle m.}An integern{\displaystyle n}isdivisibleorevenly divisibleby another integerm{\displaystyle m}ifm{\displaystyle m}is a divisor ofn{\displaystyle n}; this implies dividingn{\displaystyle n}bym{\displaystyle m}leaves no remainder.
Anintegern{\displaystyle n}is divisible by a nonzero integerm{\displaystyle m}if there exists an integerk{\displaystyle k}such thatn=km.{\displaystyle n=km.}This is written as
This may be read as thatm{\displaystyle m}dividesn,{\displaystyle n,}m{\displaystyle m}is a divisor ofn,{\displaystyle n,}m{\displaystyle m}is a factor ofn,{\displaystyle n,}orn{\displaystyle n}is a multiple ofm.{\displaystyle m.}Ifm{\displaystyle m}does not dividen,{\displaystyle n,}then the notation ism∤n.{\displaystyle m\not \mid n.}[2][3]
There are two conventions, distinguished by whetherm{\displaystyle m}is permitted to be zero:
Divisors can benegativeas well as positive, although often the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.
1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are calledeven, and integers not divisible by 2 are calledodd.
1, −1,n{\displaystyle n}and−n{\displaystyle -n}are known as thetrivial divisorsofn.{\displaystyle n.}A divisor ofn{\displaystyle n}that is not a trivial divisor is known as anon-trivial divisor(or strict divisor[6]). A nonzero integer with at least one non-trivial divisor is known as acomposite number, while theunits−1 and 1 andprime numbershave no non-trivial divisors.
There aredivisibility rulesthat allow one to recognize certain divisors of a number from the number's digits.
There are some elementary rules:
Ifa∣bc,{\displaystyle a\mid bc,}andgcd(a,b)=1,{\displaystyle \gcd(a,b)=1,}thena∣c.{\displaystyle a\mid c.}[b]This is calledEuclid's lemma.
Ifp{\displaystyle p}is a prime number andp∣ab{\displaystyle p\mid ab}thenp∣a{\displaystyle p\mid a}orp∣b.{\displaystyle p\mid b.}
A positive divisor ofn{\displaystyle n}that is different fromn{\displaystyle n}is called aproper divisoror analiquot partofn{\displaystyle n}(for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly dividen{\displaystyle n}but leaves a remainder is sometimes called analiquant partofn.{\displaystyle n.}
An integern>1{\displaystyle n>1}whose only proper divisor is 1 is called aprime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.
Any positive divisor ofn{\displaystyle n}is a product ofprime divisorsofn{\displaystyle n}raised to some power. This is a consequence of thefundamental theorem of arithmetic.
A numbern{\displaystyle n}is said to beperfectif it equals the sum of its proper divisors,deficientif the sum of its proper divisors is less thann,{\displaystyle n,}andabundantif this sum exceedsn.{\displaystyle n.}
The total number of positive divisors ofn{\displaystyle n}is amultiplicative functiond(n),{\displaystyle d(n),}meaning that when two numbersm{\displaystyle m}andn{\displaystyle n}arerelatively prime, thend(mn)=d(m)×d(n).{\displaystyle d(mn)=d(m)\times d(n).}For instance,d(42)=8=2×2×2=d(2)×d(3)×d(7){\displaystyle d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)}; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However, the number of positive divisors is not a totally multiplicative function: if the two numbersm{\displaystyle m}andn{\displaystyle n}share a common divisor, then it might not be true thatd(mn)=d(m)×d(n).{\displaystyle d(mn)=d(m)\times d(n).}The sum of the positive divisors ofn{\displaystyle n}is another multiplicative functionσ(n){\displaystyle \sigma (n)}(for example,σ(42)=96=3×4×8=σ(2)×σ(3)×σ(7)=1+2+3+6+7+14+21+42{\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42}). Both of these functions are examples ofdivisor functions.
If theprime factorizationofn{\displaystyle n}is given by
then the number of positive divisors ofn{\displaystyle n}is
and each of the divisors has the form
where0≤μi≤νi{\displaystyle 0\leq \mu _{i}\leq \nu _{i}}for each1≤i≤k.{\displaystyle 1\leq i\leq k.}
For every naturaln,{\displaystyle n,}d(n)<2n.{\displaystyle d(n)<2{\sqrt {n}}.}
Also,[7]
whereγ{\displaystyle \gamma }isEuler–Mascheroni constant.
One interpretation of this result is that a randomly chosen positive integernhas an average
number of divisors of aboutlnn.{\displaystyle \ln n.}However, this is a result from the contributions ofnumbers with "abnormally many" divisors.
In definitions that allow the divisor to be 0, the relation of divisibility turns the setN{\displaystyle \mathbb {N} }ofnon-negativeintegers into apartially ordered setthat is acomplete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation∧is given by thegreatest common divisorand the join operation∨by theleast common multiple. This lattice is isomorphic to thedualof thelattice of subgroupsof the infinitecyclic groupZ.
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Inalgebra(in particular inalgebraic geometryoralgebraic number theory), avaluationis afunctionon afieldthat provides a measure of the size or multiplicity of elements of the field. It generalizes tocommutative algebrathe notion of size inherent in consideration of the degree of apoleormultiplicityof azeroincomplex analysis, the degree of divisibility of a number by a prime number innumber theory, and the geometrical concept ofcontactbetween twoalgebraicoranalytic varietiesin algebraic geometry. A field with a valuation on it is called avalued field.
One starts with the following objects:
The ordering andgroup lawonΓare extended to the setΓ ∪ {∞}[a]by the rules
Then avaluation ofKis anymap
that satisfies the following properties for alla,binK:
A valuationvistrivialifv(a) = 0 for allainK×, otherwise it isnon-trivial.
The second property asserts that any valuation is agroup homomorphismonK×. The third property is a version of thetriangle inequalityonmetric spacesadapted to an arbitrary Γ (seeMultiplicative notationbelow). For valuations used ingeometricapplications, the first property implies that any non-emptygermof an analytic variety near a point contains that point.
The valuation can be interpreted as the order of theleading-order term.[b]The third property then corresponds to the order of a sum being the order of the larger term,[c]unless the two terms have the same order, in which case they may cancel and the sum may have larger order.
For many applications,Γis an additive subgroup of thereal numbersR{\displaystyle \mathbb {R} }[d]in which case ∞ can be interpreted as +∞ in theextended real numbers; note thatmin(a,+∞)=min(+∞,a)=a{\displaystyle \min(a,+\infty )=\min(+\infty ,a)=a}for any real numbera, and thus +∞ is the unit under the binary operation of minimum. The real numbers (extended by +∞) with the operations of minimum and addition form asemiring, called the mintropical semiring,[e]and a valuationvis almost a semiring homomorphism fromKto the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together.
The concept was developed byEmil Artinin his bookGeometric Algebrawriting the group inmultiplicative notationas(Γ, ·, ≥):[1]
Instead of ∞, we adjoin a formal symbolOto Γ, with the ordering and group law extended by the rules
Then avaluationofKis any map
satisfying the following properties for alla,b∈K:
(Note that the directions of the inequalities are reversed from those in the additive notation.)
IfΓis a subgroup of thepositive real numbersunder multiplication, the last condition is theultrametricinequality, a stronger form of thetriangle inequality|a+b|v≤|a|v+|b|v, and| ⋅ |vis anabsolute value. In this case, we may pass to the additive notation with value groupΓ+⊆(R,+){\displaystyle \Gamma _{+}\subseteq (\mathbb {R} ,+)}by takingv+(a) = −log|a|v.
Each valuation onKdefines a corresponding linearpreorder:a≼b⇔|a|v≤|b|v. Conversely, given a "≼" satisfying the required properties, we can define valuation|a|v= {b:b≼a∧a≼b}, with multiplication and ordering based onKand≼.
In this article, we use the terms defined above, in the additive notation. However, some authors use alternative terms:
There are several objects defined from a given valuationv:K→ Γ ∪ {∞};
Two valuationsv1andv2ofKwith valuation group Γ1and Γ2, respectively, are said to beequivalentif there is an order-preservinggroup isomorphismφ: Γ1→ Γ2such thatv2(a) = φ(v1(a)) for allainK×. This is anequivalence relation.
Two valuations ofKare equivalent if and only if they have the same valuation ring.
Anequivalence classof valuations of a field is called aplace.Ostrowski's theoremgives a complete classification of places of the field ofrational numbersQ:{\displaystyle \mathbb {Q} :}these are precisely the equivalence classes of valuations for thep-adiccompletionsofQ.{\displaystyle \mathbb {Q} .}
Letvbe a valuation ofKand letLbe afield extensionofK. Anextension ofv(toL) is a valuationwofLsuch that therestrictionofwtoKisv. The set of all such extensions is studied in theramification theory of valuations.
LetL/Kbe afinite extensionand letwbe an extension ofvtoL. Theindexof Γvin Γw, e(w/v) = [Γw: Γv], is called thereduced ramification indexofwoverv. It satisfies e(w/v) ≤ [L:K] (thedegreeof the extensionL/K). Therelative degreeofwovervis defined to bef(w/v) = [Rw/mw:Rv/mv] (the degree of the extension of residue fields). It is also less than or equal to the degree ofL/K. WhenL/Kisseparable, theramification indexofwovervis defined to be e(w/v)pi, wherepiis theinseparable degreeof the extensionRw/mwoverRv/mv.
When the ordered abelian groupΓis the additive group of theintegers, the associated valuation is equivalent to an absolute value, and hence induces ametricon the fieldK. IfKiscompletewith respect to this metric, then it is called acomplete valued field. IfKis not complete, one can use the valuation to construct itscompletion, as in the examples below, and different valuations can define different completion fields.
In general, a valuation induces auniform structureonK, andKis called a complete valued field if it iscompleteas a uniform space. There is a related property known asspherical completeness: it is equivalent to completeness ifΓ=Z,{\displaystyle \Gamma =\mathbb {Z} ,}but stronger in general.
The most basic example is thep-adic valuationνpassociated to a prime integerp, on the rational numbersK=Q,{\displaystyle K=\mathbb {Q} ,}with valuation ringR=Z(p),{\displaystyle R=\mathbb {Z} _{(p)},}whereZ(p){\displaystyle \mathbb {Z} _{(p)}}is the localization ofZ{\displaystyle \mathbb {Z} }at the prime ideal(p){\displaystyle (p)}. The valuation group is the additive integersΓ=Z.{\displaystyle \Gamma =\mathbb {Z} .}For an integera∈R=Z,{\displaystyle a\in R=\mathbb {Z} ,}the valuation νp(a) measures the divisibility ofaby powers ofp:
and for a fraction, νp(a/b) = νp(a) − νp(b).
Writing this multiplicatively yields thep-adic absolute value, which conventionally has as base1/p=p−1{\displaystyle 1/p=p^{-1}}, so|a|p:=p−νp(a){\displaystyle |a|_{p}:=p^{-\nu _{p}(a)}}.
ThecompletionofQ{\displaystyle \mathbb {Q} }with respect to νpis the fieldQp{\displaystyle \mathbb {Q} _{p}}ofp-adic numbers.
Let K =F(x), the rational functions on the affine lineX=F1, and take a pointa∈ X. For a polynomialf(x)=ak(x−a)k+ak+1(x−a)k+1+⋯+an(x−a)n{\displaystyle f(x)=a_{k}(x{-}a)^{k}+a_{k+1}(x{-}a)^{k+1}+\cdots +a_{n}(x{-}a)^{n}}withak≠0{\displaystyle a_{k}\neq 0}, defineva(f) = k, the order of vanishing atx=a; andva(f/g) =va(f) −va(g). Then the valuation ringRconsists of rational functions with no pole atx=a, and the completion is theformal Laurent seriesringF((x−a)). This can be generalized to the field ofPuiseux seriesK{{t}} (fractional powers), theLevi-Civita field(its Cauchy completion), and the field ofHahn series, with valuation in all cases returning the smallest exponent oftappearing in the series.
Generalizing the previous examples, letRbe aprincipal ideal domain,Kbe itsfield of fractions, andπbe anirreducible elementofR. Since every principal ideal domain is aunique factorization domain, every non-zero elementaofRcan be written (essentially) uniquely as
where thee's are non-negative integers and thepiare irreducible elements ofRthat are notassociatesofπ. In particular, the integereais uniquely determined bya.
Theπ-adic valuation ofKis then given by
If π' is another irreducible element ofRsuch that (π') = (π) (that is, they generate the same ideal inR), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called theP-adic valuation, whereP= (π).
The previous example can be generalized toDedekind domains. LetRbe a Dedekind domain,Kits field of fractions, and letPbe a non-zero prime ideal ofR. Then, thelocalizationofRatP, denotedRP, is a principal ideal domain whose field of fractions isK. The construction of the previous section applied to the prime idealPRPofRPyields theP-adic valuation ofK.
Suppose thatΓ∪ {0} is the set of non-negative real numbers under multiplication. Then we say that the valuation isnon-discreteif its range (the valuation group) is infinite (and hence has an accumulation point at 0).
Suppose thatXis a vector space overKand thatAandBare subsets ofX. Then we say thatAabsorbsBif there exists aα∈Ksuch thatλ∈Kand|λ| ≥ |α|implies thatB ⊆ λ A.Ais calledradialorabsorbingifAabsorbs every finite subset ofX. Radial subsets ofXare invariant under finite intersection. Also,Ais calledcircledifλinKand|λ| ≥ |α|impliesλ A ⊆ A. The set of circled subsets ofLis invariant under arbitrary intersections. Thecircled hullofAis the intersection of all circled subsets ofXcontainingA.
Suppose thatXandYare vector spaces over a non-discrete valuation fieldK, letA ⊆ X,B ⊆ Y, and letf : X → Ybe a linear map. IfBis circled or radial then so isf−1(B){\displaystyle f^{-1}(B)}. IfAis circled then so isf(A)but ifAis radial thenf(A)will be radial under the additional condition thatfis surjective.
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Inabstract algebraandanalysis, theArchimedean property, named after the ancient Greek mathematicianArchimedesofSyracuse, is a property held by somealgebraic structures, such as ordered or normedgroups, andfields.
The property, as typically construed, states that given two positive numbersx{\displaystyle x}andy{\displaystyle y}, there is an integern{\displaystyle n}such thatnx>y{\displaystyle nx>y}. It also means that the set ofnatural numbersis not bounded above.[1]Roughly speaking, it is the property of having noinfinitely largeorinfinitely smallelements. It wasOtto Stolzwho gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’On the Sphere and Cylinder.[2]
The notion arose from the theory ofmagnitudesof ancient Greece; it still plays an important role in modern mathematics such asDavid Hilbert'saxioms for geometry, and the theories ofordered groups,ordered fields, andlocal fields.
An algebraic structure in which any two non-zero elements arecomparable, in the sense that neither of them isinfinitesimalwith respect to the other, is said to beArchimedean.
A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to benon-Archimedean.
For example, alinearly ordered groupthat is Archimedean is anArchimedean group.
This can be made precise in various contexts with slightly different formulations.
For example, in the context ofordered fields, one has theaxiom of Archimedeswhich formulates this property, where the field ofreal numbersis Archimedean, but that ofrational functionsin real coefficients is not.
The concept was named byOtto Stolz(in the 1880s) after theancient Greekgeometer and physicistArchimedesofSyracuse.
The Archimedean property appears in Book V ofEuclid'sElementsas Definition 4:
Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
Because Archimedes credited it toEudoxus of Cnidusit is also known as the "Theorem of Eudoxus" or theEudoxus axiom.[3]
Archimedes used infinitesimalsinheuristicarguments, although he denied that those were finishedmathematical proofs.
Letxandybepositive elementsof alinearly ordered groupG.
Thenx{\displaystyle x}is infinitesimal with respect toy{\displaystyle y}(or equivalently,y{\displaystyle y}is infinite with respect tox{\displaystyle x}) if, for anynatural numbern{\displaystyle n}, the multiplenx{\displaystyle nx}is less thany{\displaystyle y}, that is, the following inequality holds:x+⋯+x⏟nterms<y.{\displaystyle \underbrace {x+\cdots +x} _{n{\text{ terms}}}<y.\,}
This definition can be extended to the entire group by taking absolute values.
The groupG{\displaystyle G}isArchimedeanif there is no pair(x,y){\displaystyle (x,y)}such thatx{\displaystyle x}is infinitesimal with respect toy{\displaystyle y}.
Additionally, ifK{\displaystyle K}is analgebraic structurewith a unit (1) — for example, aring— a similar definition applies toK{\displaystyle K}.
Ifx{\displaystyle x}is infinitesimal with respect to1{\displaystyle 1}, thenx{\displaystyle x}is aninfinitesimal element.
Likewise, ify{\displaystyle y}is infinite with respect to1{\displaystyle 1}, theny{\displaystyle y}is aninfinite element.
The algebraic structureK{\displaystyle K}is Archimedean if it has no infinite elements and no infinitesimal elements.
Ordered fieldshave some additional properties:
In this setting, an ordered fieldKis Archimedean precisely when the following statement, called theaxiom of Archimedes, holds:
Alternatively one can use the following characterization:∀ε∈K(ε>0⟹∃n∈N:1/n<ε).{\displaystyle \forall \,\varepsilon \in K{\big (}\varepsilon >0\implies \exists \ n\in N:1/n<\varepsilon {\big )}.}
The qualifier "Archimedean" is also formulated in the theory ofrank one valued fieldsand normed spaces over rank one valued fields as follows.
LetK{\displaystyle K}be a field endowed with an absolute value function, i.e., a function which associates the real number0{\displaystyle 0}with the field element 0 and associates a positive real number|x|{\displaystyle |x|}with each non-zerox∈K{\displaystyle x\in K}and satisfies|xy|=|x||y|{\displaystyle |xy|=|x||y|}and|x+y|≤|x|+|y|{\displaystyle |x+y|\leq |x|+|y|}.
Then,K{\displaystyle K}is said to beArchimedeanif for any non-zerox∈K{\displaystyle x\in K}there exists anatural numbern{\displaystyle n}such that|x+⋯+x⏟nterms|>1.{\displaystyle |\underbrace {x+\cdots +x} _{n{\text{ terms}}}|>1.}
Similarly, a normed space is Archimedean if a sum ofn{\displaystyle n}terms, each equal to a non-zero vectorx{\displaystyle x}, has norm greater than one for sufficiently largen{\displaystyle n}.
A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as theultrametrictriangle inequality,|x+y|≤max(|x|,|y|),{\displaystyle |x+y|\leq \max(|x|,|y|),}respectively.
A field or normed space satisfying the ultrametric triangle inequality is callednon-Archimedean.
The concept of a non-Archimedean normed linear space was introduced by A. F. Monna.[4]
The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function|x|=1{\displaystyle |x|=1}, whenx≠0{\displaystyle x\neq 0}, the more usual|x|=x2{\textstyle |x|={\sqrt {x^{2}}}}, and thep{\displaystyle p}-adic absolute valuefunctions.
ByOstrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or somep{\displaystyle p}-adic absolute value.
The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete.
The completion with respect to the usual absolute value (from the order) is the field of real numbers.
By this construction the field of real numbers is Archimedean both as an ordered field and as a normed field.[5]On the other hand, the completions with respect to the other non-trivial absolute values give the fields ofp-adic numbers, wherep{\displaystyle p}is a prime integer number (see below); since thep{\displaystyle p}-adic absolute values satisfy theultrametricproperty, then thep{\displaystyle p}-adic number fields are non-Archimedean as normed fields (they cannot be made into ordered fields).
In theaxiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by theleast upper bound propertyas follows.
Denote byZ{\displaystyle Z}the set consisting of all positive infinitesimals.
This set is bounded above by1{\displaystyle 1}.
Nowassume for a contradictionthatZ{\displaystyle Z}is nonempty.
Then it has aleast upper boundc{\displaystyle c}, which is also positive, soc/2<c<2c{\displaystyle c/2<c<2c}.
Sincecis anupper boundofZ{\displaystyle Z}and2c{\displaystyle 2c}is strictly larger thanc{\displaystyle c},2c{\displaystyle 2c}is not a positive infinitesimal.
That is, there is some natural numbern{\displaystyle n}for which1/n<2c{\displaystyle 1/n<2c}.
On the other hand,c/2{\displaystyle c/2}is a positive infinitesimal, since by the definition of least upper bound there must be an infinitesimalx{\displaystyle x}betweenc/2{\displaystyle c/2}andc{\displaystyle c}, and if1/k<c/2≤x{\displaystyle 1/k<c/2\leq x}thenx{\displaystyle x}is not infinitesimal.
But1/(4n)<c/2{\displaystyle 1/(4n)<c/2}, soc/2{\displaystyle c/2}is not infinitesimal, and this is a contradiction.
This means thatZ{\displaystyle Z}is empty after all: there are no positive, infinitesimal real numbers.
The Archimedean property of real numbers holds also inconstructive analysis, even though the least upper bound property may fail in that context.
For an example of anordered fieldthat is not Archimedean, take the field ofrational functionswith real coefficients.
(A rational function is any function that can be expressed as onepolynomialdivided by another polynomial; we will assume in what follows that this has been done in such a way that theleading coefficientof the denominator is positive.)
To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations.
Nowf>g{\displaystyle f>g}if and only iff−g>0{\displaystyle f-g>0}, so we only have to say which rational functions are considered positive.
Call the function positive if the leading coefficient of the numerator is positive. (One must check that this ordering is well defined and compatible with addition and multiplication.)
By this definition, the rational function1/x{\displaystyle 1/x}is positive but less than the rational function1{\displaystyle 1}.
In fact, ifn{\displaystyle n}is any natural number, thenn(1/x)=n/x{\displaystyle n(1/x)=n/x}is positive but still less than1{\displaystyle 1}, no matter how bign{\displaystyle n}is.
Therefore,1/x{\displaystyle 1/x}is an infinitesimal in this field.
This example generalizes to other coefficients.
Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field.
Taking the coefficients to be the rational functions in a different variable, sayy{\displaystyle y}, produces an example with a differentorder type.
The field of the rational numbers endowed with the p-adic metric and thep-adic numberfields which are the completions, do not have the Archimedean property as fields with absolute values.
All Archimedean valued fields are isometrically isomorphic to a subfield of the complex numbers with a power of the usual absolute value.[6]
Every linearly ordered fieldK{\displaystyle K}contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit1{\displaystyle 1}ofK{\displaystyle K}, which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an orderedmonoid.
The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers inK{\displaystyle K}.
The following are equivalent characterizations of Archimedean fields in terms of these substructures.[7]
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Inmathematics, themultiplicityof a member of amultisetis the number of times it appears in the multiset. For example, the number of times a givenpolynomialhas arootat a given point is the multiplicity of that root.
The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example,double rootscounted twice). Hence the expression, "counted with multiplicity".
If multiplicity is ignored, this may be emphasized by counting the number ofdistinctelements, as in "the number of distinct roots". However, whenever aset(as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".
Inprime factorization, themultiplicityof a prime factor is itsp{\displaystyle p}-adic valuation. For example, the prime factorization of theinteger60is
the multiplicity of the prime factor2is2, while the multiplicity of each of the prime factors3and5is1. Thus,60has four prime factors allowing for multiplicities, but only three distinct prime factors.
LetF{\displaystyle F}be afieldandp(x){\displaystyle p(x)}be apolynomialin one variable withcoefficientsinF{\displaystyle F}. An elementa∈F{\displaystyle a\in F}is arootof multiplicityk{\displaystyle k}ofp(x){\displaystyle p(x)}if there is a polynomials(x){\displaystyle s(x)}such thats(a)≠0{\displaystyle s(a)\neq 0}andp(x)=(x−a)ks(x){\displaystyle p(x)=(x-a)^{k}s(x)}. Ifk=1{\displaystyle k=1}, thenais called asimple root. Ifk≥2{\displaystyle k\geq 2}, thena{\displaystyle a}is called amultiple root.
For instance, the polynomialp(x)=x3+2x2−7x+4{\displaystyle p(x)=x^{3}+2x^{2}-7x+4}has 1 and −4 asroots, and can be written asp(x)=(x+4)(x−1)2{\displaystyle p(x)=(x+4)(x-1)^{2}}. This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of thefundamental theorem of algebra.
Ifa{\displaystyle a}is a root of multiplicityk{\displaystyle k}of a polynomial, then it is a root of multiplicityk−1{\displaystyle k-1}of thederivativeof that polynomial, unless thecharacteristicof the underlying field is a divisor ofk, in which casea{\displaystyle a}is a root of multiplicity at leastk{\displaystyle k}of the derivative.
Thediscriminantof a polynomial is zero if and only if the polynomial has a multiple root.
Thegraphof apolynomial functionftouches thex-axis at the real roots of the polynomial. The graph istangentto it at the multiple roots offand not tangent at the simple roots. The graph crosses thex-axis at roots of odd multiplicity and does not cross it at roots of even multiplicity.
A non-zero polynomial function is everywherenon-negativeif and only if all its roots have even multiplicity and there exists anx0{\displaystyle x_{0}}such thatf(x0)>0{\displaystyle f(x_{0})>0}.
For an equationf(x)=0{\displaystyle f(x)=0}with a single variable solutionx∗{\displaystyle x_{*}}, the multiplicity isk{\displaystyle k}if
In other words, the differential functional∂j{\displaystyle \partial _{j}}, defined as the derivative1j!djdxj{\displaystyle {\frac {1}{j!}}{\frac {d^{j}}{dx^{j}}}}of a function atx∗{\displaystyle x_{*}}, vanishes atf{\displaystyle f}forj{\displaystyle j}up tok−1{\displaystyle k-1}. Those differential functionals∂0,∂1,⋯,∂k−1{\displaystyle \partial _{0},\partial _{1},\cdots ,\partial _{k-1}}span a vector space, called theMacaulay dual spaceatx∗{\displaystyle x_{*}},[1]and its dimension is the multiplicity ofx∗{\displaystyle x_{*}}as a zero off{\displaystyle f}.
Letf(x)=0{\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} }be a system ofm{\displaystyle m}equations ofn{\displaystyle n}variables with a solutionx∗{\displaystyle \mathbf {x} _{*}}wheref{\displaystyle \mathbf {f} }is a mapping fromRn{\displaystyle R^{n}}toRm{\displaystyle R^{m}}or fromCn{\displaystyle C^{n}}toCm{\displaystyle C^{m}}. There is also a Macaulay dual space of differential functionals atx∗{\displaystyle \mathbf {x} _{*}}in which every functional vanishes atf{\displaystyle \mathbf {f} }. The dimension of this Macaulay dual space is the multiplicity of the solutionx∗{\displaystyle \mathbf {x} _{*}}to the equationf(x)=0{\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} }. The Macaulay dual space forms the multiplicity structure of the system at the solution.[2][3]
For example, the solutionx∗=(0,0){\displaystyle \mathbf {x} _{*}=(0,0)}of the system of equations in the form off(x)=0{\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} }with
is of multiplicity 3 because the Macaulay dual space
is of dimension 3, where∂ij{\displaystyle \partial _{ij}}denotes the differential functional1i!j!∂i+j∂x1i∂x2j{\displaystyle {\frac {1}{i!j!}}{\frac {\partial ^{i+j}}{\partial x_{1}^{i}\,\partial x_{2}^{j}}}}applied on a function at the pointx∗=(0,0){\displaystyle \mathbf {x} _{*}=(0,0)}.
The multiplicity is always finite if the solution is isolated, is perturbation invariant in the sense that ak{\displaystyle k}-fold solution becomes a cluster of solutions with a combined multiplicityk{\displaystyle k}under perturbation in complex spaces, and is identical to the intersection multiplicity on polynomial systems.
Inalgebraic geometry, the intersection of two sub-varieties of an algebraic variety is a finite union ofirreducible varieties. To each component of such an intersection is attached anintersection multiplicity. This notion islocalin the sense that it may be defined by looking at what occurs in a neighborhood of anygeneric pointof this component. It follows that without loss of generality, we may consider, in order to define the intersection multiplicity, the intersection of twoaffines varieties(sub-varieties of an affine space).
Thus, given two affine varietiesV1andV2, consider anirreducible componentWof the intersection ofV1andV2. Letdbe thedimensionofW, andPbe any generic point ofW. The intersection ofWwithdhyperplanesingeneral positionpassing throughPhas an irreducible component that is reduced to the single pointP. Therefore, thelocal ringat this component of thecoordinate ringof the intersection has only oneprime ideal, and is therefore anArtinian ring. This ring is thus afinite dimensionalvector space over the ground field. Its dimension is the intersection multiplicity ofV1andV2atW.
This definition allows us to stateBézout's theoremand its generalizations precisely.
This definition generalizes the multiplicity of a root of a polynomial in the following way. The roots of a polynomialfare points on theaffine line, which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set isR=K[X]/⟨f⟩,{\displaystyle R=K[X]/\langle f\rangle ,}whereKis analgebraically closed fieldcontaining the coefficients off. Iff(X)=∏i=1k(X−αi)mi{\displaystyle f(X)=\prod _{i=1}^{k}(X-\alpha _{i})^{m_{i}}}is the factorization off, then the local ring ofRat the prime ideal⟨X−αi⟩{\displaystyle \langle X-\alpha _{i}\rangle }isK[X]/⟨(X−α)mi⟩.{\displaystyle K[X]/\langle (X-\alpha )^{m_{i}}\rangle .}This is a vector space overK, which has the multiplicitymi{\displaystyle m_{i}}of the root as a dimension.
This definition of intersection multiplicity, which is essentially due toJean-Pierre Serrein his bookLocal Algebra, works only for the set theoretic components (also calledisolated components) of the intersection, not for theembedded components. Theories have been developed for handling the embedded case (seeIntersection theoryfor details).
Letz0be a root of aholomorphic functionf, and letnbe the least positive integer such that thenthderivative offevaluated atz0differs from zero. Then thepower seriesoffaboutz0begins with thenthterm, andfis said to have a root of multiplicity (or “order”)n. Ifn= 1, the root is called a simple root.[4]
We can also define the multiplicity of thezeroesandpolesof ameromorphic function. If we have a meromorphic functionf=gh,{\textstyle f={\frac {g}{h}},}take theTaylor expansionsofgandhabout a pointz0, and find the first non-zero term in each (denote the order of the termsmandnrespectively) then ifm=n, then the point has non-zero value. Ifm>n,{\displaystyle m>n,}then the point is a zero of multiplicitym−n.{\displaystyle m-n.}Ifm<n{\displaystyle m<n}, then the point has a pole of multiplicityn−m.{\displaystyle n-m.}
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https://en.wikipedia.org/wiki/Multiplicity_(mathematics)
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Innumber theory,Ostrowski's theorem, due toAlexander Ostrowski(1916), states that every non-trivialabsolute valueon therational numbersQ{\displaystyle \mathbb {Q} }is equivalent to either the usual real absolute value or ap-adicabsolute value.[1]
Anabsolute valueon the rational numbers is a function|⋅|∗:Q→R{\displaystyle |\cdot |_{*}:\mathbb {Q} \to \mathbb {R} }satisfying for allx,y{\displaystyle x,y}that|x|∗≥0{\displaystyle |x|_{*}\geq 0},|xy|∗=|x|∗|y|∗{\displaystyle |xy|_{*}=|x|_{*}|y|_{*}},|x+y|∗≤|x|+|y|{\displaystyle |x+y|_{*}\leq |x|+|y|}, and|x|=0{\displaystyle |x|=0}only ifx=0{\displaystyle x=0}.
Two absolute values|⋅|{\displaystyle |\cdot |}and|⋅|∗{\displaystyle |\cdot |_{*}}on the rationals are defined to beequivalentif they induce the sametopology; this can be shown to be equivalent to the existence of a positivereal numberλ∈(0,∞){\displaystyle \lambda \in (0,\infty )}such that
(Note: In general, if|x|{\displaystyle |x|}is an absolute value,|x|λ{\displaystyle |x|^{\lambda }}is not necessarily an absolute value anymore; howeveriftwo absolute values are equivalent, then each is a positive power of the other.[2]) Thetrivial absolute valueon any fieldKis defined to be
Thereal absolute valueon therationalsQ{\displaystyle \mathbb {Q} }is the standardabsolute valueon the reals, defined to be
This is sometimes written with a subscript 1 instead ofinfinity.
For aprime numberp, thep-adic absolute valueonQ{\displaystyle \mathbb {Q} }is defined as follows: any non-zero rationalxcan be written uniquely asx=pnab{\displaystyle x=p^{n}{\tfrac {a}{b}}}, whereaandbarecoprime integersnot divisible byp, andnis an integer; so we define
Let|⋅|∗:Q→R{\displaystyle |\cdot |_{*}:\mathbb {Q} \to \mathbb {R} }be any absolute value on the rational numbers. Then either|⋅|∗=|⋅|0{\displaystyle |\cdot |_{*}=|\cdot |_{0}}, or|⋅|∗{\displaystyle |\cdot |_{*}}is equivalent to|⋅|{\displaystyle |\cdot |}, or|⋅|∗{\displaystyle |\cdot |_{*}}is equivalent to|⋅|p{\displaystyle |\cdot |_{p}}.[1]
The following proof follows the one of Theorem 10.1 in Schikhof (2007).
Let|⋅|∗{\displaystyle |\cdot |_{*}}be an absolute value on the rationals. We start the proof by showing that it is entirely determined by the values it takes onprime numbers.
From the fact that1×1=1{\displaystyle 1\times 1=1}and the multiplicativity property of the absolute value, we infer that|1|∗2=|1|∗{\displaystyle |1|_{*}^{2}=|1|_{*}}. In particular,|1|∗{\displaystyle |1|_{*}}has to be 0 or 1 and since1≠0{\displaystyle 1\neq 0}, one must have|1|∗=1{\displaystyle |1|_{*}=1}. A similar argument shows that|−1|∗=1{\displaystyle |-1|_{*}=1}.
For all positive integern, the multiplicativity property entails|−n|∗=|−1|∗×|n|∗=|n|∗{\displaystyle |-n|_{*}=|-1|_{*}\times |n|_{*}=|n|_{*}}. In other words, the absolute value of a negative integer coincides with that of its opposite.
Letnbe a positive integer. From the fact thatn−1×n=1{\displaystyle n^{-1}\times n=1}and the multiplicativity property, we conclude that|n−1|∗=|n|∗−1{\displaystyle |n^{-1}|_{*}=|n|_{*}^{-1}}.
Let nowrbe a positive rational. There exist twocoprimepositive integerspandqsuch thatr=pq−1{\displaystyle r=pq^{-1}}. The properties above show that|r|∗=|p|∗|q−1|∗=|p|∗|q|∗−1{\displaystyle |r|_{*}=|p|_{*}|q^{-1}|_{*}=|p|_{*}|q|_{*}^{-1}}. Altogether, the absolute value of a positive rational is entirely determined from that of its numerator and denominator.
Finally, letP{\displaystyle \mathbb {P} }be the set of prime numbers. For all positive integern, we can write
wherevp(n){\displaystyle v_{p}(n)}is thep-adic valuationofn. The multiplicativity property enables one to compute the absolute value ofnfrom that of the prime numbers using the following relationship
We continue the proof by separating two cases:
Suppose that there exists a positive integernsuch that|n|∗>1.{\displaystyle |n|_{*}>1.}Letkbe a non-negative integer andbbe a positive integer greater than1{\displaystyle 1}. We expressnk{\displaystyle n^{k}}inbaseb: there exist a positive integermand integers(ci)0≤i<m{\displaystyle (c_{i})_{0\leq i<m}}such that for alli,0≤ci<b{\displaystyle 0\leq c_{i}<b}andnk=∑i<mcibi{\displaystyle n^{k}=\sum _{i<m}c_{i}b^{i}}. In particular,nk≥bm−1{\displaystyle n^{k}\geq b^{m-1}}som≤1+klogbn{\displaystyle m\leq 1+k\log _{b}n}.
Each term|cibi|∗{\displaystyle |c_{i}b^{i}|_{*}}is smaller than(b−1)|b|∗i{\displaystyle (b-1)|b|_{*}^{i}}. (By the multiplicative property,|cibi|∗=|ci|∗|b|∗i{\displaystyle |c_{i}b^{i}|_{*}=|c_{i}|_{*}|b|_{*}^{i}}, then using the fact thatci{\displaystyle c_{i}}is a digit, writeci=1+1+⋯+1{\displaystyle c_{i}=1+1+\cdots +1}so by the triangle inequality,|ci|≤|1|+|1|+⋯+|1|=ci≤b−1{\displaystyle |c_{i}|\leq |1|+|1|+\cdots +|1|=c_{i}\leq b-1}.) Besides,|b|∗i{\displaystyle |b|_{*}^{i}}is smaller thanmax{1,|b|∗m−1}{\displaystyle \max\{1,|b|_{*}^{m-1}\}}. By the triangle inequality and the above bound onm, it follows:
Therefore, raising both sides to the power1/k{\displaystyle 1/k}, we obtain
Finally, taking the limit asktends to infinity shows that
Together with the condition|n|∗>1,{\displaystyle |n|_{*}>1,}the above argument leads to|b|∗>1{\displaystyle |b|_{*}>1}regardless of the choice ofb(otherwise|b|∗logbn≤1{\displaystyle |b|_{*}^{\log _{b}n}\leq 1}implies|n|∗≤1{\displaystyle |n|_{*}\leq 1}). As a result, all integers greater than one have an absolute value strictly greater than one. Thus generalizing the above, for any choice of integersnandbgreater than or equal to 2, we get
i.e.
By symmetry, this inequality is an equality. In particular, for alln≥2{\displaystyle n\geq 2},logn|n|∗=log2|2|∗=λ{\displaystyle \log _{n}|n|_{*}=\log _{2}|2|_{*}=\lambda }, i.e.|n|∗=nλ=|n|∞λ{\displaystyle |n|_{*}=n^{\lambda }=|n|_{\infty }^{\lambda }}. Because thetriangle inequalityimplies that for all positive integersnwe have|n|∗≤n{\displaystyle |n|_{*}\leq n}, in this case we obtain more precisely that0<λ≤1{\displaystyle 0<\lambda \leq 1}.
As per the above result on the determination of an absolute value by its values on the prime numbers, we easily see that|r|∗=|r|∞λ{\displaystyle |r|_{*}=|r|_{\infty }^{\lambda }}for all rationalr, thus demonstrating equivalence to the real absolute value.
Suppose that for all integern, one has|n|∗≤1{\displaystyle |n|_{*}\leq 1}. As our absolute value is non-trivial, there must exist a positive integernfor which|n|∗<1.{\displaystyle |n|_{*}<1.}Decomposing|n|∗{\displaystyle |n|_{*}}on the prime numbers shows that there existsp∈P{\displaystyle p\in \mathbb {P} }such that|p|∗<1{\displaystyle |p|_{*}<1}. We claim that in fact this is so for one prime number only.
Supposeby way of contradictionthatpandqare two distinct primes with absolute value strictly less than 1. Letkbe a positive integer such that|p|∗k{\displaystyle |p|_{*}^{k}}and|q|∗k{\displaystyle |q|_{*}^{k}}are smaller than1/2{\displaystyle 1/2}. ByBézout's identity, sincepk{\displaystyle p^{k}}andqk{\displaystyle q^{k}}arecoprime, there exist two integersaandbsuch thatapk+bqk=1.{\displaystyle ap^{k}+bq^{k}=1.}This yields a contradiction, as
This means that there exists a unique primepsuch that|p|∗<1{\displaystyle |p|_{*}<1}and that for all other primeq, one has|q|∗=1{\displaystyle |q|_{*}=1}(from the hypothesis of this second case). Letλ=−logp|p|∗{\displaystyle \lambda =-\log _{p}|p|_{*}}. From|p|∗<1{\displaystyle |p|_{*}<1}, we infer that0<λ<∞{\displaystyle 0<\lambda <\infty }. (And indeed in this case, all positiveλ{\displaystyle \lambda }give absolute values equivalent to the p-adic one.)
We finally verify that|p|pλ=p−λ=|p|∗{\displaystyle |p|_{p}^{\lambda }=p^{-\lambda }=|p|_{*}}and that for all other primeq,|q|pλ=1=|q|∗{\displaystyle |q|_{p}^{\lambda }=1=|q|_{*}}. As per the above result on the determination of an absolute value by its values on the prime numbers, we conclude that|r|∗=|r|pλ{\displaystyle |r|_{*}=|r|_{p}^{\lambda }}for all rationalr, implying that this absolute value is equivalent to thep-adic one.◼{\displaystyle \blacksquare }
Another theorem states that any field, complete with respect to anArchimedean absolute value, is (algebraically and topologically) isomorphic to either thereal numbersor thecomplex numbers. This is sometimes also referred to as Ostrowski's theorem.[3]
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In mathematics,Legendre's formulagives an expression for the exponent of the largest power of aprimepthat divides thefactorialn!. It is named afterAdrien-Marie Legendre. It is also sometimes known asde Polignac's formula, afterAlphonse de Polignac.
For any prime numberpand any positive integern, letνp(n){\displaystyle \nu _{p}(n)}be the exponent of the largest power ofpthat dividesn(that is, thep-adic valuationofn). Then
where⌊x⌋{\displaystyle \lfloor x\rfloor }is thefloor function. While the sum on the right side is an infinite sum, for any particular values ofnandpit has only finitely many nonzero terms: for everyilarge enough thatpi>n{\displaystyle p^{i}>n}, one has⌊npi⌋=0{\displaystyle \textstyle \left\lfloor {\frac {n}{p^{i}}}\right\rfloor =0}. This reduces the infinite sum above to
whereL=⌊logpn⌋{\displaystyle L=\lfloor \log _{p}n\rfloor }.
Forn= 6, one has6!=720=24⋅32⋅51{\displaystyle 6!=720=2^{4}\cdot 3^{2}\cdot 5^{1}}. The exponentsν2(6!)=4,ν3(6!)=2{\displaystyle \nu _{2}(6!)=4,\nu _{3}(6!)=2}andν5(6!)=1{\displaystyle \nu _{5}(6!)=1}can be computed by Legendre's formula as follows:
Sincen!{\displaystyle n!}is the product of the integers 1 throughn, we obtain at least one factor ofpinn!{\displaystyle n!}for each multiple ofpin{1,2,…,n}{\displaystyle \{1,2,\dots ,n\}}, of which there are⌊np⌋{\displaystyle \textstyle \left\lfloor {\frac {n}{p}}\right\rfloor }. Each multiple ofp2{\displaystyle p^{2}}contributes an additional factor ofp, each multiple ofp3{\displaystyle p^{3}}contributes yet another factor ofp, etc. Adding up the number of these factors gives the infinite sum forνp(n!){\displaystyle \nu _{p}(n!)}.
One may also reformulate Legendre's formula in terms of thebase-pexpansion ofn. Letsp(n){\displaystyle s_{p}(n)}denote the sum of the digits in the base-pexpansion ofn; then
For example, writingn= 6 inbinaryas 610= 1102, we have thats2(6)=1+1+0=2{\displaystyle s_{2}(6)=1+1+0=2}and so
Similarly, writing 6 internaryas 610= 203, we have thats3(6)=2+0=2{\displaystyle s_{3}(6)=2+0=2}and so
Writen=nℓpℓ+⋯+n1p+n0{\displaystyle n=n_{\ell }p^{\ell }+\cdots +n_{1}p+n_{0}}in basep. Then⌊npi⌋=nℓpℓ−i+⋯+ni+1p+ni{\displaystyle \textstyle \left\lfloor {\frac {n}{p^{i}}}\right\rfloor =n_{\ell }p^{\ell -i}+\cdots +n_{i+1}p+n_{i}}, and therefore
Legendre's formula can be used to proveKummer's theorem. As one special case, it can be used to prove that ifnis a positive integer then 4 divides(2nn){\displaystyle {\binom {2n}{n}}}if and only ifnis not a power of 2.
It follows from Legendre's formula that thep-adic exponential functionhas radius of convergencep−1/(p−1){\displaystyle p^{-1/(p-1)}}.
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In elementarynumber theory, thelifting-the-exponent lemma(LTE lemma) provides several formulas for computing thep-adic valuationνp{\displaystyle \nu _{p}}of special forms of integers. The lemma is named as such because it describes the steps necessary to "lift" the exponent ofp{\displaystyle p}in such expressions. It is related toHensel's lemma.
The exact origins of the LTE lemma are unclear; the result, with its present name and form, has only come into focus within the last 10 to 20 years.[1]However, several key ideas used in its proof were known toGaussand referenced in hisDisquisitiones Arithmeticae.[2]Despite chiefly featuring inmathematical olympiads, it is sometimes applied to research topics, such aselliptic curves.[3][4]
For any integersx{\displaystyle x}andy{\displaystyle y}, a positive integern{\displaystyle n}, and a prime numberp{\displaystyle p}such thatp∤x{\displaystyle p\nmid x}andp∤y{\displaystyle p\nmid y}, the following statements hold:
LTE has been generalized to complex values ofx,y{\displaystyle x,y}provided that the value ofxn−ynx−y{\displaystyle {\tfrac {x^{n}-y^{n}}{x-y}}}is integer.[5]
The base caseνp(xn−yn)=νp(x−y){\displaystyle \nu _{p}(x^{n}-y^{n})=\nu _{p}(x-y)}whenp∤n{\displaystyle p\nmid n}is proven first. Becausep∣x−y⟺x≡y(modp){\displaystyle p\mid x-y\iff x\equiv y{\pmod {p}}},
The fact thatxn−yn=(x−y)(xn−1+xn−2y+xn−3y2+⋯+yn−1){\displaystyle x^{n}-y^{n}=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^{2}+\dots +y^{n-1})}completes the proof. The conditionνp(xn+yn)=νp(x+y){\displaystyle \nu _{p}(x^{n}+y^{n})=\nu _{p}(x+y)}for oddn{\displaystyle n}is similar, where we observe that the proof above holds for integersx{\displaystyle x}andy{\displaystyle y}, and therefore we can substitute−y{\displaystyle -y}fory{\displaystyle y}above to obtain the desired result.
Via thebinomial expansion, the substitutiony=x+kp{\displaystyle y=x+kp}can be used in (1) to show thatνp(xp−yp)=νp(x−y)+1{\displaystyle \nu _{p}(x^{p}-y^{p})=\nu _{p}(x-y)+1}because (1) is a multiple ofp{\displaystyle p}but notp2{\displaystyle p^{2}}.[1]Likewise,νp(xp+yp)=νp(x+y)+1{\displaystyle \nu _{p}(x^{p}+y^{p})=\nu _{p}(x+y)+1}.
Then, ifn{\displaystyle n}is written aspab{\displaystyle p^{a}b}wherep∤b{\displaystyle p\nmid b}, the base case givesνp(xn−yn)=νp((xpa)b−(ypa)b)=νp(xpa−ypa){\displaystyle \nu _{p}(x^{n}-y^{n})=\nu _{p}((x^{p^{a}})^{b}-(y^{p^{a}})^{b})=\nu _{p}(x^{p^{a}}-y^{p^{a}})}.
By induction ona{\displaystyle a},
A similar argument can be applied forνp(xn+yn){\displaystyle \nu _{p}(x^{n}+y^{n})}.
The proof for the oddp{\displaystyle p}case cannot be directly applied whenp=2{\displaystyle p=2}because the binomial coefficient(p2)=p(p−1)2{\displaystyle {\binom {p}{2}}={\frac {p(p-1)}{2}}}is only an integral multiple ofp{\displaystyle p}whenp{\displaystyle p}is odd.
However, it can be shown thatν2(xn−yn)=ν2(x−y)+ν2(n){\displaystyle \nu _{2}(x^{n}-y^{n})=\nu _{2}(x-y)+\nu _{2}(n)}when4∣x−y{\displaystyle 4\mid x-y}by writingn=2ab{\displaystyle n=2^{a}b}wherea{\displaystyle a}andb{\displaystyle b}are integers withb{\displaystyle b}odd and noting that
because sincex≡y≡±1(mod4){\displaystyle x\equiv y\equiv \pm 1{\pmod {4}}}, each factor in the difference of squares step in the formx2k+y2k{\displaystyle x^{2^{k}}+y^{2^{k}}}is congruent to 2 modulo 4.
The stronger statementν2(xn−yn)=ν2(x−y)+ν2(x+y)+ν2(n)−1{\displaystyle \nu _{2}(x^{n}-y^{n})=\nu _{2}(x-y)+\nu _{2}(x+y)+\nu _{2}(n)-1}when2∣x−y{\displaystyle 2\mid x-y}is proven analogously.[1]
The LTE lemma can be used to solve 2020AIMEI #12:
Letn{\displaystyle n}be the least positive integer for which149n−2n{\displaystyle 149^{n}-2^{n}}is divisible by33⋅55⋅77.{\displaystyle 3^{3}\cdot 5^{5}\cdot 7^{7}.}Find the number of positive integer divisors ofn{\displaystyle n}.[6]
Solution.Note that149−2=147=3⋅72{\displaystyle 149-2=147=3\cdot 7^{2}}. Using the LTE lemma, since3∤149{\displaystyle 3\nmid 149}and3∤2{\displaystyle 3\nmid 2}, but3∣147{\displaystyle 3\mid 147},ν3(149n−2n)=ν3(147)+ν3(n)=ν3(n)+1{\displaystyle \nu _{3}(149^{n}-2^{n})=\nu _{3}(147)+\nu _{3}(n)=\nu _{3}(n)+1}. Thus,33∣149n−2n⟺32∣n{\displaystyle 3^{3}\mid 149^{n}-2^{n}\iff 3^{2}\mid n}. Similarly,7∤149,2{\displaystyle 7\nmid 149,2}but7∣147{\displaystyle 7\mid 147}, soν7(149n−2n)=ν7(147)+ν7(n)=ν7(n)+2{\displaystyle \nu _{7}(149^{n}-2^{n})=\nu _{7}(147)+\nu _{7}(n)=\nu _{7}(n)+2}and77∣149n−2n⟺75∣n{\displaystyle 7^{7}\mid 149^{n}-2^{n}\iff 7^{5}\mid n}.
Since5∤147{\displaystyle 5\nmid 147}, the factors of 5 are addressed by noticing that since the residues of149n{\displaystyle 149^{n}}modulo 5 follow the cycle4,1,4,1{\displaystyle 4,1,4,1}and those of2n{\displaystyle 2^{n}}follow the cycle2,4,3,1{\displaystyle 2,4,3,1}, the residues of149n−2n{\displaystyle 149^{n}-2^{n}}modulo 5 cycle through the sequence2,2,1,0{\displaystyle 2,2,1,0}. Thus,5∣149n−2n{\displaystyle 5\mid 149^{n}-2^{n}}iffn=4k{\displaystyle n=4k}for some positive integerk{\displaystyle k}. The LTE lemma can now be applied again:ν5(1494k−24k)=ν5((1494)k−(24)k)=ν5(1494−24)+ν5(k){\displaystyle \nu _{5}(149^{4k}-2^{4k})=\nu _{5}((149^{4})^{k}-(2^{4})^{k})=\nu _{5}(149^{4}-2^{4})+\nu _{5}(k)}. Since1494−24≡(−1)4−24≡−15(mod25){\displaystyle 149^{4}-2^{4}\equiv (-1)^{4}-2^{4}\equiv -15{\pmod {25}}},ν5(1494−24)=1{\displaystyle \nu _{5}(149^{4}-2^{4})=1}. Hence55∣149n−2n⟺54∣k⟺4⋅54∣n{\displaystyle 5^{5}\mid 149^{n}-2^{n}\iff 5^{4}\mid k\iff 4\cdot 5^{4}\mid n}.
Combining these three results, it is found thatn=22⋅32⋅54⋅75{\displaystyle n=2^{2}\cdot 3^{2}\cdot 5^{4}\cdot 7^{5}}, which has(2+1)(2+1)(4+1)(5+1)=270{\displaystyle (2+1)(2+1)(4+1)(5+1)=270}positive divisors.
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Innumber theoryandcombinatorics, therankof aninteger partitionis a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced byFreeman Dysonin a paper published in the journalEureka.[1]It was presented in the context of a study of certaincongruenceproperties of thepartition functiondiscovered by the Indian mathematical geniusSrinivasa Ramanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of theDurfee squareof the partition.
By apartitionof a positive integernwe mean a finite multiset λ = { λk, λk− 1, . . . , λ1} of positive integers satisfying the following two conditions:
Ifλk, . . . ,λ2,λ1are distinct, that is, if
then the partitionλis called astrict partitionofn.
The integersλk, λk− 1, ...,λ1are thepartsof the partition. The number of parts in the partitionλiskand the largest part in the partition isλk. The rank of the partitionλ(whether ordinary or strict) is defined asλk−k.[1]
The ranks of the partitions ofntake the following values and no others:[1]
The following table gives the ranks of the various partitions of the number 5.
Ranks of the partitions of the integer 5
The following notations are used to specify how many partitions have a given rank. Letn,qbe a positive integers andmbe any integer.
For example,
Letn,qbe a positive integers andmbe any integer.[1]
Srinivasa Ramanujan in a paper published in 1919 proved the followingcongruencesinvolving the partition functionp(n):[2]
In commenting on this result, Dyson noted that " . . . although we can prove that the partitions of 5n+ 4 can be divided into five equally numerous subclasses, it is unsatisfactory to receive from the proofs no concrete idea of how the division is to be made. We require a proof which will not appeal to generating functions, . . . ".[1]Dyson introduced the idea of rank of a partition to accomplish the task he set for himself. Using this new idea, he made the following conjectures:
These conjectures were proved by Atkin and Swinnerton-Dyer in 1954.[3]
The following tables show how the partitions of the integers 4 (5 ×n+ 4 withn= 0) and 9 (5 ×n+ 4 withn= 1 ) get divided into five equally numerous subclasses.
Partitions of the integer 4
Partitions of the integer 9
In combinatorics, the phraserank of a partitionis sometimes used to describe a different concept: the rank of a partition λ is the largest integerisuch that λ has at leastiparts each of which is no smaller thani.[7]Equivalently, this is the length of the main diagonal in theYoung diagramorFerrers diagramfor λ, or the side-length of theDurfee squareof λ.
The table of ranks (under this alternate definition) of partitions of 5 is given below.
Ranks of the partitions of the integer 5
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Innumber theory, thecrankof aninteger partitionis a certain number associated with the partition. It was first introduced without a definition byFreeman Dyson, who hypothesised its existence in a 1944 paper.[1]Dyson gave a list of properties this yet-to-be-defined quantity should have. In 1988,George E. AndrewsandFrank Garvandiscovered a definition for the crank satisfying the properties hypothesized for it by Dyson.[2]
Letnbe a non-negative integer and letp(n) denote the number of partitions ofn(p(0) is defined to be 1).Srinivasa Ramanujanin a paper[3]published in 1918 stated and proved the following congruences for thepartition functionp(n), since known asRamanujan congruences.
These congruences imply that partitions of numbers of the form 5n+ 4 (respectively, of the forms 7n+ 5 and 11n+ 6 ) can be divided into 5 (respectively, 7 and 11) subclasses of equal size. The then known proofs of these congruences were based on the ideas of generating functions and they did not specify a method for the division of the partitions into subclasses of equal size.
In his Eureka paper Dyson proposed the concept of therank of a partition. The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = { 4, 2, 1, 1, 1 } of 9 is 4 − 5 = −1. Denoting byN(m,q,n), the number of partitions ofnwhose ranks are congruent tommoduloq, Dyson consideredN(m, 5, 5n+ 4) andN(m, 7, 7n+ 5) for various values ofnandm. Based on empirical evidences Dyson formulated the following conjectures known asrank conjectures.
For all non-negative integersnwe have:
Assuming that these conjectures are true, they provided a way of splitting up all partitions of numbers of the form 5n+ 4 into five classes of equal size: Put in one class all those partitions whose ranks are congruent to each other modulo 5. The same idea can be applied to divide the partitions of integers of the form 7n+ 5 into seven equally numerous classes. But the idea fails to divide partitions of integers of the form 11n+ 6 into 11 classes of the same size, as the following table shows.
Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote,
I hold in fact :
Whether these guesses are warranted by evidence, I leave it to the reader to decide. Whatever the final verdict of posterity may be, I believe the "crank" is unique among arithmetical functions in having been named before it was discovered. May it be preserved from the ignominious fate of the planetVulcan.
In a paper[2]published in 1988 George E. Andrews and F. G. Garvan defined the crank of a partition as follows:
The cranks of the partitions of the integers 4, 5, 6 are computed in the following tables.
For all integersn≥ 0 and all integersm, the number of partitions ofnwith crank equal tomis denoted byM(m,n) except forn= 1 whereM(−1,1) = −M(0,1) =M(1,1) = 1 as given by the following generating function. The number of partitions ofnwith crank equal tommoduloqis denoted byM(m,q,n).
The generating function forM(m,n) is given below:
Andrews and Garvan proved the following result[2]which shows that the crank as defined above does meet the conditions given by Dyson.
The concepts of rank and crank can both be used to classify partitions of certain integers into subclasses of equal size. However the two concepts produce different subclasses of partitions. This is illustrated in the following two tables.
Recent work byBruce C. Berndtand his coauthors argued that Ramanujan knew about the crank, although not in the form that Andrews and Garvan have defined. In a systematic study of the Lost Notebook of Ramanujan, Berndt and his coauthors have given substantial evidence that Ramanujan knew about the dissections of the crank generating function.[4][5]
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Indiscrete mathematics,dominance order(synonyms:dominance ordering,majorization order,natural ordering) is apartial orderon the set ofpartitionsof a positive integernthat plays an important role inalgebraic combinatoricsandrepresentation theory, especially in the context ofsymmetric functionsandrepresentation theory of the symmetric group.
Ifp= (p1,p2,...) andq= (q1,q2,...) are partitions ofn, with the parts arranged in the weakly decreasing order, thenpprecedesqin the dominance order if for anyk≥ 1, the sum of theklargest parts ofpis less than or equal to the sum of theklargest parts ofq:
In this definition, partitions are extended by appending zero parts at the end as necessary.
Partitions ofnform alatticeunder the dominance ordering, denotedLn, and the operation of conjugation is anantiautomorphismof this lattice. To explicitly describe the lattice operations, for each partitionpconsider theassociated (n+ 1)-tuple:
The partitionpcan be recovered from its associated (n+1)-tuple by applying the step 1difference,pi=p^i−p^i−1.{\displaystyle p_{i}={\hat {p}}_{i}-{\hat {p}}_{i-1}.}Moreover, the (n+1)-tuples associated to partitions ofnare characterized among all integer sequences of lengthn+ 1 by the following three properties:
By the definition of the dominance ordering, partitionpprecedes partitionqif and only if the associated (n+ 1)-tuple ofpis term-by-term less than or equal to the associated (n+ 1)-tuple ofq. Ifp,q,rare partitions thenr⊴p,r⊴q{\displaystyle r\trianglelefteq p,r\trianglelefteq q}if and only ifr^≤p^,r^≤q^.{\displaystyle {\hat {r}}\leq {\hat {p}},{\hat {r}}\leq {\hat {q}}.}The componentwise minimum of two nondecreasing concave integer sequences is also nondecreasing and concave. Therefore, for any two partitions ofn,pandq, theirmeetis the partition ofnwhose associated (n+ 1)-tuple has componentsmin(p^i,q^i).{\displaystyle \operatorname {min} ({\hat {p}}_{i},{\hat {q}}_{i}).}The natural idea to use a similar formula for thejoinfails, because the componentwise maximum of two concave sequences need not be concave. For example, forn= 6, the partitions [3,1,1,1] and [2,2,2] have associated sequences (0,3,4,5,6,6,6) and (0,2,4,6,6,6,6), whose componentwise maximum (0,3,4,6,6,6,6) does not correspond to any partition. To show that any two partitions ofnhave a join, one uses the conjugation antiautomorphism: the join ofpandqis the conjugate partition of the meet ofp′ andq′:
For the two partitionspandqin the preceding example, their conjugate partitions are [4,1,1] and [3,3] with meet [3,2,1], which is self-conjugate; therefore, the join ofpandqis [3,2,1].
Thomas Brylawskihas determined many invariants of the latticeLn, such as the minimal height and the maximal covering number, and classified theintervalsof small length. WhileLnis notdistributiveforn≥ 7, it shares some properties with distributive lattices: for example, itsMöbius functiontakes on only values 0, 1, −1.
Partitions ofncan be graphically represented byYoung diagramsonnboxes.
StandardYoung tableauxare certain ways to fill Young diagrams with numbers, and a partial order on them (sometimes called thedominance order on Young tableaux) can be defined in terms of the dominance order on the Young diagrams. For a Young tableauTto dominate another Young tableauS, the shape ofTmust dominate that ofSas a partition, and moreover the same must hold wheneverTandSare first truncated to their sub-tableaux containing entries up to a given valuek, for each choice ofk.
Similarly, there is a dominance order on the set of standard Young bitableaux, which plays a role in the theory ofstandard monomials.
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Inmathematics, apartition of a setis a grouping of its elements intonon-emptysubsets, in such a way that every element is included in exactly one subset.
Everyequivalence relationon asetdefines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called asetoid, typically intype theoryandproof theory.
A partition of a setXis a set of non-empty subsets ofXsuch that every elementxinXis in exactly one of these subsets[2](i.e., the subsets are nonempty mutuallydisjoint sets).
Equivalently, afamily of setsPis a partition ofXif and only if all of the following conditions hold:[3]
The sets inP{\displaystyle P}are called theblocks,parts, orcells, of the partition.[4]Ifa∈X{\displaystyle a\in X}then we represent the cell containinga{\displaystyle a}by[a]{\displaystyle [a]}. That is to say,[a]{\displaystyle [a]}is notation for the cell inP{\displaystyle P}which containsa{\displaystyle a}.
Every partitionP{\displaystyle P}may be identified with an equivalence relation onX{\displaystyle X}, namely the relation∼P{\displaystyle \sim _{\!P}}such that for anya,b∈X{\displaystyle a,b\in X}we havea∼Pb{\displaystyle a\sim _{\!P}b}if and only ifa∈[b]{\displaystyle a\in [b]}(equivalently, if and only ifb∈[a]{\displaystyle b\in [a]}). The notation∼P{\displaystyle \sim _{\!P}}evokes the idea that the equivalence relation may be constructed from the partition. Conversely every equivalence relation may be identified with a partition. This is why it is sometimes said informally that "an equivalence relation is the same as a partition". IfPis the partition identified with a given equivalence relation∼{\displaystyle \sim }, then some authors writeP=X/∼{\displaystyle P=X/{\sim }}. This notation is suggestive of the idea that the partition is the setXdivided in to cells. The notation also evokes the idea that, from the equivalence relation one may construct the partition.
TherankofP{\displaystyle P}is|X|−|P|{\displaystyle |X|-|P|}, ifX{\displaystyle X}isfinite.
For anyequivalence relationon a setX, the set of itsequivalence classesis a partition ofX. Conversely, from any partitionPofX, we can define an equivalence relation onXby settingx~yprecisely whenxandyare in the same part inP. Thus the notions of equivalence relation and partition are essentially equivalent.[5]
Theaxiom of choiceguarantees for any partition of a setXthe existence of a subset ofXcontaining exactly one element from each part of the partition. This implies that given an equivalence relation on a set one can select acanonical representative elementfrom every equivalence class.
A partitionαof a setXis arefinementof a partitionρofX—and we say thatαisfinerthanρand thatρiscoarserthanα—if every element ofαis a subset of some element ofρ. Informally, this means thatαis a further fragmentation ofρ. In that case, it is written thatα≤ρ.
This "finer-than" relation on the set of partitions ofXis apartial order(so the notation "≤" is appropriate). Each set of elements has aleast upper bound(their "join") and agreatest lower bound(their "meet"), so that it forms alattice, and more specifically (for partitions of a finite set) it is ageometricandsupersolvablelattice.[6][7]Thepartition latticeof a 4-element set has 15 elements and is depicted in theHasse diagramon the left.
The meet and join of partitions α and ρ are defined as follows. Themeetα∧ρ{\displaystyle \alpha \wedge \rho }is the partition whose blocks are the intersections of a block ofαand a block ofρ, except for the empty set. In other words, a block ofα∧ρ{\displaystyle \alpha \wedge \rho }is the intersection of a block ofαand a block ofρthat are not disjoint from each other. To define thejoinα∨ρ{\displaystyle \alpha \vee \rho }, form a relation on the blocksAofαand the blocksBofρbyA~BifAandBare not disjoint. Thenα∨ρ{\displaystyle \alpha \vee \rho }is the partition in which each blockCis the union of a family of blocks connected by this relation.
Based on the equivalence between geometric lattices andmatroids, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of theatomsof the lattice, namely, the partitions withn−2{\displaystyle n-2}singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of acomplete graph. Thematroid closureof a set of atomic partitions is the finest common coarsening of them all; in graph-theoretic terms, it is the partition of theverticesof the complete graph into theconnected componentsof the subgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the lattice of flats of thegraphic matroidof the complete graph.
Another example illustrates refinement of partitions from the perspective of equivalence relations. IfDis the set of cards in a standard 52-card deck, thesame-color-asrelation onD– which can be denoted ~C– has two equivalence classes: the sets {red cards} and {black cards}. The 2-part partition corresponding to ~Chas a refinement that yields thesame-suit-asrelation ~S, which has the four equivalence classes {spades}, {diamonds}, {hearts}, and {clubs}.
A partition of the setN= {1, 2, ...,n} with corresponding equivalence relation ~ isnoncrossingif it has the following property: If four elementsa,b,canddofNhavinga<b<c<dsatisfya~candb~d, thena~b~c~d. The name comes from the following equivalent definition: Imagine the elements 1, 2, ...,nofNdrawn as thenvertices of a regularn-gon (in counterclockwise order). A partition can then be visualized by drawing each block as a polygon (whose vertices are the elements of the block). The partition is then noncrossing if and only if these polygons do not intersect.
The lattice of noncrossing partitions of a finite set forms a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.
The noncrossing partition lattice has taken on importance because of its role infree probabilitytheory.
The total number of partitions of ann-element set is theBell numberBn. The first several Bell numbers areB0= 1,B1= 1,B2= 2,B3= 5,B4= 15,B5= 52, andB6= 203 (sequenceA000110in theOEIS). Bell numbers satisfy therecursion
and have theexponential generating function
The Bell numbers may also be computed using theBell trianglein which the first value in each row is copied from the end of the previous row, and subsequent values are computed by adding two numbers, the number to the left and the number to the above left of the position. The Bell numbers are repeated along both sides of this triangle. The numbers within the triangle count partitions in which a given element is the largestsingleton.
The number of partitions of ann-element set into exactlyk(non-empty) parts is theStirling number of the second kindS(n,k).
The number of noncrossing partitions of ann-element set is theCatalan number
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Incombinatorics,stars and bars(also called "sticks and stones",[1]"balls and bars",[2]and "dots and dividers"[3]) is a graphical aid for deriving certaincombinatorialtheorems. It can be used to solve a variety ofcounting problems, such as how many ways there are to putnindistinguishable balls intokdistinguishable bins.[4]The solution to this particular problem is given by the binomial coefficient(n+k−1k−1){\displaystyle {\tbinom {n+k-1}{k-1}}}, which is the number of subsets of sizek− 1that can be formed from a set of sizen+k− 1.
If, for example, there are two balls and three bins, then the number of ways of placing the balls is(2+3−13−1)=(42)=6{\displaystyle {\tbinom {2+3-1}{3-1}}={\tbinom {4}{2}}=6}. The table shows the six possible ways of distributing the two balls, the strings of stars and bars that represent them (with stars indicating balls and bars separating bins from one another), and the subsets that correspond to the strings. As two bars are needed to separate three bins and there are two balls, each string contains two bars and two stars. Each subset indicates which of the four symbols in the corresponding string is a bar.
The stars and bars method is often introduced specifically to prove the following two theorems of elementary combinatorics concerning the number of solutions to an equation.
For any pair ofpositive integersnandk, the number ofk-tuplesofpositiveintegers whose sum isnis equal to the number of(k− 1)-element subsets of a set withn− 1elements.
For example, ifn= 10andk= 4, the theorem gives the number of solutions tox1+x2+x3+x4= 10(withx1,x2,x3,x4> 0) as thebinomial coefficient
where(n−1k−1){\displaystyle {\tbinom {n-1}{k-1}}}is the number ofcombinationsofn− 1elements takenk− 1at a time.
This corresponds tocompositionsof an integer.
For any pair of positive integersnandk, the number ofk-tuplesofnon-negativeintegers whose sum isnis equal to the number ofmultisetsof sizek− 1taken from a set of sizen+ 1, or equivalently, the number of multisets of sizentaken from a set of sizek, and is given by
For example, ifn= 10andk= 4, the theorem gives the number of solutions tox1+x2+x3+x4= 10(withx1,x2,x3,x4≥0{\displaystyle \geq 0}) as
where themultiset coefficient((kn)){\displaystyle \left(\!\!{\binom {k}{n}}\!\!\right)}is the number of multisets of sizen, with elements taken from a set of sizek.
This corresponds toweak compositionsof an integer. Withkfixed, the numbers forn= 0, 1, 2, 3, ...are those in the(k− 1)st diagonal ofPascal's triangle. For example, whenk= 3thenth number is the(n+ 1)sttriangular number, which falls on the second diagonal, 1, 3, 6, 10, ....
The problem of enumeratingk-tuples whose sum isnis equivalent to the problem of counting configurations of the following kind: let there benobjects to be placed intokbins, so that all bins contain at least one object. The bins are distinguished (say they are numbered 1 tok) but thenobjects are not (so configurations are only distinguished by thenumber of objectspresent in each bin). A configuration is thus represented by ak-tuple of positive integers.
Thenobjects are now represented as a row ofnstars; adjacent bins are separated by bars. The configuration will be specified by indicating the boundary between the first and second bin, the boundary between the second and third bin, and so on. Hencek− 1bars need to be placed between stars. Because no bin is allowed to be empty, there is at most one bar between any pair of stars. There aren− 1gaps between stars and hencen− 1positions in which a bar may be placed. A configuration is obtained by choosingk− 1of these gaps to contain a bar; therefore there are(n−1k−1){\displaystyle {\tbinom {n-1}{k-1}}}configurations.
Withn= 7andk= 3, start by placing seven stars in a line:
Now indicate the boundaries between the bins:
In general two of the six possible bar positions must be chosen. Therefore there are(62)=15{\displaystyle {\tbinom {6}{2}}=15}such configurations.
In this case, the weakened restriction of non-negativity instead of positivity means that we can place multiple bars between stars and that one or more bars also be placed before the first star and after the last star. In terms of configurations involving objects and bins, bins are now allowed to be empty.
Rather than a(k− 1)-set of bar positions taken from a set of sizen− 1as in the proof of Theorem one, we now have a(k− 1)-multiset of bar positions taken from a set of sizen+ 1(since bar positions may repeat and since the ends are now allowed bar positions). An alternative interpretation in terms of multisets is the following: there is a set ofkbin labels from which a multiset of sizenis to be chosen, the multiplicity of a bin label in this multiset indicating the number of objects placed in that bin. The equality((n+1k−1))=((kn)){\displaystyle \left(\!\!{n+1 \choose k-1}\!\!\right)=\left(\!\!{k \choose n}\!\!\right)}can also be understood as an equivalence of different counting problems: the number ofk-tuples of non-negative integers whose sum isnequals the number of(n+ 1)-tuples of non-negative integers whose sum isk− 1, which follows by interchanging the roles of bars and stars in the diagrams representing configurations.
To see the expression(n+k−1k−1){\displaystyle {\tbinom {n+k-1}{k-1}}}directly, observe that any arrangement of stars and bars consists of a total ofn+k− 1symbols,nof which are stars andk− 1of which are bars. Thus, we may lay outn+k− 1slots and choosek− 1of these to contain bars (or, equivalently, choosenof the slots to contain stars).
Whenn= 7andk= 5, the tuple (4, 0, 1, 2, 0) may be represented by the following diagram:
If possible bar positions are labeled 1, 2, 3, 4, 5, 6, 7, 8 with labeli≤7corresponding to a bar preceding theith star and following any previous star and 8 to a bar following the last star, then this configuration corresponds to the(k− 1)-multiset{5,5,6,8}, as described in the proof of Theorem two. If bins are labeled 1, 2, 3, 4, 5, then it also corresponds to then-multiset{1,1,1,1,3,4,4}, also as described in the proof of Theorem two.
Theorem one can be restated in terms of Theorem two, because the requirement that each variable be positive can be imposed by shifting each variable by −1, and then requiring only that each variable be non-negative.
For example:
withx1,x2,x3,x4>0{\displaystyle x_{1},x_{2},x_{3},x_{4}>0}
is equivalent to:
withx1′,x2′,x3′,x4′≥0,{\displaystyle x'_{1},x'_{2},x'_{3},x'_{4}\geq 0,}
wherexi′=xi−1{\displaystyle x'_{i}=x_{i}-1}for eachi∈{1,2,3,4}{\displaystyle i\in \{1,2,3,4\}}.
If one wishes to count the number of ways to distribute seven indistinguishable one dollar coins among Amber, Ben, and Curtis so that each of them receives at least one dollar, one may observe that distributions are essentially equivalent to tuples of three positive integers whose sum is 7. (Here the first entry in the tuple is the number of coins given to Amber, and so on.) Thus Theorem 1 applies, withn= 7andk= 3, and there are(7−13−1)=15{\displaystyle {\tbinom {7-1}{3-1}}=15}ways to distribute the coins.
Ifn= 5,k= 4, and thekbin labels area,b,c,d, then ★|★★★||★ could represent either the 4-tuple(1, 3, 0, 1), or the multiset of bar positions{2, 5, 5}, or the multiset of bin labels{a,b,b,b,d}. The solution of this problem should use Theorem 2 withn= 5stars andk– 1 = 3bars to give(5+4−14−1)=(83)=56{\displaystyle {\tbinom {5+4-1}{4-1}}={\tbinom {8}{3}}=56}configurations.
In the proof of Theorem two there can be more bars than stars, which cannot happen in the proof of Theorem one.
So, for example, 10 balls into 7 bins gives(166){\displaystyle {\tbinom {16}{6}}}configurations, while 7 balls into 10 bins gives(169){\displaystyle {\tbinom {16}{9}}}configurations, and 6 balls into 11 bins gives(1610)=(166){\displaystyle {\tbinom {16}{10}}={\tbinom {16}{6}}}configurations.
The graphical method was used byPaul EhrenfestandHeike Kamerlingh Onnes—with symbolε(quantum energy element) in place of a star and the symbol0in place of a bar—as a simple derivation ofMax Planck's expression for the number of "complexions" for a system of "resonators" of a single frequency.[5][6]
By complexions (microstates) Planck meant distributions ofPenergy elementsεoverNresonators.[7][8]The numberRof complexions is
The graphical representation of each possible distribution would containPcopies of the symbolεandN– 1copies of the symbol0. In their demonstration, Ehrenfest and Kamerlingh Onnes tookN= 4andP= 7(i.e.,R= 120combinations). They chose the 4-tuple (4, 2, 0, 1) as the illustrative example for this symbolic representation:εεεε0εε00ε.
The enumerations of Theorems one and two can also be found usinggenerating functionsinvolving simple rational expressions. The two cases are very similar; we will look at the case whenxi≥0{\displaystyle x_{i}\geq 0}, that is, Theorem two first. There is only one configuration for a single bin and any given number of objects (because the objects are not distinguished). This is represented by the generating function
The series is a geometric series, and the last equality holds analytically for|x| < 1, but is better understood in this context as a manipulation offormal power series. The exponent ofxindicates how many objects are placed in the bin.
Each additional bin is represented by another factor of11−x{\displaystyle {\frac {1}{1-x}}}; the generating function forkbins is
where the multiplication is theCauchy productof formal power series.
To find the number of configurations withnobjects, we want the coefficient ofxn{\displaystyle x^{n}}(denoted by prefixing the expression for the generating function with[xn]{\displaystyle [x^{n}]}), that is,
This coefficient can be found usingbinomial seriesand agrees with the result of Theorem two, namely(n+k−1k−1){\displaystyle {\tbinom {n+k-1}{k-1}}}.
This Cauchy product expression is justified via stars and bars: the coefficient ofxn{\displaystyle x^{n}}in the expansion of the product
is the number of ways of obtaining thenth power ofxby multiplying one power ofxfrom each of thekfactors. So the stars representxs and a bar separates thexs coming from one factor from those coming from the next factor.
For the case whenxi>0{\displaystyle x_{i}>0}, that is, Theorem one, no configuration has an empty bin, and so the generating function for a single bin is
The Cauchy product is thereforexk(1−x)k{\displaystyle {\frac {x^{k}}{(1-x)^{k}}}}, and the coefficient ofxn{\displaystyle x^{n}}is found using binomial series to be(n−1k−1){\displaystyle {\tbinom {n-1}{k-1}}}.
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Inmathematicsand especially incombinatorics, aplane partitionis a two-dimensional array of nonnegative integersπi,j{\displaystyle \pi _{i,j}}(withpositiveintegerindicesiandj) that is nonincreasing in both indices. This means that
Moreover, only finitely many of theπi,j{\displaystyle \pi _{i,j}}may be nonzero. Plane partitions are a generalization ofpartitions of an integer.
A plane partition may be represented visually by the placement of a stack ofπi,j{\displaystyle \pi _{i,j}}unit cubesabove the point (i,j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form
Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subsetP{\displaystyle {\mathcal {P}}}of positive integer lattice points (i,j,k) inN3{\displaystyle \mathbb {N} ^{3}}, such that if (r,s,t) lies inP{\displaystyle {\mathcal {P}}}and if(i,j,k){\displaystyle (i,j,k)}satisfies1≤i≤r{\displaystyle 1\leq i\leq r},1≤j≤s{\displaystyle 1\leq j\leq s}, and1≤k≤t{\displaystyle 1\leq k\leq t}, then (i,j,k) also lies inP{\displaystyle {\mathcal {P}}}.
Thesumof a plane partition is
The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns theenumerationof plane partitions in various classes. The number of plane partitions with sumnis denoted by PL(n). For example, there are six plane partitions with sum 3
so PL(3) = 6.
Plane partitions may be classified by how symmetric they are. Many symmetric classes of plane partitions are enumerated by simple product formulas.
Thegenerating functionfor PL(n) is[1]
It is sometimes referred to as theMacMahon function, as it was discovered byPercy A. MacMahon.
This formula may be viewed as the 2-dimensional analogue ofEuler'sproduct formulafor the number ofinteger partitionsofn. There is no analogous formula known for partitions in higher dimensions (i.e., forsolid partitions).[2]The asymptotics for plane partitions were first calculated byE. M. Wright.[3]One obtains, for largen{\displaystyle n}, that[a]
Evaluating numerically yields
Around 1896, MacMahon set up the generating function of plane partitions that are subsets of ther×s×t{\displaystyle r\times s\times t}boxB(r,s,t)={(i,j,k)|1≤i≤r,1≤j≤s,1≤k≤t}{\displaystyle {\mathcal {B}}(r,s,t)=\{(i,j,k)|1\leq i\leq r,1\leq j\leq s,1\leq k\leq t\}}in his first paper on plane partitions.[5]The formula is given by∑π∈B(r,s,t)q|π|=∏i=1r∏j=1s1−qi+j+t−11−qi+j−1{\displaystyle \sum _{\pi \in {\mathcal {B}}(r,s,t)}q^{|\pi |}=\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {1-q^{i+j+t-1}}{1-q^{i+j-1}}}}
A proof of this formula can be found in the bookCombinatory Analysiswritten by MacMahon.[6]MacMahon also mentions the generating functions of plane partitions.[7]The formula for the generating function can be written in an alternative way, which is given by∑π∈B(r,s,t)q|π|=∏i=1r∏j=1s∏k=1t1−qi+j+k−11−qi+j+k−2{\displaystyle \sum _{\pi \in {\mathcal {B}}(r,s,t)}q^{|\pi |}=\prod _{i=1}^{r}\prod _{j=1}^{s}\prod _{k=1}^{t}{\frac {1-q^{i+j+k-1}}{1-q^{i+j+k-2}}}}
Multiplying each component by1−q1−q{\displaystyle \textstyle {\frac {1-q}{1-q}}}, and settingq= 1 in the formulas above yields that the total numberN1(r,s,t){\displaystyle N_{1}(r,s,t)}of plane partitions that fit in ther×s×t{\displaystyle r\times s\times t}boxB(r,s,t){\displaystyle {\mathcal {B}}(r,s,t)}is equal to the following product formula:[8]N1(r,s,t)=∏(i,j,k)∈B(r,s,t)i+j+k−1i+j+k−2=∏i=1r∏j=1si+j+t−1i+j−1.{\displaystyle N_{1}(r,s,t)=\prod _{(i,j,k)\in {\mathcal {B}}(r,s,t)}{\frac {i+j+k-1}{i+j+k-2}}=\prod _{i=1}^{r}\prod _{j=1}^{s}{\frac {i+j+t-1}{i+j-1}}.}The planar case (whent= 1) yields thebinomial coefficients:
The general solution is
Theisometric projectionof the unit cubes representing a plane partition in a box gives a bijection between these plane partitions and rhombus tilings of a hexagon with the same edge lengths as the box.[9]
Special plane partitions include symmetric, cyclic and self-complementary plane partitions, and combinations of these properties.
In the subsequent sections, the enumeration of special sub-classes of plane partitions inside a box are considered.
These articles use the notationNi(r,s,t){\displaystyle N_{i}(r,s,t)}for the number of such plane partitions, wherer,s, andtare the dimensions of the box under consideration, andiis the index for the case being considered.
S2{\displaystyle {\mathcal {S}}_{2}}is the group ofpermutationsacting on the first two coordinates of a point. This group contains the identity, which sends (i,j,k) to itself, and the transposition (i,j,k) → (j,i,k). The number of elements in an orbitη{\displaystyle \eta }is denoted by|η|{\displaystyle |\eta |}.B/S2{\displaystyle {\mathcal {B}}/{\mathcal {S}}_{2}}denotes the set of orbits of elements ofB{\displaystyle {\mathcal {B}}}under the action ofS2{\displaystyle {\mathcal {S}}_{2}}. The height of an element (i,j,k) is defined byht(i,j,k)=i+j+k−2.{\displaystyle ht(i,j,k)=i+j+k-2.}The height increases by one for each step away from the back right corner. For example, the corner position (1, 1, 1) has height 1 andht(2, 1, 1) = 2. The height of an orbit is defined to be the height of any element in the orbit. This notation of the height differs from the notation ofIan G. Macdonald.[10]
There is a natural action of the permutation groupS3{\displaystyle {\mathcal {S}}_{3}}on a Ferrers diagram of a plane partition—this corresponds to simultaneously permuting the three coordinates of all nodes. This generalizes the conjugation operation forinteger partitions. The action ofS3{\displaystyle {\mathcal {S}}_{3}}can generate new plane partitions starting from a given plane partition. Below there are shown six plane partitions of 4 that are generated by theS3{\displaystyle {\mathcal {S}}_{3}}action. Only the exchange of the first two coordinates is manifest in the representation given below.
C3{\displaystyle {\mathcal {C}}_{3}}is called the group of cyclic permutations and consists of
A plane partitionπ{\displaystyle \pi }is called symmetric ifπi,j=πj,ifor alli,j. In other words, a plane partition is symmetric if(i,j,k)∈B(r,s,t){\displaystyle (i,j,k)\in {\mathcal {B}}(r,s,t)}if and only if(j,i,k)∈B(r,s,t){\displaystyle (j,i,k)\in {\mathcal {B}}(r,s,t)}. Plane partitions of this type are symmetric with respect to the planex=y. Below is an example of a symmetric plane partition and its visualisation.
In 1898, MacMahon formulated his conjecture about the generating function for symmetric plane partitions which are subsets ofB(r,r,t){\displaystyle {\mathcal {B}}(r,r,t)}.[11]This conjecture is calledThe MacMahon conjecture. The generating function is given by∑π∈B(r,r,t)/S2q|π|=∏i=1r[1−qt+2i−11−q2i−1∏j=i+1r1−q2(i+j+t−1)1−q2(i+j−1)]{\displaystyle \sum _{\pi \in {\mathcal {B}}(r,r,t)/{\mathcal {S}}_{2}}q^{|\pi |}=\prod _{i=1}^{r}\left[{\frac {1-q^{t+2i-1}}{1-q^{2i-1}}}\prod _{j=i+1}^{r}{\frac {1-q^{2(i+j+t-1)}}{1-q^{2(i+j-1)}}}\right]}
Macdonald[10]pointed out that Percy A. MacMahon's conjecture reduces to
In 1972 Edward A. Bender andDonald E. Knuthconjectured[12]a simple closed form for the generating function for plane partition which have at mostrrows and strict decrease along the rows.George Andrewsshowed[13]that the conjecture of Bender and Knuth and the MacMahon conjecture are equivalent. MacMahon's conjecture was proven almost simultaneously by George Andrews in 1977[14]and later Ian G. Macdonald presented an alternative proof.[15]When settingq= 1 yields the counting functionN2(r,r,t){\displaystyle N_{2}(r,r,t)}which is given by
For a proof of the caseq= 1 please refer to George Andrews' paperMacMahon's conjecture on symmetric plane partitions.[16]
πis called cyclically symmetric, if thei-th row ofπ{\displaystyle \pi }is conjugate to thei-th column for alli. Thei-th row is regarded as an ordinary partition. The conjugate of a partitionπ{\displaystyle \pi }is the partition whose diagram is the transpose of partitionπ{\displaystyle \pi }.[10]In other words, the plane partition is cyclically symmetric if whenever(i,j,k)∈B(r,s,t){\displaystyle (i,j,k)\in {\mathcal {B}}(r,s,t)}then (k,i,j) and (j,k,i) also belong toB(r,s,t){\displaystyle {\mathcal {B}}(r,s,t)}. Below an example of a cyclically symmetric plane partition and its visualization is given.
Macdonald's conjecture provides a formula for calculating the number of cyclically symmetric plane partitions for a given integerr. This conjecture is calledThe Macdonald conjecture. The generating function for cyclically symmetric plane partitions which are subsets ofB(r,r,r){\displaystyle {\mathcal {B}}(r,r,r)}is given by
This equation can also be written in another way
In 1979, Andrews proved Macdonald's conjecture for the caseq= 1 as the"weak" Macdonald conjecture.[17]Three years later William H. Mills,David Robbinsand Howard Rumsey proved the general case of Macdonald's conjecture in their paperProof of the Macdonald conjecture.[18]The formula forN3(r,r,r){\displaystyle N_{3}(r,r,r)}is given by the"weak" Macdonald conjecture
A totally symmetric plane partitionπ{\displaystyle \pi }is a plane partition which is symmetric and cyclically symmetric. This means that the diagram is symmetric at all three diagonal planes, or in other words that if(i,j,k)∈B(r,s,t){\displaystyle (i,j,k)\in {\mathcal {B}}(r,s,t)}then all six permutations of (i,j,k) are also inB(r,s,t){\displaystyle {\mathcal {B}}(r,s,t)}. Below an example of a matrix for a totally symmetric plane partition is given. The picture shows the visualisation of the matrix.
Macdonald found the total number of totally symmetric plane partitions that are subsets ofB(r,r,r){\displaystyle {\mathcal {B}}(r,r,r)}. The formula is given by
In 1995John R. Stembridgefirst proved the formula forN4(r,r,r){\displaystyle N_{4}(r,r,r)}[19]and later in 2005 it was proven by George Andrews,Peter Paule, and Carsten Schneider.[20]Around 1983 Andrews and Robbins independently stated an explicit product formula for the orbit-counting generating function for totally symmetric plane partitions.[21][22]This formula already alluded to in George E. Andrews' paperTotally symmetric plane partitionswhich was published 1980.[23]The conjecture is calledTheq-TSPPconjectureand it is given by:
LetS3{\displaystyle {\mathcal {S}}_{3}}be the symmetric group. The orbit counting function for totally symmetric plane partitions that fit insideB(r,r,r){\displaystyle {\mathcal {B}}(r,r,r)}is given by the formula
This conjecture was proved in 2011 byChristoph Koutschan,Manuel KauersandDoron Zeilberger.[24]
Ifπi,j+πr−i+1,s−j+1=t{\displaystyle \pi _{i,j}+\pi _{r-i+1,s-j+1}=t}for all1≤i≤r{\displaystyle 1\leq i\leq r},1≤j≤s{\displaystyle 1\leq j\leq s}, then the plane partition is called self-complementary. It is necessary that the productr⋅s⋅t{\displaystyle r\cdot s\cdot t}is even. Below an example of a self-complementary symmetric plane partition and its visualisation is given.
Richard P. Stanley[25]conjectured formulas for the total number of self-complementary plane partitionsN5(r,s,t){\displaystyle N_{5}(r,s,t)}. According to Stanley, Robbins also formulated formulas for the total number of self-complementary plane partitions in a different but equivalent form. The total number of self-complementary plane partitions that are subsets ofB(r,s,t){\displaystyle {\mathcal {B}}(r,s,t)}is given by
It is necessary that the product ofr,sandtis even. A proof can be found in the paperSymmetries of Plane Partitionswhich was written by Stanley.[26][25]The proof works with Schur functionsssr(x){\displaystyle s_{s^{r}}(x)}. Stanley's proof of the ordinary enumeration of self-complementary plane partitions yields theq-analogue by substitutingxi=qi{\displaystyle x_{i}=q^{i}}fori=1,…,n{\displaystyle i=1,\ldots ,n}.[27]This is a special case of Stanley's hook-content formula.[28]The generating function for self-complementary plane partitions is given by
Substituting this formula in
supplies the desiredq-analogue case.
A plane partitionπ{\displaystyle \pi }is called cyclically symmetric self-complementary if it iscyclically symmetricandself-complementary. The figure presents a cyclically symmetric self-complementary plane partition and the according matrix is below.
In a private communication with Stanley, Robbins conjectured that the total number of cyclically symmetric self-complementary plane partitions is given byN6(2r,2r,2r){\displaystyle N_{6}(2r,2r,2r)}.[22][25]The total number of cyclically symmetric self-complementary plane partitions is given by
Dr{\displaystyle D_{r}}is the number ofr×r{\displaystyle r\times r}alternating sign matrices. A formula forDr{\displaystyle D_{r}}is given by
Greg Kuperbergproved the formula forN6(r,r,r){\displaystyle N_{6}(r,r,r)}in 1994.[9]
A totally symmetric self-complementary plane partition is a plane partition that is bothtotally symmetricandself-complementary. For instance, the matrix below is such a plane partition; it is visualised in the accompanying picture.
The formulaN7(r,r,r){\displaystyle N_{7}(r,r,r)}was conjectured by William H. Mills, Robbins and Howard Rumsey in their workSelf-Complementary Totally Symmetric Plane Partitions.[29]The total number of totally symmetric self-complementary plane partitions is given by
Andrews proves this formula in 1994 in his paperPlane Partitions V: The TSSCPP Conjecture.[30]
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Innumber theory, apolite numberis apositive integerthat can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is calledimpolite.[1][2]The impolite numbers are exactly thepowers of two, and the polite numbers are thenatural numbersthat are not powers of two.
Polite numbers have also been calledstaircase numbersbecause theYoung diagramswhich represent graphically thepartitionsof a polite number into consecutive integers (in theFrench notationof drawing these diagrams) resemblestaircases.[3][4][5]If all numbers in the sum are strictly greater than one, the numbers so formed are also calledtrapezoidal numbersbecause they represent patterns of points arranged in atrapezoid.[6][7][8][9][10][11][12]
The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied bySylvester,[13]Mason,[14][15]Leveque,[16]and many other more recent authors.[1][2][17][18][19][20][21][22][23]The polite numbers describe the possible numbers of sides of theReinhardt polygons.[24]
The first few polite numbers are
The impolite numbers are exactly thepowers of two.[13]It follows from theLambek–Moser theoremthat thenth polite number isf(n+ 1), where
Thepolitenessof a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For everyx, the politeness ofxequals the number ofodddivisorsofxthat are greater than one.[13]The politeness of the numbers 1, 2, 3, ... is
For instance, the politeness of 9 is 2 because it has two odd divisors, 3 and 9, and two polite representations
the politeness of 15 is 3 because it has three odd divisors, 3, 5, and 15, and (as is familiar tocribbageplayers)[25]three polite representations
An easy way of calculating the politeness of a positive number by decomposing the number into itsprime factors, taking the powers of all prime factors greater than 2, adding 1 to all of them, multiplying the numbers thus obtained with each other and subtracting 1. For instance 90 has politeness 5 because90=2×32×51{\displaystyle 90=2\times 3^{2}\times 5^{1}}; the powers of 3 and 5 are respectively 2 and 1, and applying this method(2+1)×(1+1)−1=5{\displaystyle (2+1)\times (1+1)-1=5}.
To see the connection between odd divisors and polite representations, suppose a numberxhas the odd divisory> 1. Thenyconsecutive integers centered onx/y(so that their average value isx/y) havexas their sum:
Some of the terms in this sum may be zero or negative. However, if a term is zero it can be omitted and any negative terms may be used to cancel positive ones, leading to a polite representation forx. (The requirement thaty> 1 corresponds to the requirement that a polite representation have more than one term; applying the same construction fory= 1 would just lead to the trivial one-term representationx=x.)
For instance, the polite numberx= 14 has a single nontrivial odd divisor, 7. It is therefore the sum of 7 consecutive numbers centered at 14/7 = 2:
The first term, −1, cancels a later +1, and the second term, zero, can be omitted, leading to the polite representation
Conversely, every polite representation ofxcan be formed from this construction. If a representation has an odd number of terms,x/yis the middle term, while if it has anevennumber of terms and its minimum value ismit may be extended in a unique way to a longer sequence with the same sum and an odd number of terms, by including the 2m− 1 numbers −(m− 1), −(m− 2), ..., −1, 0, 1, ...,m− 2,m− 1.
After this extension, again,x/yis the middle term. By this construction, the polite representations of a number and its odd divisors greater than one may be placed into aone-to-one correspondence, giving abijective proofof the characterization of polite numbers and politeness.[13][26]More generally, the same idea gives a two-to-one correspondence between, on the one hand, representations as a sum of consecutive integers (allowing zero, negative numbers, and single-term representations) and on the other hand odd divisors (including 1).[15]
Another generalization of this result states that, for anyn, the number of partitions ofninto odd numbers havingkdistinct values equals the number of partitions ofninto distinct numbers havingkmaximal runs of consecutive numbers.[13][27][28]Here a run is one or more consecutive values such that the next larger and the next smaller consecutive values are not part of the partition; for instance the partition 10 = 1 + 4 + 5 has two runs, 1 and 4 + 5.
A polite representation has a single run, and a partition with one valuedis equivalent to a factorization ofnas the productd⋅ (n/d), so the special casek= 1 of this result states again the equivalence between polite representations and odd factors (including in this case the trivial representationn=nand the trivial odd factor 1).
If a polite representation starts with 1, the number so represented is atriangular number
Otherwise, it is the difference of two nonconsecutive triangular numbers
This second case is called a trapezoidal number.[12]One can also consider polite numbers that aren't trapezoidal. The only such numbers are the triangular numbers with only one nontrivial odd divisor, because for those numbers, according to thebijectiondescribed earlier, the odd divisor corresponds to the triangular representation and there can be no other polite representations. Thus, non-trapezoidal polite number must have the form of a power of two multiplied by an odd prime. As Jones and Lord observe,[12]there are exactly two types of triangular numbers with this form:
(sequenceA068195in theOEIS). For instance, the perfect number 28 = 23 − 1(23− 1) and the number 136 = 24 − 1(24+ 1) are both this type of polite number. It is conjectured that there are infinitely many Mersenne primes, in which case there are also infinitely many polite numbers of this type.
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Incombinatorics, thetwelvefold wayis a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems ofcountingpermutations,combinations,multisets, and partitions eitherof a setorof a number. The idea of the classification is credited toGian-Carlo Rota, and the name was suggested byJoel Spencer.[1]
LetNandXbefinite sets. Letn=|N|{\displaystyle n=|N|}andx=|X|{\displaystyle x=|X|}be thecardinalitiesof the sets. ThusNis a set withnelements, andXis a set withxelements.
The general problem we consider is the enumeration ofequivalence classesoffunctionsf:N→X{\displaystyle f:N\to X}.
The functions are subject to one of the three following restrictions:
(The condition "fisbijective" is only an option whenn=x{\displaystyle n=x}; but then it is equivalent to both "fis injective" and "fis surjective".)
There are four differentequivalence relationswhich may be defined on the set of functionsffromNtoX:
The three conditions on the functions and the four equivalence relations can be paired in3 × 4 = 12ways.
The twelve problems of counting equivalence classes of functions do not involve the same difficulties, and there is not one systematic method for solving them. Two of the problems are trivial (the number of equivalence classes is 0 or 1), five problems have an answer in terms of a multiplicative formula ofnandx, and the remaining five problems have an answer in terms of combinatorial functions (Stirling numbersand thepartition functionfor a given number of parts).
The incorporation of classical enumeration problems into this setting is as follows.
The various problems in the twelvefold way may be considered from different points of view.
Traditionally many of the problems in the twelvefold way have been formulated in terms of placing balls in boxes (or some similar visualization) instead of defining functions. The setNcan be identified with a set of balls, andXwith a set of boxes; the functionf:N→X{\displaystyle f:N\to X}then describes a way to distribute the balls into the boxes, namely by putting each ballainto boxf(a){\displaystyle f(a)}. A function ascribes a unique image to each value in its domain; this property is reflected by the property that any ball can go into only one box (together with the requirement that no ball should remain outside of the boxes), whereas any box can accommodate an arbitrary number of balls. Requiring in additionf{\displaystyle f}to be injective means to forbid putting more than one ball in any one box, while requiringf{\displaystyle f}to be surjective means insisting that every box contain at least one ball.
Countingmodulopermutations ofNorXis reflected by calling the balls or the boxes, respectively, "indistinguishable". This is an imprecise formulation, intended to indicate that different configurations are not to be counted separately if one can be transformed into the other by some interchange of balls or of boxes. This possibility of transformation is formalized by the action by permutations.
Another way to think of some of the cases is in terms ofsampling, instatistics. Imagine a population ofXitems (or people), of which we chooseN. Two different schemes are normally described, known as "sampling with replacement" and "sampling without replacement". In the former case (sampling with replacement), once we've chosen an item, we put it back in the population, so that we might choose it again. The result is that each choice isindependentof all the other choices, and the set of samples is technically referred to asindependent identically distributed. In the latter case, however, once we have chosen an item, we put it aside so that we can not choose it again. This means that the act of choosing an item has an effect on all the following choices (the particular item can not be seen again), so our choices are dependent on one another.
A second distinction among sampling schemes is whether ordering matters. For example, if we have ten items, of which we choose two, then the choice (4, 7) is different from (7, 4) if ordering matters; on the other hand, if ordering does not matter, then the choices (4, 7) and (7, 4) are equivalent.
The first two rows and columns of the table below correspond to sampling with and without replacement, with and without consideration of order. The cases of sampling with replacement are found in the column labeled "Anyf{\displaystyle f}", while the cases of sampling without replacement are found in the column labeled "Injectivef{\displaystyle f}". The cases where ordering matters are found in the row labeled "Distinct," and the cases where ordering does not matter are found in the row labeled "Snorbits". Each table entry indicates how many different sets of choices there are, in a particular sampling scheme. Three of these table entries also correspond toprobability distributions. Sampling with replacement where ordering matters is comparable to describing thejoint distributionofNseparaterandom variables, each with anX-foldcategorical distribution. Sampling with replacement where ordering does not matter, however, is comparable to describing a singlemultinomial distributionofNdraws from anX-fold category, where only the number seen of each category matters. Sampling without replacement where ordering does not matter is comparable to a singlemultivariate hypergeometric distribution. Sampling without replacement where order does matter does not seem to correspond to a probability distribution.[2]In all the injective cases (sampling without replacement), the number of sets of choices is zero unlessN≤X. ("Comparable" in the above cases means that each element of thesample spaceof the corresponding distribution corresponds to a separate set of choices, and hence the number in the appropriate box indicates the size of the sample space for the given distribution.)
From the perspective of sampling, the column labeled "Surjectivef{\displaystyle f}" is somewhat strange: Essentially, we keep sampling with replacement until we have chosen each item at least once. Then, we count how many choices we have made, and if it is not equal toN, throw out the entire set and repeat. This is vaguely comparable to thecoupon collector's problem, where the process involves "collecting" (by sampling with replacement) a set ofXcoupons until each coupon has been seen at least once. In all surjective cases, the number of sets of choices is zero unlessN≥X.
A functionf:N→X{\displaystyle f:N\to X}can be considered from the perspective ofXor ofN. This leads to different views:
These points of view are not equally suited to all cases. The labelling and selection points of view are not well compatible with permutation of the elements ofX, since this changes the labels or the selection; on the other hand the grouping point of view does not give complete information about the configurationunlessthe elements ofXmay be freely permuted. The labelling and selection points of view are more or less equivalent whenNis not permuted, but when it is, the selection point of view is more suited. The selection can then be viewed as an unordered selection: a single choice of a (multi-)set ofnelements fromXis made.
When viewingf{\displaystyle f}as a labelling of the elements ofN, the latter may be thought of as arranged in a sequence, and the labels fromXas being successively assigned to them. A requirement thatf{\displaystyle f}be injective means that no label can be used a second time; the result is a sequence of labelswithout repetition. In the absence of such a requirement, the terminology "sequences with repetition" is used, meaning that labels may be used more than once (although sequences that happen to be without repetition are also allowed).
When viewingf{\displaystyle f}as an unordered selection of the elements ofX, the same kind of distinction applies. Iff{\displaystyle f}must be injective, then the selection must involvendistinct elements ofX, so it is a subset ofXof sizen, also called ann-combination. Without the requirement, one and the same element ofXmay occur multiple times in the selection, and the result is amultisetof sizenof elements fromX, also called ann-multicombinationorn-combination with repetition.
The requirement thatf{\displaystyle f}be surjective, from the viewpoint of labelling elements ofN, means that every label is to be used at least once; from the viewpoint of selection fromX, it means that every element ofXmust be included in the selection at least once. Labelling with surjection is equivalent to a grouping of elements ofNfollowed by labeling each group by an element ofX, and is accordingly somewhat more complicated to describe mathematically.
When viewingf{\displaystyle f}as a grouping of the elements ofN(which assumes one identifies under permutations ofX), requiringf{\displaystyle f}to be surjective means the number of groups must be exactlyx. Without this requirement the number of groups can be at mostx. The requirement of injectivef{\displaystyle f}means each element ofNmust be a group in itself, which leaves at most one valid grouping and therefore gives a rather uninteresting counting problem.
When in addition one identifies under permutations ofN, this amounts to forgetting the groups themselves but retaining only their sizes. These sizes moreover do not come in any definite order, while the same size may occur more than once; one may choose to arrange them into a weakly decreasing list of numbers, whose sum is the numbern. This gives the combinatorial notion of apartitionof the numbern, into exactlyx(for surjectivef{\displaystyle f}) or at mostx(for arbitraryf{\displaystyle f}) parts.
Formulas for the different cases of the twelvefold way are summarized in the following table; each table entry links to a subsection below explaining the formula.
The particular notations used are:
This is a quick summary of what the different cases mean. The cases are described in detail below.
Think of a set ofXnumbered items (numbered from 1 tox), from which we choosen, yielding an ordered list of the items: e.g. if there arex=10{\displaystyle x=10}items of which we choosen=3{\displaystyle n=3}, the result might be the list (5, 2, 10). We then count how many different such lists exist, sometimes first transforming the lists in ways that reduce the number of distinct possibilities.
Then the columns mean:
And the rows mean:
The chart below is similar to the chart above, but instead of showing the formulas, it gives an intuitive understanding of their meaning using the familiar balls and boxes example. The rows represent the distinctness of the balls and boxes. The columns represent if multi-packs (more than one ball in one box), or empty boxes are allowed. The cells in the chart show the question that is answered by solving the formula given in the formula chart above.
(no rules on placement)
(no multi-packs allowed)
(no empty boxes allowed)
How many ways can you placenmarked balls intoxmarked boxes,with no other rules on placement?
How many ways can you placenmarked balls intoxmarked boxes,with no multi-packs allowed?
How many ways can you placenmarked balls intoxmarked boxes,with no empty boxes allowed?
How many ways can you placenplain balls intoxmarked boxes,with no other rules on placement?
How many ways can you placenplain balls intoxmarked boxes,with no multi-packs allowed?
How many ways can you placenplain balls intoxmarked boxes,with no empty boxes allowed?
How many ways can you placenmarked balls intoxplain boxes,with no other rules on placement?
How many ways can you placenmarked balls intoxplain boxes,with no multi-packs allowed?
How many ways can you placenmarked balls intoxplain boxes,with no empty boxes allowed?
How many ways can you placenplain balls intoxplain boxes,with no other rules on placement?
How many ways can you placenplain balls intoxplain boxes,with no multi-packs allowed?
How many ways can you placenplain balls intoxplain boxes,with no empty boxes allowed?
The cases below are ordered in such a way as to group those cases for which the arguments used in counting are related, which is not the ordering in the table given.
This case is equivalent to countingsequences ofnelementsofXwith no restriction: a functionf:N→Xis determined by thenimages of the elements ofN, which can each be independently chosen among the elements ofx. This gives a total ofxnpossibilities.
Example:
X={a,b,c},N={1,2}, then{\displaystyle X=\{a,b,c\},N=\{1,2\}{\text{, then }}}
|{(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)}|=32=9{\displaystyle \left\vert \{(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)\}\right\vert =3^{2}=9}
This case is equivalent to counting sequences ofndistinctelements ofX, also calledn-permutationsofX, orsequences without repetitions; again this sequence is formed by thenimages of the elements ofN. This case differs from the one of unrestricted sequences in that there is one choice fewer for the second element, two fewer for the third element, and so on. Therefore instead of by an ordinary power ofx, the value is given by afalling factorial powerofx, in which each successive factor is one fewer than the previous one. The formula is
Note that ifn>xthen one obtains a factor zero, so in this case there are no injective functionsN→Xat all; this is just a restatement of thepigeonhole principle.
Example:
X={a,b,c,d},N={1,2}, then{\displaystyle X=\{a,b,c,d\},N=\{1,2\}{\text{, then }}}
|{(a,b),(a,c),(a,d),(b,a),(b,c),(b,d),(c,a),(c,b),(c,d),(d,a),(d,b),(d,c)}|=42_=4×3=12{\displaystyle \left\vert \{(a,b),(a,c),(a,d),(b,a),(b,c),(b,d),(c,a),(c,b),(c,d),(d,a),(d,b),(d,c)\}\right\vert =4^{\underline {2}}=4\times 3=12}
This case is equivalent to countingsubsets withnelementsofX, also calledn-combinations ofX: among the sequences ofndistinct elements ofX, those that differ only in the order of their terms are identified by permutations ofN. Since in all cases this groups together exactlyn! different sequences, we can divide the number of such sequences byn! to get the number ofn-combinations ofX. This number is known as thebinomial coefficient(xn){\displaystyle {\tbinom {x}{n}}}, which is therefore given by
Example:
X={a,b,c,d},N={1,2}, then{\displaystyle X=\{a,b,c,d\},N=\{1,2\}{\text{, then }}}
|{{a,b},{a,c},{a,d},{b,c},{b,d},{c,d}}|=42_2!=4×32=6{\displaystyle \left\vert \{\{a,b\},\{a,c\},\{a,d\},\{b,c\},\{b,d\},\{c,d\}\}\right\vert ={\frac {4^{\underline {2}}}{2!}}={\frac {4\times 3}{2}}=6}
This case is equivalent to countingmultisets withnelementsfromX(also calledn-multicombinations). The reason is that for each element ofXit is determined how many elements ofNare mapped to it byf, while two functions that give the same such "multiplicities" to each element ofXcan always be transformed into another by a permutation ofN. The formula counting all functionsN→Xis not useful here, because the number of them grouped together by permutations ofNvaries from one function to another. Rather, as explained undercombinations, the number ofn-multicombinations from a set withxelements can be seen to be the same as the number ofn-combinations from a set withx+n− 1elements. This reduces the problem toanother onein the twelvefold way, and gives as result
Example:
X={a,b,c},N={1,2}, then{\displaystyle X=\{a,b,c\},N=\{1,2\}{\text{, then }}}
|{{a,a},{a,b},{a,c},{b,b},{b,c},{c,c}}|=32¯2!=4×32=6{\displaystyle \left\vert \{\{a,a\},\{a,b\},\{a,c\},\{b,b\},\{b,c\},\{c,c\}\}\right\vert ={\frac {3^{\overline {2}}}{2!}}={\frac {4\times 3}{2}}=6}
This case is equivalent to countingmultisetswithnelements fromX, for which each element ofXoccurs at least once. This is also equivalent to counting thecompositionsofnwithx(non-zero) terms, by listing the multiplicities of the elements ofxin order. The correspondence between functions and multisets is the same as in the previous case, and the surjectivity requirement means that all multiplicities are at least one. By decreasing all multiplicities by 1, this reduces to the previous case; since the change decreases the value ofnbyx, the result is
Note that whenn<xthere are no surjective functionsN→Xat all (a kind of "empty pigeonhole" principle); this is taken into account in the formula, by the convention that binomial coefficients are always 0 if the lower index is negative. The same value is also given by the expression
except in the extreme casen=x= 0, where with the former expression correctly gives(−10)=1{\displaystyle {\tbinom {-1}{0}}=1}, while the latter incorrectly gives(−1−1)=0{\displaystyle {\tbinom {-1}{-1}}=0}.
The form of the result suggests looking for a manner to associate a class of surjective functionsN→Xdirectly to a subset ofn−xelements chosen from a total ofn− 1, which can be done as follows. First choose atotal orderingof the setsNandX, and note that by applying a suitable permutation ofN, every surjective functionN→Xcan be transformed into a uniqueweakly increasing(and of course still surjective) function. If one connects the elements ofNin order byn− 1arcs into alinear graph, then choosing any subset ofn−xarcs and removing the rest, one obtains a graph withxconnected components, and by sending these to the successive elements ofX, one obtains a weakly increasing surjective functionN→X; also the sizes of the connected components give a composition ofnintoxparts. This argument is basically the one given atstars and bars, except that there the complementary choice ofx− 1"separations" is made.
Example:
X={a,b},N={1,2,3}, then{\displaystyle X=\{a,b\},N=\{1,2,3\}{\text{, then }}}
|{{a,a,b},{a,b,b}}|=(3−13−2)=(21)=2!1!×(2−1)!=2{\displaystyle \left\vert \{\{a,a,b\},\{a,b,b\}\}\right\vert ={\binom {3-1}{3-2}}={\binom {2}{1}}={\frac {2!}{1!\times (2-1)!}}=2}
In this case we consider sequences ofndistinct elements fromX, but identify those obtained from one another by applying to each element a permutation ofX. It is easy to see that two different such sequences can always be identified: the permutation must map termiof the first sequence to termiof the second sequence, and since no value occurs twice in either sequence these requirements do not contradict each other; it remains to map the elements not occurring in the first sequence bijectively to those not occurring in the second sequence in an arbitrary way. The only fact that makes the result depend onnandxat all is that the existence of any such sequences to begin with requiresn≤x, by the pigeonhole principle. The number is therefore expressed as[n≤x]{\displaystyle [n\leq x]}, using theIverson bracket.
This case is reduced to the previous one: since all sequences ofndistinct elements fromXcan already be transformed into each other by applying a permutation ofXto each of their terms, also allowing reordering of the terms does not give any new identifications; the number remains[n≤x]{\displaystyle [n\leq x]}.
This case is equivalent to countingpartitions ofNintox(non-empty) subsets, or countingequivalence relationsonNwith exactlyxclasses. Indeed, for any surjective functionf:N→X, the relation of having the same image underfis such an equivalence relation, and it does not change when a permutation ofXis subsequently applied; conversely one can turn such an equivalence relation into a surjective function by assigning the elements ofXin some manner to thexequivalence classes. The number of such partitions or equivalence relations is by definition theStirling number of the second kindS(n,x), also written{nx}{\displaystyle \textstyle \{{n \atop x}\}}. Its value can be described using a recursion relation or usinggenerating functions, but unlike binomial coefficients there is noclosed formulafor these numbers that does not involve asummation.
For each surjective functionf:N→X, its orbit under permutations ofXhasx! elements, since composition (on the left) with two distinct permutations ofXnever gives the same function onN(the permutations must differ at some element ofX, which can always be written asf(i){\displaystyle f(i)}for somei∈N, and the compositions will then differ ati). It follows that the number for this case isx! times the number for the previous case, that isx!{nx}.{\displaystyle \textstyle x!\{{n \atop x}\}.}
Example:
X={a,b},N={1,2,3}, then{\displaystyle X=\{a,b\},N=\{1,2,3\}{\text{, then }}}
|{(a,a,b),(a,b,a),(a,b,b),(b,a,a),(b,a,b),(b,b,a)}|=2!{32}=2×3=6{\displaystyle \left\vert \{(a,a,b),(a,b,a),(a,b,b),(b,a,a),(b,a,b),(b,b,a)\}\right\vert =2!\left\{{3 \atop 2}\right\}=2\times 3=6}
This case is like thecorresponding onefor surjective functions, but some elements ofxmight not correspond to any equivalence class at all (since one considers functions up to a permutation ofX, it does not matterwhichelements are concerned, just how many). As a consequence one is counting equivalence relations onNwith at mostxclasses, and the result is obtained from the mentioned case by summation over values up tox, giving∑k=0x{nk}{\displaystyle \textstyle \sum _{k=0}^{x}\{{n \atop k}\}}. In casex≥n, the size ofxposes no restriction at all, and one is countingallequivalence relations on a set ofnelements (equivalently all partitions of such a set); therefore∑k=0n{nk}{\displaystyle \textstyle \sum _{k=0}^{n}\{{n \atop k}\}}gives anexpressionfor theBell numberBn.
This case is equivalent to countingpartitions of the numbernintoxnon-zero parts. Compared to the case of countingsurjective functions up to permutations ofXonly ({nx}{\displaystyle \textstyle \{{n \atop x}\}}), one only retains the sizes of the equivalence classes that the function partitionsNinto (including the multiplicity of each size), since two equivalence relations can be transformed into one another by a permutation ofNif and only if the sizes of their classes match. This is precisely what distinguishes the notion of partition ofnfrom that of partition ofN, so as a result one gets by definition the numberpx(n) of partitions ofnintoxnon-zero parts.
This case is equivalent to countingpartitions of the numberninto ≤xparts. The association is the same as for the previous case, except that now some parts of the partition may be equal to 0. (Specifically, they correspond to elements ofXnot in the image of the function.) Each partition ofninto at mostxnon-zero parts can be extended to such a partition by adding the required number of zeroes, and this accounts for all possibilities exactly once, so the result is given by∑k=0xpk(n){\displaystyle \textstyle \sum _{k=0}^{x}p_{k}(n)}. By adding 1 to each of thexparts, one obtains a partition ofn+xintoxnonzero parts, and this correspondence is bijective; hence the expression given can be simplified by writing it aspx(n+x){\displaystyle p_{x}(n+x)}.
The above formulas give the proper values for all finite setsNandX. In some cases there are alternative formulas which are almost equivalent, but which do not give the correct result in some extremal cases, such as whenNorXare empty. The following considerations apply to such cases.
In particular in the case ofcounting multisetswithnelements taken fromX, the given expression(n+x−1n){\displaystyle {\tbinom {n+x-1}{n}}}is equivalent in most cases to(n+x−1x−1){\displaystyle {\tbinom {n+x-1}{x-1}}}, but the latter expression would give 0 for the casen=x= 0(by the usual convention that binomial coefficients with a negative lower index are always 0). Similarly, for the case ofcounting compositionsofnwithxnon-zero parts, the given expression(n−1n−x){\displaystyle {\tbinom {n-1}{n-x}}}is almost equivalent to the expression(n−1x−1){\displaystyle {\tbinom {n-1}{x-1}}}given by thestars and barsargument, but the latter gives incorrect values forn= 0andallvalues ofx. For the cases where the result involves a summation, namely those of countingpartitions ofNinto at mostxnon-empty subsets orpartitions ofninto at mostxnon-zero parts, the summation index is taken to start at 0; although the corresponding term is zero whenevern> 0, it is the unique non-zero term whenn= 0, and the result would be wrong for those cases if the summation were taken to start at 1.
We can generalize further by allowing othergroupsof permutations to act onNandX. IfGis a group of permutations ofN, andHis a group of permutations ofX, then we count equivalence classes of functionsf:N→X{\displaystyle f\colon N\rightarrow X}. Two functionsfandFare considered equivalent if, and only if, there existg∈G,h∈H{\displaystyle g\in G,h\in H}so thatF=h∘f∘g{\displaystyle F=h\circ f\circ g}. This extension leads to notions such ascyclicanddihedralpermutations, as well as cyclic and dihedral partitions of numbers and sets.
Another generalization calledthe twentyfold waywas developed byKenneth P. Bogartin his book "Combinatorics Through Guided Discovery". In the problem of distributing objects to boxes both the objects and the boxes may be identical or distinct. Bogart identifies 20 cases.[3]Robert A. Proctor has constructed the thirtyfold way.[4]
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https://en.wikipedia.org/wiki/Twelvefold_way
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Inpopulation genetics,Ewens's sampling formuladescribes theprobabilitiesassociated with counts of how many differentallelesare observed a given number of times in thesample.
Ewens's sampling formula, introduced byWarren Ewens, states that under certain conditions (specified below), if a random sample ofngametesis taken from a population and classified according to thegeneat a particularlocusthen theprobabilitythat there area1allelesrepresented once in the sample, anda2alleles represented twice, and so on, is
for some positive numberθrepresenting thepopulation mutation rate, whenevera1,…,an{\displaystyle a_{1},\ldots ,a_{n}}is a sequence of nonnegative integers such that
The phrase "under certain conditions" used above is made precise by the following assumptions:
This is aprobability distributionon the set of allpartitions of the integern. Among probabilists and statisticians it is often called themultivariate Ewens distribution.
Whenθ= 0, the probability is 1 that allngenes are the same. Whenθ= 1, then the distribution is precisely that of the integer partition induced by a uniformly distributedrandom permutation. Asθ→ ∞, the probability that no two of thengenes are the same approaches 1.
This family of probability distributions enjoys the property that if after the sample ofnis taken,mof thengametes are chosen without replacement, then the resulting probability distribution on the set of all partitions of the smaller integermis just what the formula above would give ifmwere put in place ofn.
The Ewens distribution arises naturally from theChinese restaurant process.
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https://en.wikipedia.org/wiki/Ewens%27s_sampling_formula
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Faà di Bruno's formulais an identity inmathematicsgeneralizing thechain ruleto higher derivatives. It is named afterFrancesco Faà di Bruno(1855,1857), although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematicianLouis François Antoine Arbogasthad stated the formula in a calculus textbook,[1]which is considered to be the first published reference on the subject.[2]
Perhaps the most well-known form of Faà di Bruno's formula says that
dndxnf(g(x))=∑n!m1!1!m1m2!2!m2⋯mn!n!mn⋅f(m1+⋯+mn)(g(x))⋅∏j=1n(g(j)(x))mj,{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,1!^{m_{1}}\,m_{2}!\,2!^{m_{2}}\,\cdots \,m_{n}!\,n!^{m_{n}}}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left(g^{(j)}(x)\right)^{m_{j}},}
where the sum is over alln{\displaystyle n}-tuplesof nonnegative integers(m1,…,mn){\displaystyle (m_{1},\ldots ,m_{n})}satisfying the constraint
1⋅m1+2⋅m2+3⋅m3+⋯+n⋅mn=n.{\displaystyle 1\cdot m_{1}+2\cdot m_{2}+3\cdot m_{3}+\cdots +n\cdot m_{n}=n.}
Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:
Combining the terms with the same value ofm1+m2+⋯+mn=k{\displaystyle m_{1}+m_{2}+\cdots +m_{n}=k}and noticing thatmj{\displaystyle m_{j}}has to be zero forj>n−k+1{\displaystyle j>n-k+1}leads to a somewhat simpler formula expressed in terms of partial (or incomplete) exponentialBell polynomialsBn,k(x1,…,xn−k+1){\displaystyle B_{n,k}(x_{1},\ldots ,x_{n-k+1})}:
This formula works for alln≥0{\displaystyle n\geq 0}, however forn>0{\displaystyle n>0}the polynomialsBn,0{\displaystyle B_{n,0}}are zero and thus summation in the formula can start withk=1{\displaystyle k=1}.
The formula has a "combinatorial" form:
where
The following is a concrete explanation of the combinatorial form for then=4{\displaystyle n=4}case.
The pattern is:
The factorg″(x)g′(x)2{\displaystyle g''(x)g'(x)^{2}}corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factorf‴(g(x)){\displaystyle f'''(g(x))}that goes with it corresponds to the fact that there arethreesummands in that partition. The coefficient 6 that goes with those factors corresponds to the fact that there are exactly six partitions of a set of four members that break it into one part of size 2 and two parts of size 1.
Similarly, the factorg″(x)2{\displaystyle g''(x)^{2}}in the third line corresponds to the partition 2 + 2 of the integer 4, (4, because we are finding the fourth derivative), whilef″(g(x)){\displaystyle f''(g(x))}corresponds to the fact that there aretwosummands (2 + 2) in that partition. The coefficient 3 corresponds to the fact that there are12(42)=3{\displaystyle {\tfrac {1}{2}}{\tbinom {4}{2}}=3}ways of partitioning 4 objects into groups of 2. The same concept applies to the others.
A memorizable scheme is as follows:
Lety=g(x1,…,xn){\displaystyle y=g(x_{1},\dots ,x_{n})}. Then the following identity holds regardless of whether then{\displaystyle n}variables are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables (if it seems opaque, see the very concrete example below):[3]
where (as above)
|B|{\displaystyle |B|}is the size of the blockB{\displaystyle B}).
More general versions hold for cases where the all functions are vector- and evenBanach-space-valued. In this case one needs to consider theFréchet derivativeorGateaux derivative.
The five terms in the following expression correspond in the obvious way to the five partitions of the set{1,2,3}{\displaystyle \{1,2,3\}}, and in each case the order of the derivative off{\displaystyle f}is the number of parts in the partition:
If the three variables are indistinguishable from each other, then three of the five terms above are also indistinguishable from each other, and then we have the classic one-variable formula.
Supposef(x)=∑n=0∞anxn{\displaystyle f(x)=\sum _{n=0}^{\infty }{a_{n}}x^{n}}andg(x)=∑n=0∞bnxn{\displaystyle g(x)=\sum _{n=0}^{\infty }{b_{n}}x^{n}}areformal power seriesandb0=0{\displaystyle b_{0}=0}.
Then the compositionf∘g{\displaystyle f\circ g}is again a formalpower series,
wherec0=a0{\displaystyle c_{0}=a_{0}}and the other coefficientcn{\displaystyle c_{n}}forn≥1{\displaystyle n\geq 1}can be expressed as a sum overcompositionsofn{\displaystyle n}or as an equivalent sum overinteger partitionsofn{\displaystyle n}:
where
is the set of compositions ofn{\displaystyle n}withk{\displaystyle k}denoting the number of parts,
or
where
is the set of partitions ofn{\displaystyle n}intok{\displaystyle k}parts, in frequency-of-parts form.
The first form is obtained by picking out the coefficient ofxn{\displaystyle x^{n}}in(b1x+b2x2+⋯)k{\displaystyle (b_{1}x+b_{2}x^{2}+\cdots )^{k}}"by inspection", and the second form
is then obtained by collecting like terms, or alternatively, by applying themultinomial theorem.
The special casef(x)=ex{\displaystyle f(x)=e^{x}},g(x)=∑n≥11n!anxn{\displaystyle g(x)=\sum _{n\geq 1}{\frac {1}{n!}}a_{n}x^{n}}gives theexponential formula.
The special casef(x)=1/(1−x){\displaystyle f(x)=1/(1-x)},g(x)=∑n≥1(−an)xn{\displaystyle g(x)=\sum _{n\geq 1}(-a_{n})x^{n}}gives an expression for thereciprocalof the formal power series∑n≥0anxn{\displaystyle \sum _{n\geq 0}a_{n}x^{n}}in the casea0=1{\displaystyle a_{0}=1}.
Stanley[4]gives a version for exponential power series.
In theformal power series
we have then{\displaystyle n}th derivative at 0:
This should not be construed as the value of a function, since these series are purely formal; there is no such thing as convergence or divergence in this context.
If
and
and
then the coefficientcn{\displaystyle c_{n}}(which would be then{\displaystyle n}th derivative ofh{\displaystyle h}evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by
whereπ{\displaystyle \pi }runs through the set of all partitions of the set{1,…,n}{\displaystyle \{1,\ldots ,n\}}andB1,…,Bk{\displaystyle B_{1},\ldots ,B_{k}}are the blocks of the partitionπ{\displaystyle \pi }, and|Bj|{\displaystyle |B_{j}|}is the number of members of thej{\displaystyle j}th block, forj=1,…,k{\displaystyle j=1,\ldots ,k}.
This version of the formula is particularly well suited to the purposes ofcombinatorics.
We can also write with respect to the notation above
whereBn,k(a1,…,an−k+1){\displaystyle B_{n,k}(a_{1},\ldots ,a_{n-k+1})}areBell polynomials.
Iff(x)=ex{\displaystyle f(x)=e^{x}}, then all of the derivatives off{\displaystyle f}are the same and are a factor common to every term:
whereBn(x){\displaystyle B_{n}(x)}is thenthcomplete exponential Bell polynomial.
In caseg(x){\displaystyle g(x)}is acumulant-generating function, thenf(g(x)){\displaystyle f(g(x))}is amoment-generating function, and the polynomial in various derivatives ofg{\displaystyle g}is the polynomial that expresses themomentsas functions of thecumulants.
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Innumber theoryandcombinatorics, amultipartitionof a positiveintegernis a way of writingnas asum, each element of which is in turn aninteger partition. The concept is also found in the theory ofLie algebras.
Anr-component multipartition of an integernis anr-tuple of partitionsλ(1), ...,λ(r)where eachλ(i)is a partition of someaiand theaisum ton. The number ofr-component multipartitions ofnis denotedPr(n). Congruences for the functionPr(n) have been studied byA. O. L. Atkin.
Thiscombinatorics-related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Multipartition
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Inmathematics,Newton's identities, also known as theGirard–Newton formulae, give relations between two types ofsymmetric polynomials, namely betweenpower sumsandelementary symmetric polynomials. Evaluated at therootsof a monicpolynomialPin one variable, they allow expressing the sums of thek-thpowersof all roots ofP(counted with their multiplicity) in terms of the coefficients ofP, without actually finding those roots. These identities were found byIsaac Newtonaround 1666, apparently in ignorance of earlier work (1629) byAlbert Girard. They have applications in many areas of mathematics, includingGalois theory,invariant theory,group theory,combinatorics, as well as further applications outside mathematics, includinggeneral relativity.
Letx1, ...,xnbe variables, denote fork≥ 1 bypk(x1, ...,xn) thek-thpower sum:
and fork≥ 0 denote byek(x1, ...,xn) theelementary symmetric polynomial(that is, the sum of all distinct products ofkdistinct variables), so
Then Newton's identities can be stated as
valid for alln≥k≥ 1.
Also, one has
for allk>n≥ 1.
Concretely, one gets for the first few values ofk:
The form and validity of these equations do not depend on the numbernof variables (although the point where the left-hand side becomes 0 does, namely after then-th identity), which makes it possible to state them as identities in thering of symmetric functions. In that ring one has
and so on; here the left-hand sides never become zero.
These equations allow to recursively express theeiin terms of thepk; to be able to do the inverse, one may rewrite them as
In general, we have
valid for alln≥k≥ 1.
Also, one has
for allk>n≥ 1.
The polynomial with rootsximay be expanded as
where thecoefficientsek(x1,…,xn){\displaystyle e_{k}(x_{1},\ldots ,x_{n})}are the symmetric polynomials defined above.
Given thepower sumsof the roots
the coefficients of the polynomial with rootsx1,…,xn{\displaystyle x_{1},\ldots ,x_{n}}may be expressed recursively in terms of the power sums as
Formulating polynomials in this way is useful in using the method of Delves and Lyness[1]to find the zeros of an analytic function.
When the polynomial above is thecharacteristic polynomialof amatrixA{\displaystyle \mathbf {A} }(in particular whenA{\displaystyle \mathbf {A} }is thecompanion matrixof the polynomial), the rootsxi{\displaystyle x_{i}}are theeigenvaluesof the matrix, counted with their algebraic multiplicity. For any positive integerk{\displaystyle k}, the matrixAk{\displaystyle \mathbf {A} ^{k}}has as eigenvalues the powersxik{\displaystyle x_{i}^{k}}, and each eigenvaluexi{\displaystyle x_{i}}ofA{\displaystyle \mathbf {A} }contributes its multiplicity to that of the eigenvaluexik{\displaystyle x_{i}^{k}}ofAk{\displaystyle \mathbf {A} ^{k}}. Then the coefficients of the characteristic polynomial ofAk{\displaystyle \mathbf {A} ^{k}}are given by the elementary symmetric polynomialsin those powersxik{\displaystyle x_{i}^{k}}. In particular, the sum of thexik{\displaystyle x_{i}^{k}}, which is thek{\displaystyle k}-th power sumpk{\displaystyle p_{k}}of the roots of the characteristic polynomial ofA{\displaystyle \mathbf {A} }, is given by itstrace:
pk=tr(Ak).{\displaystyle p_{k}=\operatorname {tr} (\mathbf {A} ^{k})\,.}
The Newton identities now relate the traces of the powersAk{\displaystyle \mathbf {A} ^{k}}to the coefficients of the characteristic polynomial ofA{\displaystyle \mathbf {A} }. Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can be used to find the characteristic polynomial by computing only the powersAk{\displaystyle \mathbf {A} ^{k}}and their traces.
This computation requires computing the traces of matrix powersAk{\displaystyle \mathbf {A} ^{k}}and solving a triangular system of equations. Both can be done in complexity classNC(solving a triangular system can be done by divide-and-conquer). Therefore, characteristic polynomial of a matrix can be computed in NC. By theCayley–Hamilton theorem, every matrix satisfies its characteristic polynomial, anda simple transformationallows to find theadjugate matrixin NC.
Rearranging the computations into an efficient form leads to theFaddeev–LeVerrier algorithm(1840), a fast parallel implementation of it is due to L. Csanky (1976). Its disadvantage is that it requires division by integers, so in general the field should have characteristic 0.
For a givenn, the elementary symmetric polynomialsek(x1,...,xn) fork= 1,...,nform an algebraic basis for the space of symmetric polynomials inx1,....xn: every polynomial expression in thexithat is invariant under all permutations of those variables is given by apolynomialexpression in those elementary symmetric polynomials, and this expression is unique up to equivalence of polynomial expressions. This is a general fact known as thefundamental theorem of symmetric polynomials, and Newton's identities provide explicit formulae in the case of power sum symmetric polynomials. Applied to the monic polynomialtn+∑k=1n(−1)kaktn−k{\textstyle t^{n}+\sum _{k=1}^{n}(-1)^{k}a_{k}t^{n-k}}with all coefficientsakconsidered as free parameters, this means that every symmetric polynomial expressionS(x1,...,xn) in its roots can be expressed instead as a polynomial expressionP(a1,...,an) in terms of its coefficients only, in other words without requiring knowledge of the roots. This fact also follows from general considerations inGalois theory(one views theakas elements of a base field with roots in an extension field whose Galois group permutes them according to the full symmetric group, and the field fixed under all elements of the Galois group is the base field).
The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the firstnpower sums also form an algebraic basis for the space of symmetric polynomials.
There are a number of (families of) identities that, while they should be distinguished from Newton's identities, are very closely related to them.
Denoting byhkthecomplete homogeneous symmetric polynomial(that is, the sum of allmonomialsof degreek), the power sum polynomials also satisfy identities similar to Newton's identities, but not involving any minus signs. Expressed as identities of in thering of symmetric functions, they read
valid for all n ≥k≥ 1. Contrary to Newton's identities, the left-hand sides do not become zero for largek, and the right-hand sides contain ever more non-zero terms. For the first few values ofk, one has
These relations can be justified by an argument analogous to the one by comparing coefficients inpower seriesgiven above, based in this case on the generating function identity
Proofs of Newton's identities, like these given below, cannot be easily adapted to prove these variants of those identities.
As mentioned, Newton's identities can be used to recursively express elementary symmetric polynomials in terms of power sums. Doing so requires the introduction of integer denominators, so it can be done in the ring ΛQof symmetric functions with rational coefficients:
and so forth.[2]The general formula can be conveniently expressed as
where theBnis the complete exponentialBell polynomial. This expression also leads to the following identity for generating functions:
Applied to a monic polynomial, these formulae express the coefficients in terms of the power sums of the roots: replace eacheibyaiand eachpkbysk.
The analogous relations involving complete homogeneous symmetric polynomials can be similarly developed, giving equations
and so forth, in which there are only plus signs. In terms of the complete Bell polynomial,
These expressions correspond exactly to thecycle indexpolynomials of thesymmetric groups, if one interprets the power sumspias indeterminates: the coefficient in the expression forhkof any monomialp1m1p2m2...plmlis equal to the fraction of all permutations ofkthat havem1fixed points,m2cycles of length 2, ..., andmlcycles of lengthl. Explicitly, this coefficient can be written as1/N{\displaystyle 1/N}whereN=∏i=1l(mi!imi){\textstyle N=\prod _{i=1}^{l}(m_{i}!\,i^{m_{i}})}; thisNis the number permutations commuting with any given permutationπof the given cycle type. The expressions for the elementary symmetric functions have coefficients with the same absolute value, but a sign equal to the sign ofπ, namely (−1)m2+m4+....
It can be proved by considering the following inductive step:
By analogy with the derivation of the generating function of theen{\displaystyle e_{n}}, we can also obtain the generating function of thehn{\displaystyle h_{n}}, in terms of the power sums, as:
This generating function is thus theplethystic exponentialofp1t=(x1+⋯+xn)t{\displaystyle p_{1}t=(x_{1}+\cdots +x_{n})t}.
One may also use Newton's identities to express power sums in terms of elementary symmetric polynomials, which does not introduce denominators:
The first four formulas were obtained byAlbert Girardin 1629 (thus before Newton).[3]
The general formula (for all positive integersm) is:
This can be conveniently stated in terms ofordinary Bell polynomialsas
or equivalently as thegenerating function:[4]
which is analogous to theBell polynomialexponentialgenerating function given in theprevious subsection.
The multiple summation formula above can be proved by considering the following inductive step:
Finally one may use the variant identities involving complete homogeneous symmetric polynomials similarly to express power sums in term of them:
and so on. Apart from the replacement of eacheiby the correspondinghi, the only change with respect to the previous family of identities is in the signs of the terms, which in this case depend just on the number of factors present: the sign of the monomial∏i=1lhimi{\textstyle \prod _{i=1}^{l}h_{i}^{m_{i}}}is −(−1)m1+m2+m3+.... In particular the above description of the absolute value of the coefficients applies here as well.
The general formula (for all non-negative integersm) is:
One can obtain explicit formulas for the above expressions in the form of determinants, by considering the firstnof Newton's identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which the elementary symmetric functions are known and the power sums are unknowns (or vice versa), and applyCramer's ruleto find the solution for the final unknown. For instance taking Newton's identities in the form
we considerp1,−p2,p3,…,(−1)npn−1{\displaystyle p_{1},-p_{2},p_{3},\ldots ,(-1)^{n}p_{n-1}}andpn{\displaystyle p_{n}}as unknowns, and solve for the final one, giving
Solving foren{\displaystyle e_{n}}instead of forpn{\displaystyle p_{n}}is similar, as the analogous computations for the complete homogeneous symmetric polynomials; in each case the details are slightly messier than the final results, which are (Macdonald 1979, p. 20):
Note that the use of determinants makes that the formula forhn{\displaystyle h_{n}}has additional minus signs compared to the one foren{\displaystyle e_{n}}, while the situation for the expanded form given earlier is opposite. As remarked in (Littlewood 1950, p. 84) one can alternatively obtain the formula forhn{\displaystyle h_{n}}by taking thepermanentof the matrix foren{\displaystyle e_{n}}instead of the determinant, and more generally an expression for anySchur polynomialcan be obtained by taking the correspondingimmanantof this matrix.
Each of Newton's identities can easily be checked by elementary algebra; however, their validity in general needs a proof. Here are some possible derivations.
One can obtain thek-th Newton identity inkvariables by substitution into
as follows. Substitutingxjfortgives
Summing over alljgives
where the terms fori= 0 were taken out of the sum becausep0is (usually) not defined. This equation immediately gives thek-th Newton identity inkvariables. Since this is an identity of symmetric polynomials (homogeneous) of degreek, its validity for any number of variables follows from its validity forkvariables. Concretely, the identities inn<kvariables can be deduced by settingk−nvariables to zero. Thek-th Newton identity inn>kvariables contains more terms on both sides of the equation than the one inkvariables, but its validity will be assured if the coefficients of any monomial match. Because no individual monomial involves more thankof the variables, the monomial will survive the substitution of zero for some set ofn−k(other) variables, after which the equality of coefficients is one that arises in thek-th Newton identity ink(suitably chosen) variables.
Another derivation can be obtained by computations in the ring offormal power seriesR[[t]], whereRisZ[x1,...,xn], thering of polynomialsinnvariablesx1,...,xnover the integers.
Starting again from the basic relation
and "reversing the polynomials" by substituting 1/tfortand then multiplying both sides bytnto remove negative powers oft, gives
(the above computation should be performed in thefield of fractionsofR[[t]]; alternatively, the identity can be obtained simply by evaluating the product on the left side)
Swapping sides and expressing theaias the elementary symmetric polynomials they stand for gives the identity
Oneformally differentiatesboth sides with respect tot, and then (for convenience) multiplies byt, to obtain
where the polynomial on the right hand side was first rewritten as arational functionin order to be able to factor out a product out of the summation, then the fraction in the summand was developed as a series int, using the formula
and finally the coefficient of eachtjwas collected, giving a power sum. (The series intis a formal power series, but may alternatively be thought of as a series expansion fortsufficiently close to 0, for those more comfortable with that; in fact one is not interested in the function here, but only in the coefficients of the series.) Comparing coefficients oftkon both sides one obtains
which gives thek-th Newton identity.
The following derivation, given essentially in (Mead, 1992), is formulated in thering of symmetric functionsfor clarity (all identities are independent of the number of variables). Fix somek> 0, and define the symmetric functionr(i) for 2 ≤i≤kas the sum of all distinctmonomialsof degreekobtained by multiplying one variable raised to the poweriwithk−idistinct other variables (this is themonomial symmetric functionmγwhere γ is a hook shape (i,1,1,...,1)). In particularr(k) =pk; forr(1) the description would amount to that ofek, but this case was excluded since here monomials no longer have any distinguished variable. All productspiek−ican be expressed in terms of ther(j) with the first and last case being somewhat special. One has
since each product of terms on the left involving distinct variables contributes tor(i), while those where the variable frompialready occurs among the variables of the term fromek−icontributes tor(i+ 1), and all terms on the right are so obtained exactly once. Fori=kone multiplies bye0= 1, giving trivially
Finally the productp1ek−1fori= 1 gives contributions tor(i+ 1) =r(2) like for other valuesi<k, but the remaining contributions producektimes each monomial ofek, since any one of the variables may come from the factorp1; thus
Thek-th Newton identity is now obtained by taking the alternating sum of these equations, in which all terms of the formr(i) cancel out.
A shortcombinatorial proofof Newton's identities was given byDoron Zeilbergerin 1984.[5]
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Thespt function(smallest parts function) is a function innumber theorythat counts the sum of the number of smallest parts in eachinteger partitionof a positive integer. It is related to thepartition function.[1]
The first few values of spt(n) are:
For example, there are five partitions of 4 (with smallest parts underlined):
These partitions have 1, 1, 2, 2, and 4 smallest parts, respectively. So spt(4) = 1 + 1 + 2 + 2 + 4 = 10.
Like the partition function, spt(n) has agenerating function. It is given by
where(q)∞=∏n=1∞(1−qn){\displaystyle (q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{n})}.
The functionS(q){\displaystyle S(q)}is related to amock modular form. LetE2(z){\displaystyle E_{2}(z)}denote the weight 2 quasi-modularEisenstein seriesand letη(z){\displaystyle \eta (z)}denote theDedekind eta function. Then forq=e2πiz{\displaystyle q=e^{2\pi iz}}, the function
is amock modular formof weight 3/2 on the fullmodular groupSL2(Z){\displaystyle SL_{2}(\mathbb {Z} )}with multiplier systemχη−1{\displaystyle \chi _{\eta }^{-1}}, whereχη{\displaystyle \chi _{\eta }}is the multiplier system forη(z){\displaystyle \eta (z)}.
While a closed formula is not known for spt(n), there are Ramanujan-likecongruencesincluding
Thisnumber theory-related article is astub. You can help Wikipedia byexpanding it.
Thiscombinatorics-related article is astub. You can help Wikipedia byexpanding it.
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Goldbach's conjectureis one of the oldest and best-knownunsolved problemsinnumber theoryand all ofmathematics. It states that everyevennatural numbergreater than 2 is the sum of twoprime numbers.
The conjecture has been shown to hold for all integers less than4×1018but remains unproven despite considerable effort.
On 7 June 1742, thePrussianmathematicianChristian Goldbachwrote a letter toLeonhard Euler(letter XLIII),[2]in which he proposed the following conjecture:
Goldbach was following the now-abandoned convention ofconsidering 1to be aprime number,[3]so that a sum of units would be a sum of primes.
He then proposed a second conjecture in the margin of his letter, which implies the first:[4]
Es scheinet wenigstens, dass eine jede Zahl, die grösser ist als 2, ein aggregatum trium numerorum primorum sey.It seems at least, that every integer greater than 2 can be written as the sum of three primes.
Euler replied in a letter dated 30 June 1742[5]and reminded Goldbach of an earlier conversation they had had ("... so Ew vormals mit mir communicirt haben ..."), in which Goldbach had remarked that the first of those two conjectures would follow from the statement
This is in fact equivalent to his second, marginal conjecture.
In the letter dated 30 June 1742, Euler stated:[6][7]
Dass ... ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann.That ... every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.
René Descarteswrote that "Every even number can be expressed as the sum of at most three primes."[8]The proposition is similar to, but weaker than, Goldbach's conjecture.Paul Erdőssaid that "Descartes actually discovered this before Goldbach... but it is better that the conjecture was named for Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor."[9]
The strong Goldbach conjecture is much more difficult than theweak Goldbach conjecture, which says that every odd integer greater than 5 is the sum of three primes. UsingVinogradov's method,Nikolai Chudakov,[10]Johannes van der Corput,[11]andTheodor Estermann[12]showed (1937–1938) thatalmost alleven numbers can be written as the sum of two primes (in the sense that the fraction of even numbers up to someNwhich can be so written tends towards 1 asNincreases). In 1930,Lev Schnirelmannproved that anynatural numbergreater than 1 can be written as the sum of not more thanCprime numbers, whereCis an effectively computable constant; seeSchnirelmann density.[13][14]Schnirelmann's constant is the lowest numberCwith this property. Schnirelmann himself obtainedC<800000. This result was subsequently enhanced by many authors, such asOlivier Ramaré, who in 1995 showed that every even numbern≥ 4is in fact the sum of at most 6 primes. The best known result currently stems from the proof of the weak Goldbach conjecture byHarald Helfgott,[15]which directly implies that every even numbern≥ 4is the sum of at most 4 primes.[16][17]
In 1924, Hardy and Littlewood showed under the assumption of thegeneralized Riemann hypothesisthat the number of even numbers up toXviolating the Goldbach conjecture ismuch less thanX1⁄2+cfor smallc.[18]
In 1948, usingsieve theorymethods,Alfréd Rényishowed that every sufficiently large even number can be written as the sum of a prime and analmost primewith at mostKfactors.[19]Chen Jingrunshowed in 1973 using sieve theory that everysufficiently largeeven number can be written as the sum of either two primes, or a prime and asemiprime(the product of two primes).[20]SeeChen's theoremfor further information.
In 1975,Hugh Lowell MontgomeryandBob Vaughanshowed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constantscandCsuch that for all sufficiently large numbersN, every even number less thanNis the sum of two primes, with at mostCN1 −cexceptions. In particular, the set of even integers that are not the sum of two primes hasdensityzero.
In 1951,Yuri Linnikproved the existence of a constantKsuch that every sufficiently large even number is the sum of two primes and at mostKpowers of 2.János PintzandImre Ruzsafound in 2020 thatK= 8works.[21]Assuming thegeneralized Riemann hypothesis,K= 7also works, as shown byRoger Heath-BrownandJan-Christoph Schlage-Puchtain 2002.[22]
A proof for the weak conjecture was submitted in 2013 byHarald HelfgotttoAnnals of Mathematics Studiesseries. Although the article was accepted, Helfgott decided to undertake the major modifications suggested by the referee. Despite several revisions, Helfgott's proof has not yet appeared in a peer-reviewed publication.[23][24][25]The weak conjecture is implied by the strong conjecture, as ifn− 3is a sum of two primes, thennis a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture would remain unproven if Helfgott's proof is correct.
For small values ofn, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, in 1938, Nils Pipping laboriously verified the conjecture up ton=100000.[26]With the advent of computers, many more values ofnhave been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture forn≤4×1018(and double-checked up to4×1017) as of 2013. One record from this search is that3325581707333960528is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.[27]
Goldbach's Conjecture(Chinese:哥德巴赫猜想) is the title of the biography of Chinese mathematician and number theoristChen Jingrun, written byXu Chi.
The conjecture is a central point in the plot of the 1992 novelUncle Petros and Goldbach's Conjectureby Greek authorApostolos Doxiadis, in the short story "Sixty Million Trillion Combinations" byIsaac Asimovand also in the 2008 mystery novelNo One You KnowbyMichelle Richmond.[28]
Goldbach's conjecture is part of the plot of the 2007 Spanish filmFermat's Room.
Goldbach's conjecture is featured as the main topic of research of the titular character Marguerite in the 2023 French-Swiss filmMarguerite's Theorem.[29]
Each of the three conjectures has a natural analog in terms of the modern definition of a prime, under which 1 is excluded. A modern version of the first conjecture is:
A modern version of the marginal conjecture is:
And a modern version of Goldbach's older conjecture of which Euler reminded him is:
These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integerN=p+ 1larger than 4, forpa prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integern, could not possibly rule out the existence of such a specific counterexampleN). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.
The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that
Statistical considerations that focus on theprobabilistic distribution of prime numberspresent informal evidence in favour of the conjecture (in both the weak and strong forms) forsufficiently largeintegers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes.
A very crude version of theheuristicprobabilistic argument (for the strong form of the Goldbach conjecture) is as follows. Theprime number theoremasserts that an integermselected at random has roughly a1/lnmchance of being prime. Thus ifnis a large even integer andmis a number between 3 andn/2, then one might expect the probability ofmandn−msimultaneously being prime to be1/lnmln(n−m). If one pursues this heuristic, one might expect the total number of ways to write a large even integernas the sum of two odd primes to be roughly
Sincelnn≪√n, this quantity goes to infinity asnincreases, and one would expect that every large even integer has not just one representation as the sum of two primes, but in fact very many such representations.
This heuristic argument is actually somewhat inaccurate because it assumes that the events ofmandn−mbeing prime arestatistically independentof each other. For instance, ifmis odd, thenn−mis also odd, and ifmis even, thenn−mis even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, ifnis divisible by 3, andmwas already a prime other than 3, thenn−mwould also becoprimeto 3 and thus be slightly more likely to be prime than a general number. Pursuing this type of analysis more carefully,G. H. HardyandJohn Edensor Littlewoodin 1923 conjectured (as part of theirHardy–Littlewood prime tuple conjecture) that for any fixedc≥ 2, the number of representations of a large integernas the sum ofcprimesn=p1+ ⋯ +pcwithp1≤ ⋯ ≤pcshould beasymptoticallyequal to
where the product is over all primesp, andγc,p(n)is the number of solutions to the equationn=q1+ ⋯ +qcmodpinmodular arithmetic, subject to theconstraintsq1, …,qc≠ 0 modp. This formula has been rigorously proven to be asymptotically valid forc≥ 3from the work ofIvan Matveevich Vinogradov, but is still only a conjecture whenc= 2.[citation needed]In the latter case, the above formula simplifies to 0 whennis odd, and to
whennis even, whereΠ2isHardy–Littlewood's twin prime constant
This is sometimes known as theextended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to thetwin primeconjecture, and the two conjectures are believed to be of roughly comparable difficulty.
TheGoldbach partition functionis the function that associates to each even integer the number of ways it can be decomposed into a sum of two primes. Its graph looks like acometand is therefore calledGoldbach's comet.[30]
Goldbach's comet suggests tight upper and lower bounds on the number of representations of an even number as the sum of two primes, and also that the number of these representations depend strongly on the value modulo 3 of the number.
Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using agreedy algorithmthat uses the largest possible prime at each step. ThePillai sequencetracks the numbers requiring the largest number of primes in their greedy representations.[31]
Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares:
Goldbach's conjecture is used when studying computation complexity.[37]The connection is made through theBusy Beaverfunction, where BB(n) is the maximum number of steps taken by anynstateTuring machinethat halts. There is a 27-state Turing machine that halts if and only if Goldbach's conjecture is false.[37]Hence if BB(27) was known, and the Turing machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(27) will be very hard to compute, at least as difficult as settling the Goldbach conjecture.
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Inrepresentation theory, a branch of mathematics, theKostant partition function, introduced byBertram Kostant(1958,1959), of aroot systemΔ{\displaystyle \Delta }is the number of ways one can represent a vector (weight) as a non-negative integer linear combination of thepositive rootsΔ+⊂Δ{\displaystyle \Delta ^{+}\subset \Delta }. Kostant used it to rewrite theWeyl character formulaas a formula (theKostant multiplicity formula) for themultiplicityof a weight of anirreducible representationof asemisimple Lie algebra. An alternative formula, that is more computationally efficient in some cases, isFreudenthal's formula.
The Kostant partition function can also be defined forKac–Moody algebrasand has similar properties.
Consider the A2 root system, with positive rootsα1{\displaystyle \alpha _{1}},α2{\displaystyle \alpha _{2}}, andα3:=α1+α2{\displaystyle \alpha _{3}:=\alpha _{1}+\alpha _{2}}. If an elementμ{\displaystyle \mu }can be expressed as a non-negative integer linear combination ofα1{\displaystyle \alpha _{1}},α2{\displaystyle \alpha _{2}}, andα3{\displaystyle \alpha _{3}}, then sinceα3=α1+α2{\displaystyle \alpha _{3}=\alpha _{1}+\alpha _{2}}, it can also be expressed as a non-negative integer linear combination of the positivesimple rootsα1{\displaystyle \alpha _{1}}andα2{\displaystyle \alpha _{2}}:
withn1{\displaystyle n_{1}}andn2{\displaystyle n_{2}}being non-negative integers. This expression givesoneway to writeμ{\displaystyle \mu }as a non-negative integer combination of positive roots; other expressions can be obtained by replacingα1+α2{\displaystyle \alpha _{1}+\alpha _{2}}withα3{\displaystyle \alpha _{3}}some number of times. We can do the replacementk{\displaystyle k}times, where0≤k≤min(n1,n2){\displaystyle 0\leq k\leq \mathrm {min} (n_{1},n_{2})}. Thus, if the Kostant partition function is denoted byp{\displaystyle p}, we obtain the formula
This result is shown graphically in the image at right. If an elementμ{\displaystyle \mu }is not of the formμ=n1α1+n2α2{\displaystyle \mu =n_{1}\alpha _{1}+n_{2}\alpha _{2}}, thenp(μ)=0{\displaystyle p(\mu )=0}.
The partition function for the other rank 2 root systems are more complicated but are known explicitly.[1][2]
For B2, the positive simple roots areα1=(1,0),α2=(0,1){\displaystyle \alpha _{1}=(1,0),\alpha _{2}=(0,1)}, and the positive roots are the simple roots together withα3=(1,1){\displaystyle \alpha _{3}=(1,1)}andα4=(2,1){\displaystyle \alpha _{4}=(2,1)}. The partition function can be viewed as a function of two non-negative integersn1{\displaystyle n_{1}}andn2{\displaystyle n_{2}}, which represent the elementn1α1+n2α2{\displaystyle n_{1}\alpha _{1}+n_{2}\alpha _{2}}. Then the partition functionP(n1,n2){\displaystyle P(n_{1},n_{2})}can be defined piecewise with the help of two auxiliary functions.
Ifn1≤n2{\displaystyle n_{1}\leq n_{2}}, thenP(n1,n2)=b(n1){\displaystyle P(n_{1},n_{2})=b(n_{1})}. Ifn2≤n1≤2n2{\displaystyle n_{2}\leq n_{1}\leq 2n_{2}}, thenP(n1,n2)=q2(n2)−b(2n2−n1−1)=b(n1)−q2(n1−n2−1){\displaystyle P(n_{1},n_{2})=q_{2}(n_{2})-b(2n_{2}-n_{1}-1)=b(n_{1})-q_{2}(n_{1}-n_{2}-1)}. If2n2≤n1{\displaystyle 2n_{2}\leq n_{1}}, thenP(n1,n2)=q2(n2){\displaystyle P(n_{1},n_{2})=q_{2}(n_{2})}. The auxiliary functions are defined forn≥1{\displaystyle n\geq 1}and are given byq2(n)=12(n+1)(n+2){\displaystyle q_{2}(n)={\frac {1}{2}}(n+1)(n+2)}andb(n)=14(n+2)2{\displaystyle b(n)={\frac {1}{4}}(n+2)^{2}}forn{\displaystyle n}even,14(n+1)(n+3){\displaystyle {\frac {1}{4}}(n+1)(n+3)}forn{\displaystyle n}odd.
For G2, the positive roots are(1,0),(0,1),(1,1),(2,1),(3,1){\displaystyle (1,0),(0,1),(1,1),(2,1),(3,1)}and(3,2){\displaystyle (3,2)}, with(1,0){\displaystyle (1,0)}denoting the short simple root and(0,1){\displaystyle (0,1)}denoting the long simple root.
The partition function is defined piecewise with the domain divided into five regions, with the help of two auxiliary functions.
For each rootα{\displaystyle \alpha }and eachH∈h{\displaystyle H\in {\mathfrak {h}}}, we canformallyapply the formula for the sum of a geometric series to obtain
where we do not worry about convergence—that is, the equality is understood at the level of formalpower series. UsingWeyl's denominator formula
we obtain a formal expression for the reciprocal of the Weyl denominator:[3]
Here, the first equality is by taking a product over the positive roots of the geometric series formula and the second equality is by counting all the ways a given exponentialeμ(H){\displaystyle e^{\mu (H)}}can occur in the product. The functionℓ(w){\displaystyle \ell (w)}is zero if the argument is a rotation and one if the argument is a reflection.
This argument shows that we can convert theWeyl character formulafor the irreducible representation with highest weightλ{\displaystyle \lambda }:
from a quotient to a product:
Using the preceding rewriting of the character formula, it is relatively easy to write the character as a sum of exponentials. The coefficients of these exponentials are the multiplicities of the corresponding weights. We thus obtain a formula for the multiplicity of a given weightμ{\displaystyle \mu }in the irreducible representation with highest weightλ{\displaystyle \lambda }:[4]
This result is theKostant multiplicity formula.
The dominant term in this formula is the termw=1{\displaystyle w=1}; the contribution of this term isp(λ−μ){\displaystyle p(\lambda -\mu )}, which is just the multiplicity ofμ{\displaystyle \mu }in theVerma modulewith highest weightλ{\displaystyle \lambda }. Ifλ{\displaystyle \lambda }is sufficiently far inside the fundamental Weyl chamber andμ{\displaystyle \mu }is sufficiently close toλ{\displaystyle \lambda }, it may happen that all other terms in the formula are zero. Specifically, unlessw⋅(λ+ρ){\displaystyle w\cdot (\lambda +\rho )}is higher thanμ+ρ{\displaystyle \mu +\rho }, the value of the Kostant partition function onw⋅(λ+ρ)−(μ+ρ){\displaystyle w\cdot (\lambda +\rho )-(\mu +\rho )}will be zero. Thus, although the sum is nominally over the whole Weyl group, in most cases, the number of nonzero terms is smaller than the order of the Weyl group.
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The termsmedical record,health recordandmedical chartare used somewhat interchangeably to describe the systematic documentation of a singlepatient'smedical historyandcareacross time within one particular health care provider's jurisdiction.[1]A medical record includes a variety of types of "notes" entered over time byhealthcare professionals, recording observations and administration of drugs and therapies, orders for the administration of drugs and therapies, test results,X-rays, reports, etc. The maintenance of complete and accurate medical records is a requirement of health care providers and is generally enforced as a licensing or certification prerequisite.
The terms are used for the written (paper notes), physical (image films) and digital records that exist for each individual patient and for the body of information found therein.
Medical records have traditionally been compiled and maintained by health care providers, but advances in online data storage have led to the development ofpersonal health records(PHR) that are maintained by patients themselves, often on third-party websites.[2]This concept is supported by US national health administration entities[3]and byAHIMA, the American Health Information Management Association.[4]
Because many consider the information in medical records to be sensitive private information covered by expectations ofprivacy, manyethicalandlegalissues are implicated in their maintenance, such as third-party access and appropriate storage and disposal.[5]Although the storage equipment for medical records generally is the property of the health care provider, the actual record is considered in most jurisdictions to be the property of the patient, who may obtain copies upon request.[6]
The information contained in the medical record allowshealth care providersto determine the patient's medical history and provide informed care. The medical record serves as the central repository for planning patient care and documenting communication among patient and health care provider and professionals contributing to the patient's care. An increasing purpose of the medical record is to ensure documentation of compliance with institutional, professional or governmental regulation.
The traditional medical record for inpatient care can includeadmission notes,on-service notes,progress notes(SOAP notes),preoperative notes,operative notes,postoperative notes,procedure notes,delivery notes,postpartum notes, anddischarge notes.
Personal health recordscombine many of the above features with portability, thus allowing a patient to share medical records across providers and health care systems.[7]
Electronic medical records could also be studied to quantify disease burdens – such as the number of deaths fromantimicrobial resistance[8]– or help identify causes of, factors of and contributors to diseases,[9][10]especially when combined withgenome-wide association studies.[11][12]For such purposes, electronic medical records could potentially be made available in securely anonymized or pseudonymized[13]forms to ensure patients' privacy is maintained.[14][12][15][16]
A patient's individual medical record identifies the patient and contains information regarding the patient's case history at a particular provider. The health record as well as any electronically stored variant of the traditional paper files contain proper identification of the patient.[17]Further information varies with the individual medical history of the patient.
The contents are generally written with other healthcare professionals in mind. This can result in confusion and hurt feelings when patients read these notes.[18]For example, some abbreviations, such as forshortness of breath, are similar to the abbreviations for profanities, and taking "time out" to followa surgical safety protocolmight be misunderstood asa disciplinary technique for children.[18]
Traditionally, medical records were written on paper and maintained in folders often divided into sections for each type of note (progress note, order, test results), with new information added to each section chronologically. Active records are usually housed at the clinical site, but older records are often archived offsite.
The advent ofelectronic medical recordshas not only changed the format of medical records but has increased accessibility of files. The use of an individual dossier style medical record, where records are kept on each patient by name and illness type originated at theMayo Clinicout of a desire to simplify patient tracking and to allow for medical research.[citation needed]
Maintenance of medical records requires security measures to prevent from unauthorized access or tampering with the records.[citation needed]
Themedical historyis alongitudinalrecord of what has happened to the patient since birth. It chroniclesdiseases, major and minorillnesses, as well asgrowth landmarks. It gives the clinician a feel for what has happened before to the patient. As a result, it may often give clues to current disease state. It includes several subsets detailed below.
Within the medical record, individual medical encounters are marked by discrete summations of a patient's medical history by a physician, nurse practitioner, or physician assistant and can take several forms. Hospital admission documentation (i.e., when a patient requires hospitalization) or consultation by aspecialistoften take an exhaustive form, detailing the entirety of prior health and health care. Routine visits by a provider familiar to the patient, however, may take a shorter form such as theproblem-oriented medical record(POMR), which includes a problem list of diagnoses or a "SOAP" method of documentation for each visit. Each encounter will generally contain the aspects below:
Written orders by medical providers are included in the medical record. These detail the instructions given to other members of the health care team by the primary providers.
When a patient is hospitalized, daily updates are entered into the medical record documenting clinical changes, new information, etc. These often take the form of aSOAP noteand are entered by all members of the health-care team (doctors, nurses, physical therapists, dietitians, clinical pharmacists,respiratory therapists, etc.). They are kept in chronological order and document the sequence of events leading to the current state of health.
The results of testing, such as blood tests (e.g.,complete blood count)radiologyexaminations (e.g.,X-rays),pathology(e.g.,biopsyresults), or specialized testing (e.g.,pulmonary function testing) are included. Often, as in the case of X-rays, a written report of thefindingsis included in lieu of the actual film.
Many other items are variably kept within the medical record. Digital images of the patient, flowsheets from operations/intensive care units,informed consentforms,EKGtracings, outputs from medical devices (such aspacemakers),chemotherapyprotocols, and numerous other important pieces of information form part of the record depending on the patient and his or her set of illnesses/treatments.
Medical records arelegal documentsthat can be used as evidence via asubpoena duces tecum,[20]and are thus subject to the laws of the country/state in which they are produced. As such, there is great variability in rules governing production, ownership, accessibility, and destruction. There is some controversy regarding proof verifying the facts, or absence of facts in the record, apart from the medical record itself.[citation needed]
In 2009, Congress authorized and funded legislation known as theHealth Information Technology for Economic and Clinical Health Act[21]to stimulate the conversion of paper medical records into electronic charts. While many hospitals and doctor's offices have since done this successfully, electronic health vendors' proprietary systems are sometimes incompatible.[22]
Demographicsinclude patient information that is not medical in nature. It is often information to locate the patient, including identifying numbers, addresses, and contact numbers. It may contain information aboutraceandreligionas well as workplace and type ofoccupation. It also contains information regarding the patient'shealth insurance. It is common to also find emergency contact information located in this section of the medical chart.
In theUnited States, written records must be marked with the date and time and scribed with indelible pens without use of corrective paper. Errors in the record should be struck out with a single line (so that the initial entry remains legible) and initialed by the author.[20]Orders and notes must be signed by the author. Electronic versions require anelectronic signature.
Ownership and keeping of patient's records varies from country to country.
In theUnited States, the data contained within the medical record belongs to the patient, whereas the physical form the data takes belongs to the entity responsible for maintaining the record[23]per theHealth Insurance Portability and Accountability Act.[24]Patients have the right to ensure that the information contained in their record is accurate, and can petition their health care provider to amend factually incorrect information in their records.[20][25]
There is no consensus regarding medical record ownership in theUnited States. Factors complicating questions of ownership include the form and source of the information, custody of the information, contract rights, and variation in state law.[26]There is no federal law regarding ownership of medical records.HIPAAgives patients the right to access and amend their own records, but it has no language regarding ownership of the records.[27]Twenty-eight states andWashington, D.C., have no laws that define ownership of medical records. Twenty-one states have laws stating that the providers are the owners of the records. Only one state,New Hampshire, has a law ascribing ownership of medical records to the patient.[28]
UnderCanadian federal law, the patient owns the information contained in a medical record, but the healthcare provider owns the records themselves.[29]The same is true for both nursing home and dental records. In cases where the provider is an employee of a clinic or hospital, it is the employer that has ownership of the records. By law, all providers must keep medical records for a period of 15 years beyond the last entry.[30]
The precedent for the law is the 1992Canadian Supreme Courtruling in McInerney v MacDonald. In that ruling, an appeal by a physician, Dr. Elizabeth McInerney, challenging a patient's access to their own medical record was denied. The patient, Margaret MacDonald, won a court order granting her full access to her own medical record.[31]The case was complicated by the fact that the records were in electronic form and contained information supplied by other providers. McInerney maintained that she didn't have the right to release records she herself did not author. The courts ruled otherwise. Legislation followed, codifying into law the principles of the ruling. It is that legislation which deems providers the owner of medical records, but requires thataccessto the records be granted to the patient themselves.[32]
In theUnited Kingdom, ownership of theNHS's medical records has in the past generally been described as belonging to the Secretary of State for Health[33]and this is taken by some to mean copyright also belongs to the authorities.[34]
In Germany, a relatively new law,[35]which has been established in 2013, strengthens the rights of patients. It states, amongst other things, the statutory duty of medical personnel to document the treatment of the patient in either hard copy or within theelectronic patient record(EPR). This documentation must happen in a timely manner and encompass each and every form of treatment the patient receives, as well as other necessary information, such as the patient's case history, diagnoses, findings, treatment results, therapies and their effects, surgical interventions and their effects, as well as informed consents. The information must include virtually everything that is of functional importance for the actual, but also for future treatment. This documentation must also include the medical report and must be archived by the attending physician for at least 10 years. The law clearly states that these records are not only memory aids for the physicians, but also should be kept for the patient and must be presented on request.
In addition, an electronic health insurance card was issued in January 2014 which is applicable in Germany (Elektronische Gesundheitskarteor eGK), but also in the other member states of the European Union (European Health Insurance Card). It contains data such as: the name of the health insurance company, the validity period of the card, and personal information about the patient (name, date of birth, sex, address, health insurance number) as well information about the patient's insurance status and additional charges. Furthermore, it can contain medical data if agreed to by the patient. This data can include information concerning emergency care, prescriptions, an electronic medical record, and electronic physician's letters. However, due to the limited storage space (32kB), some information is deposited on servers.
In theUnited States, the most basic rules governing access to a medical record dictate that only the patient and the health-care providers directly involved in delivering care have the right to view the record. The patient, however, may grantconsentfor any person or entity to evaluate the record. The full rules regarding access and security for medical records are set forth under the guidelines of theHealth Insurance Portability and Accountability Act(HIPAA).
The rules become more complicated in special situations. A 2018 study found discrepancies in how major hospitals handle record requests, with forms displaying limited information relative to phone conversations.[36]
In the 1992 Canadian Supreme Court ruling in McInerney v. MacDonald gave patients the right to copy and examine all information in their medical records, while the records themselves remained the property of thehealthcare provider.[31]The 2004Personal Health Information Protection Act (PHIPA)contains regulatory guidelines to protect the confidentiality of patient information for healthcare organizations acting as stewards of their medical records.[37]Despite legal precedent for access nationwide, there is still some variance in laws depending on the province. There is also some confusion among providers as to the scope of the patient information they have to give access to, but the language in the supreme court ruling gives patient access rights to their entire record.[38]
In theUnited Kingdom, theData Protection Actsand later theFreedom of Information Act 2000gave patients or their representatives the right to a copy of their record, except where information breaches confidentiality (e.g., information from another family member or where a patient has asked for information not to be disclosed to third parties) or would be harmful to the patient's wellbeing (e.g., some psychiatric assessments). Also, the legislation gives patients the right to check for any errors in their record and insist that amendments be made if required.
In general, entities in possession of medical records are required to maintain those records for a given period. In theUnited Kingdom, medical records are required for the lifetime of a patient and legally for as long as that complaint action can be brought. Generally in the UK, any recorded information should be kept legally for 7 years, but for medical records additional time must be allowed for any child to reach the age of responsibility (20 years). Medical records are required many years after a patient's death to investigate illnesses within a community (e.g., industrial or environmental disease or even deaths at the hands of doctors committing murders, as in theHarold Shipmancase).[39]
Theoutsourcingof medical record transcription and storage has the potential to violate patient–physician confidentiality by possibly allowing unaccountable persons access to patient data. With the increase of clinical notes being shared as a result of the21st Century Cures Act, the increase in sensitive terms used in the records of all patients, including minors, are increasingly shared amongst care teams making privacy more complicated.[40]Intersexpeople have historically had their medical records intentionallyfalsified/concealed, to hidebirth sex, andintersex medical procedures.Christiane Völlingbecame the first intersex person in Europe to successfully sue formedical malpractice.[41]
Falsification of a medical record by a medical professional is afelonyin most United States jurisdictions. Governments have often refused to disclose medical records of military personnel who have been used as experimental subjects.
Given the series of medicaldata breachesand the lack of public trust, some countries have enacted laws requiring safeguards to be put in place to protect the security and confidentiality of medical information as it is shared electronically and to give patients some important rights to monitor their medical records and receive notification for loss and unauthorized acquisition of health information. The United States and the EU have imposed mandatory medical data breach notifications.[42]
Patients' medical information can be shared by a number of people both within the health care industry and beyond. The Health Insurance Portability and Accessibility Act (HIPAA) is a United States federal law pertaining tomedical privacythat went into effect in 2003. This law established standards for patient privacy in all 50 states, including the right of patients to access to their own records. HIPAA provides some protection, but does not resolve the issues involving medical records privacy.[43]
Medical and health care providers experienced 767 security breaches resulting in the compromised confidential health information of 23,625,933 patients during the period of 2006–2012.[44]
The federal Health Insurance Portability and Accessibility Act (HIPAA) addresses the issue of privacy by providing medical information handling guidelines.[45]Not only is it bound by the Code of Ethics of its profession (in the case of doctors and nurses), but also by the legislation on data protection and criminal law. Professional secrecy applies to practitioners, psychologists, nursing, physiotherapists, occupational therapists, nursing assistants, chiropodists, and administrative personnel, as well as auxiliary hospital staff. The maintenance of the confidentiality and privacy of patients implies first of all in the medical history, which must be adequately guarded, remaining accessible only to the authorized personnel. However, the precepts of privacy must be observed in all fields of hospital life: privacy at the time of the conduct of theanamnesisand physical exploration, the privacy at the time of the information to the relatives, the conversations between healthcare providers in the corridors, maintenance of adequate patient data collection in hospital nursing controls (planks, slates), telephone conversations, open intercoms etc.
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TheInternational Certificate of Vaccination or Prophylaxis(ICVP), also known as theCarte JauneorYellow Card, is an officialvaccinationreport created by theWorld Health Organization(WHO).[1]As atravel document, it is a kind ofmedicalpassportthat is recognised internationally and may be required for entry to certain countries where there are increased health risks for travellers.[1]
The ICVP is not animmunity passport; the primary difference is that vaccination certificates such as the ICVP incentivise individuals to obtain vaccination against a disease, while immunity passports incentivise individuals to get infected with and recover from a disease.[2]
Various schemes forhealth passportsorvaccination certificateshave been proposed for people who have beenvaccinated against COVID-19.
The ICVP'snicknameYellow Cardor itsFrenchequivalentCarte Jaunederives from theyellowcolour of the document. The fact thatyellow feveris a commonly required vaccination for travel has contributed to the document's association with the colour yellow, even though the ICVP can cover a wide range of vaccinations and booster shots, not just yellow fever.[1]
The International Certificate of Inoculation and Vaccination was established by theInternational Sanitary Convention for Aerial Navigation (1933)inThe Hague, which came into force on 1 August 1935 and was amended in 1944.[3]Afterthe 1944 amendment, in addition to Personal, Aircraft and Maritime Declarations of Health, the Convention covered five certificates:[4][5]
TheWorld Health Organization(WHO) was formed by its constitution on 22 July 1946, effective on 7 April 1948. The WHO Constitution included stipulationsto stimulate and advance work to eradicate epidemic, endemic and other diseases(Article 2.g) and that theWorld Health Assemblywouldhave authority to adopt regulations concerning sanitary and quarantine requirements and other procedures designed to prevent the international spread of disease(Article 21.a).[6]The Fourth World Health Assembly adopted the International Sanitary Regulations (alias WHO Regulations No. 2) on 25 May 1951,replacing and completingthe earlier International Sanitary Conventions. It confirmed the validity and use of international certificates of vaccination (Article 115), and updated the old model with a new version (Appendices 2, 3, 4).[7]The certificates mentioned were used for proof of vaccination against diseases such as cholera, yellow fever and smallpox; the terminoculationwas no longer used.[7][8]The old International Certificates of Inoculation and Vaccination remained valid until they expired, after which they were replaced by the new ICV.[8]On 23 May 1956, the Ninth World Health Assembly amended the form of the International Certificate of Vaccination or Revaccination against Smallpox per 1 October 1956.[9]
The WHO'sWorld Health Assemblyadopted theInternational Health Regulations(IHR) in 1969, succeeding the previous International Sanitary Conventions/Regulations.[10]IHR Article 79 introduced a model International Certificate of Vaccination, and Appendix 2 and Annex VI stipulated a number of conditions that had to be fulfilled in order for it to be considered valid, such as being printed and filled out in English and French (a third language, relevant to the territory in which it is issued, could be added).[10]The 1969 IHR focused on four diseases: cholera,plague, smallpox, and yellow fever; however, Article 51 specified that vaccination against plague wouldnot be required as a condition of admission of any person to a territory.[10]The World Health Assembly determined in 1973 thatvaccination against cholerawas unable to prevent the introduction of cholera from one country to another,[11]and removed this requirement from the 1973 revision of the IHR;[10][11]it was also removed from the ICV.[11]
The ICV was most successful in the case of smallpox. The mandatory possession of vaccination certificates significantly increased the number of travellers who were vaccinated, and thus contributed to preventing the spread of smallpox, especially when therapid expansion of air travelin the 1960s and 1970s reduced the travelling time from endemic countries to all other countries to just a few hours.[12]After smallpox was successfully eradicated in 1980, the International Certificate of Vaccination against Smallpox was cancelled in 1981, and the new 1983 form lacked any provision for smallpox vaccination.[10][12]Thus, only yellow fever remained as vaccination requirement for international travel for which the ICV was used.[citation needed]
By 1994,Saudi Arabialegally required pilgrims going toMeccafor the annualHajjtovaccinate againstmeningococcal meningitis, while theCenter for Disease Controlalso advisedAmericanstravelling to theAfrican meningitis beltorKenya,TanzaniaandBurundito take the vaccine, especially when visiting during thedry season(November–April).[11]
The2002–2004 SARS outbreakwas the driving force behind the 23 May 2005 revision of the International Health Regulations, which entered into force on 15 June 2007.[13]: 1On that day, the model International Certificate of Vaccination or Prophylaxis contained in Annex 6 of the International Health Regulations (as amended in 2005) replaced the International Certificate of Vaccination or Revaccination against Yellow Fever contained in appendix 2 of the International Health Regulations (1969).[14]
The main portion of the ICVP is a form for physicians to fill out when administering a vaccine. This section is mandated by the WHO's 2005International Health Regulations, in which they provide a model of the document. It includes places for the traveller's name, date of birth, sex, nationality, national identification document, and signature. Below that is a row for each vaccine administered, in which the physician must include the prophylaxis or vaccine administered, date, signature, manufacturer and batch number, dates valid, and an official stamp from the administering centre.[15][13][16]
Below this, the document outlines requirements for validity. The ICVP is only valid for vaccines approved by the WHO.[citation needed]The form must be fully completed in English or French by a medical practitioner or authorized health worker and must include the official stamp of the administering centre. The certificate is valid for as long as the vaccines included are valid.[15][13]
The form may include additional information. In 2007, the WHO prepared a booklet that included the following additional sections.[17]
The notes section includes information aboutyellow fever, since it is the only disease included in the International Health Regulations. It also specifies that the same certificate can be used if any future regulations require vaccination for another disease.[15]
The information for travellers section recommends that travellers consult their physicians to determine appropriate vaccinations before international travel and inform their physician of international travel if they fall ill after their trip.[15]
Malariais a serious disease with no vaccine available. The ICVP recommends that travellers protect against mosquitos through mosquito nets or repellent, as mosquitos can transmit malaria. Travellers can also consult their physician forantimalarial medication, which must be taken regularly for the full duration of the prescription.[15]
The ICVP gives instructions for filling out the certificate. It also gives physicians guidelines for documentingcontraindicationsin cases where a traveller has a medical reason that prevents them from getting a particular vaccine. This section also reminds physicians to consider travel-associated illnesses when treating a patient who has fallen ill after traveling.[15]
Yellow fever is the most common vaccine required for international travel. Many countries require the vaccine for all travellers or only for travellers coming from countries with risk of yellow fever transmission.[19]Exceptions are typically made for newborns until 9 months or one year of age, depending on the country.[20]The ICVP form is valid for yellow fever starting 10 days after vaccination. As of 2016, the vaccine is valid for the life of the traveller. No changes need to be made for those who received their vaccine or ICVP prior to 2016.[21]
In the event that a traveller cannot be vaccinated for a particular disease for medical reasons, their physician can provide them with documentation indicating their condition. They may be subject to additional requirements, such as isolation, quarantine, or observation. A traveller who refuses a vaccine or prophylaxis that is required may be subject to similar requirements or denied entry. In some cases, equivalent military-issued forms are accepted in place of the ICVP, provided the forms include the same information.[13]
Due to the prevalence of counterfeit certificates in some places, several countries, including Zimbabwe, Zambia, and Nigeria, are developing digital certificates that can authenticate an ICVP.[22][23]As of July 2019, Nigeria requires its citizens to have its digital "e-Yellow Card" for travel outside the country. The card has a QR code that can be scanned to verify its validity. This requirement does not affect travellers from other countries with valid ICVPs, but those arriving in Nigeria who haven't been vaccinated for yellow fever may receive the vaccine and the e-Yellow Card upon arrival.[24][25][26]As of September 2023, Ecuador started handing out digital certificates too and is no longer going to issue yellow booklets after they are out of stock.[27]
Similar schemes have been proposed for travellers who have been vaccinated againstCOVID-19.[citation needed]
Multiple agencies and countries were creating different forms of documentation for people who have been vaccinated against COVID-19.[28]Agencies attempting this include non-profit organisations such asWorld Economic Forumand theCommons Project Foundation, technology companies such asIBM, travel companies such asVeriFly, and theInternational Air Transport Association.[28]As of March 2021[update], standards for digital documentation, such as an app on a smartphone, had not been established.[28]On 12 March 2021,Ecma Internationalannounced its intention to create international standards which guard against counterfeiting and protects private data as much as possible in a "Call for Participation on Vaccine Passports International Standardization".[29]
WithCOVID-19 vaccinesshowing promising results, several industry organizations including global airline lobbyIATAand theWorld Economic Forumhave announced pilots.[30]IATA's solution, "Travel Pass", is a mobile app that can display test results, proof of inoculation and will be integrated with the existingTIMATICsystem.[31]
Israel employed a digital "green pass" to allow individuals fully vaccinated against COVID-19 to dine out, attend concerts, and travel to other nations.[32]It has been the subject of several privacy and data security concerns. Shortly after the scheme was rolled out, theKnessetpassed a law allowing local authorities to compile data on citizens who have refused to get vaccinated.[33]
Work has been started to established and standardize at Ecma International, allow for an open interoperability ecosystem so that multiple COVID-19 immunity verification systems can work together and effectively across borders.[34]
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Vaccination requirements for international travelare the aspect ofvaccination policythat concerns themovement of peopleacrossborders. Countries around the world require travellers departing to other countries, or arriving from other countries, to bevaccinatedagainst certaininfectiousdiseasesin order to preventepidemics. Atborder checks, these travellers are required to show proof of vaccination against specific diseases; the most widely used vaccination record is theInternational Certificate of Vaccination or Prophylaxis (ICVP or Carte Jaune/Yellow Card). Some countries require information about a passenger's vaccination status in apassenger locator form.[citation needed]
The first International Certificate of Vaccination againstSmallpoxwas developed by the 1944 International Sanitary Convention[1](itself an amendment of the 1926 International Sanitary Convention on Maritime Navigation and the1933 International Sanitary Convention for Aerial Navigation).[2]The initial certificate was valid for a maximum of three years.[1]
The policy had a few flaws: the smallpox vaccination certificates were not always checked by qualified airport personnel, or when passengers transferred at airports in smallpox-free countries. Travel agencies mistakenly provided certificates to some unvaccinated customers, and there were some instances of falsified documents. Lastly, a small number of passengers carrying valid certificates still contracted smallpox because they were improperly vaccinated. However, all experts agree that the mandatory possession of vaccination certificates significantly increased the number of travellers who were vaccinated, and thus contributed to preventing the spread of smallpox, especially when therapid expansion of air travelin the 1960s and 1970s reduced the travelling time from endemic countries to all other countries to just a few hours.[1]
After smallpox was successfully eradicated in 1980, the International Certificate of Vaccination against Smallpox was cancelled in 1981, and the new 1983 form lacked any provision for smallpox vaccination.[1]
Travellers who wish to enter certain countries or territories must be vaccinated against yellow fever ten days before crossing the border, and be able to present a vaccination record/certificate at the border checks.[3]: 45In most cases, this travel requirement depends on whether the country they are travelling from has been designated by the World Health Organization as being a "country with risk of yellow fever transmission". In a few countries, it does not matter which country the traveller comes from: everyone who wants to enter these countries must be vaccinated against yellow fever. There are exemptions for newborn children; in most cases, any child who is at least nine months or one year old needs to be vaccinated.[4]
Travellers who wish to enter or leave certain countries must be vaccinated against polio, usually at most twelve months and at least four weeks before crossing the border, and be able to present a vaccination record/certificate at the border checks.[3]: 25–27Most requirements apply only to travel to or from so-called polio-endemic, polio-affected, polio-exporting, polio-transmission, or "high-risk" countries.[4]As of August 2020, Afghanistan and Pakistan are the only polio-endemic countries in the world (wherewild polio has not yet been eradicated).[5]Several countries have additional precautionary polio vaccination travel requirements, for example to and from "key at-risk countries", which as of December 2020 include China, Indonesia, Mozambique, Myanmar, and Papua New Guinea.[4][6]
Travellers who wish to enter or leave certain countries or territories must be vaccinated against meningococcal meningitis, preferably 10–14 days before crossing the border, and be able to present a vaccination record/certificate at the border checks.[3]: 21–24Countries with required meningococcal vaccination for travellers includeThe Gambia,Indonesia,Lebanon,Libya, thePhilippines, and most importantly and extensivelySaudi Arabiafor Muslims visiting or working inMeccaandMedinaduring theHajjorUmrahpilgrimages.[4]For some countries inAfrican meningitis belt, vaccinations prior to entry are not required, but highly recommended.[3]: 21–24
During theCOVID-19 pandemic, severalCOVID-19 vaccineswere developed, and in December 2020 the first vaccination campaign was planned.[8]
Anticipating the vaccine, on 23 November 2020,Qantasannounced that the company would ask for proof of COVID-19 vaccination from international travellers. According to Alan Joyce, the firm's CEO, a coronavirus vaccine would become a "necessity" when travelling, "We are looking at changing our terms and conditions to say for international travellers, we will ask people to have a vaccination before they can get on the aircraft."[9]Australian Prime Minister Scott Morrison subsequently announced that all international travellers who fly to Australia without proof of a COVID-19 vaccination will be required to quarantine at their own expense.[7]Victoria PremierDaniel Andrewsand the CEOs ofMelbourne Airport,Brisbane AirportandFlight Centreall supported the Morrison government's "no jab, no fly" policy, with onlySydney Airport's CEO suggesting advanced testing might also be sufficient to eliminate quarantine in the future.[10]TheInternational Air Transport Association(IATA) announced that it was almost finished with developing a digital health pass which states air passengers' COVID-19 testing and vaccination information to airlines and governments.[11]
Korean AirandAir New Zealandwere seriously considering mandatory vaccination as well, but would negotiate it with their respective governments.[12]KLM CEO Pieter Elbers responded on 24 November that KLM does not yet have any plans for mandatory vaccination on its flights.[13]Brussels Airlines and Lufthansa said they had no plans yet on requiring passengers to present proof of vaccination before boarding, but Brussels Airport CEO Arnaud Feist agreed with Qantas' policy, stating: "Sooner or later, having proof of vaccination or a negative test will become compulsory."[14]Ryanair announced it would not require proof of vaccination for air travel within the EU, EasyJet stated it would not require any proof at all.The Irish Timescommented that a vaccination certificate for flying was quite common in countries around the world for other diseases, such as foryellow feverin many African countries.[15]
On 25 November, separately from IATA's digital health pass initiative, five major airlines –United Airlines,Lufthansa,Virgin Atlantic,Swiss International Air Lines, andJetBlue– announced the 1December 2020 introduction of the CommonPass, which shows the results of passengers' COVID-19 tests. It was designed as an international standard by theWorld Economic Forumand The Commons Project, and set up in such a way that it could also be used to record vaccination results in the future. It standardises test results and aims to prevent forgery of vaccination records, while storing only limited data on a passenger's phone to safeguard their privacy. The CommonPass had already successfully undergone a trial period in October with United Airlines andCathay Pacific Airways.[16][17]
On 26 November, the Danish Ministry of Health confirmed that it was working on a COVID-19 "vaccine passport" or simply Vaccination card[18]which would likely not only work as proof of vaccination for air travel, but also for other activities such as concerts, private parties and access to various businesses, a perspective welcomed by theConfederation of Danish Industry. The Danish College of General Practitioners also welcomed the project, saying that it doesn't force anyone to vaccinate, but encourages them to do so if they want to enjoy certain privileges in society.[19]
Irish Foreign MinisterSimon Coveneysaid on 27 November 2020 that, although he "currently has no plans" for a passport vaccination stamp, his government was working on changing thepassenger locator formto include proof of PCR negative tests for the coronavirus, and that it was likely to be further adjusted to include vaccination data when a COVID-19 vaccine would become available. Coveney stressed that "We do not want, following enormous efforts and sacrifices from people, to reintroduce the virus again through international travel, which is a danger if it is not managed right."[20]
TheIATA Travel Passapplication for smartphone has been developed by the International Air Transport Association (IATA) in early 2021. Themobile appstandardizes the health verification process confirming whether passengers have been vaccinated against, or tested negative for, COVID-19 prior to travel. Passengers will use the app to create a digital passport linked to their e-passport, receive test results and vaccination details from laboratories, and share that information with airlines and authorities. The application is intended to replace the existing paper-based method of providing proof of vaccination in international travel, colloquially known as theYellow Card. Trials of the application are carried out by a number of airlines includingSingapore Airlines,Emirates,Qatar Airways,EtihadandAir New Zealand.[21][22]
It has been opined that many countries will increasingly consider the vaccination status of travellers[23]when deciding to allow them entry or not or require them toquarantine[24]since recently published research shows that thePfizer vaccineeffect lasts for at least six months.[25]
Various vaccines are not legally required for travellers, but highly recommended by the World Health Organization.[3]For example, for areas with risk of meningococcal meningitis infection in countries inAfrican meningitis belt, vaccinations prior to entry are not required by these countries, but nevertheless highly recommended by the WHO.[3]: 21–24
As of July 2019,ebola vaccinesandmalaria vaccineswere still in development and not yet recommended for travellers.[3]: 4Instead, the WHO recommends various other means of prevention, including several forms ofchemoprophylaxis, in areas where there is a significant risk of becoming infected with malaria.[26]: 4–5
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https://en.wikipedia.org/wiki/Vaccination_requirements_for_international_travel
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Vaccination requirements for international travelare the aspect ofvaccination policythat concerns themovement of peopleacrossborders. Countries around the world require travellers departing to other countries, or arriving from other countries, to bevaccinatedagainst certaininfectiousdiseasesin order to preventepidemics. Atborder checks, these travellers are required to show proof of vaccination against specific diseases; the most widely used vaccination record is theInternational Certificate of Vaccination or Prophylaxis (ICVP or Carte Jaune/Yellow Card). Some countries require information about a passenger's vaccination status in apassenger locator form.[citation needed]
The first International Certificate of Vaccination againstSmallpoxwas developed by the 1944 International Sanitary Convention[1](itself an amendment of the 1926 International Sanitary Convention on Maritime Navigation and the1933 International Sanitary Convention for Aerial Navigation).[2]The initial certificate was valid for a maximum of three years.[1]
The policy had a few flaws: the smallpox vaccination certificates were not always checked by qualified airport personnel, or when passengers transferred at airports in smallpox-free countries. Travel agencies mistakenly provided certificates to some unvaccinated customers, and there were some instances of falsified documents. Lastly, a small number of passengers carrying valid certificates still contracted smallpox because they were improperly vaccinated. However, all experts agree that the mandatory possession of vaccination certificates significantly increased the number of travellers who were vaccinated, and thus contributed to preventing the spread of smallpox, especially when therapid expansion of air travelin the 1960s and 1970s reduced the travelling time from endemic countries to all other countries to just a few hours.[1]
After smallpox was successfully eradicated in 1980, the International Certificate of Vaccination against Smallpox was cancelled in 1981, and the new 1983 form lacked any provision for smallpox vaccination.[1]
Travellers who wish to enter certain countries or territories must be vaccinated against yellow fever ten days before crossing the border, and be able to present a vaccination record/certificate at the border checks.[3]: 45In most cases, this travel requirement depends on whether the country they are travelling from has been designated by the World Health Organization as being a "country with risk of yellow fever transmission". In a few countries, it does not matter which country the traveller comes from: everyone who wants to enter these countries must be vaccinated against yellow fever. There are exemptions for newborn children; in most cases, any child who is at least nine months or one year old needs to be vaccinated.[4]
Travellers who wish to enter or leave certain countries must be vaccinated against polio, usually at most twelve months and at least four weeks before crossing the border, and be able to present a vaccination record/certificate at the border checks.[3]: 25–27Most requirements apply only to travel to or from so-called polio-endemic, polio-affected, polio-exporting, polio-transmission, or "high-risk" countries.[4]As of August 2020, Afghanistan and Pakistan are the only polio-endemic countries in the world (wherewild polio has not yet been eradicated).[5]Several countries have additional precautionary polio vaccination travel requirements, for example to and from "key at-risk countries", which as of December 2020 include China, Indonesia, Mozambique, Myanmar, and Papua New Guinea.[4][6]
Travellers who wish to enter or leave certain countries or territories must be vaccinated against meningococcal meningitis, preferably 10–14 days before crossing the border, and be able to present a vaccination record/certificate at the border checks.[3]: 21–24Countries with required meningococcal vaccination for travellers includeThe Gambia,Indonesia,Lebanon,Libya, thePhilippines, and most importantly and extensivelySaudi Arabiafor Muslims visiting or working inMeccaandMedinaduring theHajjorUmrahpilgrimages.[4]For some countries inAfrican meningitis belt, vaccinations prior to entry are not required, but highly recommended.[3]: 21–24
During theCOVID-19 pandemic, severalCOVID-19 vaccineswere developed, and in December 2020 the first vaccination campaign was planned.[8]
Anticipating the vaccine, on 23 November 2020,Qantasannounced that the company would ask for proof of COVID-19 vaccination from international travellers. According to Alan Joyce, the firm's CEO, a coronavirus vaccine would become a "necessity" when travelling, "We are looking at changing our terms and conditions to say for international travellers, we will ask people to have a vaccination before they can get on the aircraft."[9]Australian Prime Minister Scott Morrison subsequently announced that all international travellers who fly to Australia without proof of a COVID-19 vaccination will be required to quarantine at their own expense.[7]Victoria PremierDaniel Andrewsand the CEOs ofMelbourne Airport,Brisbane AirportandFlight Centreall supported the Morrison government's "no jab, no fly" policy, with onlySydney Airport's CEO suggesting advanced testing might also be sufficient to eliminate quarantine in the future.[10]TheInternational Air Transport Association(IATA) announced that it was almost finished with developing a digital health pass which states air passengers' COVID-19 testing and vaccination information to airlines and governments.[11]
Korean AirandAir New Zealandwere seriously considering mandatory vaccination as well, but would negotiate it with their respective governments.[12]KLM CEO Pieter Elbers responded on 24 November that KLM does not yet have any plans for mandatory vaccination on its flights.[13]Brussels Airlines and Lufthansa said they had no plans yet on requiring passengers to present proof of vaccination before boarding, but Brussels Airport CEO Arnaud Feist agreed with Qantas' policy, stating: "Sooner or later, having proof of vaccination or a negative test will become compulsory."[14]Ryanair announced it would not require proof of vaccination for air travel within the EU, EasyJet stated it would not require any proof at all.The Irish Timescommented that a vaccination certificate for flying was quite common in countries around the world for other diseases, such as foryellow feverin many African countries.[15]
On 25 November, separately from IATA's digital health pass initiative, five major airlines –United Airlines,Lufthansa,Virgin Atlantic,Swiss International Air Lines, andJetBlue– announced the 1December 2020 introduction of the CommonPass, which shows the results of passengers' COVID-19 tests. It was designed as an international standard by theWorld Economic Forumand The Commons Project, and set up in such a way that it could also be used to record vaccination results in the future. It standardises test results and aims to prevent forgery of vaccination records, while storing only limited data on a passenger's phone to safeguard their privacy. The CommonPass had already successfully undergone a trial period in October with United Airlines andCathay Pacific Airways.[16][17]
On 26 November, the Danish Ministry of Health confirmed that it was working on a COVID-19 "vaccine passport" or simply Vaccination card[18]which would likely not only work as proof of vaccination for air travel, but also for other activities such as concerts, private parties and access to various businesses, a perspective welcomed by theConfederation of Danish Industry. The Danish College of General Practitioners also welcomed the project, saying that it doesn't force anyone to vaccinate, but encourages them to do so if they want to enjoy certain privileges in society.[19]
Irish Foreign MinisterSimon Coveneysaid on 27 November 2020 that, although he "currently has no plans" for a passport vaccination stamp, his government was working on changing thepassenger locator formto include proof of PCR negative tests for the coronavirus, and that it was likely to be further adjusted to include vaccination data when a COVID-19 vaccine would become available. Coveney stressed that "We do not want, following enormous efforts and sacrifices from people, to reintroduce the virus again through international travel, which is a danger if it is not managed right."[20]
TheIATA Travel Passapplication for smartphone has been developed by the International Air Transport Association (IATA) in early 2021. Themobile appstandardizes the health verification process confirming whether passengers have been vaccinated against, or tested negative for, COVID-19 prior to travel. Passengers will use the app to create a digital passport linked to their e-passport, receive test results and vaccination details from laboratories, and share that information with airlines and authorities. The application is intended to replace the existing paper-based method of providing proof of vaccination in international travel, colloquially known as theYellow Card. Trials of the application are carried out by a number of airlines includingSingapore Airlines,Emirates,Qatar Airways,EtihadandAir New Zealand.[21][22]
It has been opined that many countries will increasingly consider the vaccination status of travellers[23]when deciding to allow them entry or not or require them toquarantine[24]since recently published research shows that thePfizer vaccineeffect lasts for at least six months.[25]
Various vaccines are not legally required for travellers, but highly recommended by the World Health Organization.[3]For example, for areas with risk of meningococcal meningitis infection in countries inAfrican meningitis belt, vaccinations prior to entry are not required by these countries, but nevertheless highly recommended by the WHO.[3]: 21–24
As of July 2019,ebola vaccinesandmalaria vaccineswere still in development and not yet recommended for travellers.[3]: 4Instead, the WHO recommends various other means of prevention, including several forms ofchemoprophylaxis, in areas where there is a significant risk of becoming infected with malaria.[26]: 4–5
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https://en.wikipedia.org/wiki/Vaccination_requirements_for_international_travel#COVID-19
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Apassenger name record(PNR) is a record in the database of acomputer reservation system(CRS) that contains the itinerary for a passenger or a group of passengers travelling together. The concept of a PNR was first introduced byairlinesthat needed to exchange reservation information in case passengers required flights of multiple airlines to reach their destination ("interlining"). For this purpose,IATAandATAhave defined standards for interline messaging of PNR and other data through the "ATA/IATA Reservations Interline Message Procedures - Passenger" (AIRIMP). There is no general industry standard for the layout and content of a PNR. In practice, each CRS or hosting system has its own proprietary standards, although common industry needs, including the need to map PNR data easily to AIRIMP messages, has resulted in many general similarities in data content and format between all of the major systems.
When a passenger books an itinerary, the travel agent or travel website user will create a PNR in the computer reservation system it uses. This is typically one of the largeglobal distribution systems, such asAmadeus,Sabre, orTravelport(Apollo, Galileo, and Worldspan) but if the booking is made directly with an airline the PNR can also be in the database of the airline's CRS. This PNR is called the Master PNR for the passenger and the associated itinerary. The PNR is identified in the particular database by arecord locator.
When portions of the travel are not provided by the holder of the master PNR, then copies of the PNR information are sent to the CRSs of the airlines that will be providing transportation. These CRSs will open copies of the original PNR in their own database to manage the portion of the itinerary for which they are responsible. Many airlines have their CRS hosted by one of the GDSs, which allows sharing of the PNR.
The record locators of the copied PNRs are communicated back to the CRS that owns the Master PNR, so all records remain tied together. This allows exchanging updates of the PNR when the status of trip changes in any of the CRSs.
Although PNRs were originally introduced for air travel, airlines systems can now also be used for bookings ofhotels,car rental, airport transfers, andtraintrips.
From a technical point of view, there are five parts of a PNR required before the booking can be completed. They are:
Other information, such as a timestamp and the agency'spseudo-city code, will go into the booking automatically. All entered information will be retained in the "history" of the booking.
Once the booking has been completed to this level, the CRS will issue a unique all alpha or alpha-numeric record locator, which will remain the same regardless of any further changes made (except if a multi-person PNR is split). Each airline will create their own booking record with a unique record locator, which, depending on service level agreement between the CRS and the airline(s) involved, will be transmitted to the CRS and stored in the booking. If an airline uses the same CRS as the travel agency, the record locator will be the same for both.
A considerable amount of other information is often desired by both the airlines and the travel agent to ensure efficient travel. This includes:
In more recent times,[when?]many governments now require the airline to provide further information included assisting investigators tracing criminals or terrorists. These include:
The components of a PNR are identified internally in a CRS by a one-character code. This code is often used when creating a PNR via direct entry into a terminal window (as opposed to using a graphical interface). The following codes are standard across all CRSs based on the original PARS system:
The majority of airlines and travel agencies choose to host theirPNRdatabases with acomputer reservations system(CRS) orglobal distribution system(GDS) company such asSabre,Galileo,WorldspanandAmadeus.[2]
Some privacy organizations are concerned at the amount of personal data that a PNR might contain. While the minimum data for completing a booking is quite small, a PNR will typically contain much more information of a sensitive nature.
This will include the passenger's full name, date of birth, home and work address, telephone number, e-mail address, credit card details, IP address if booked online, as well as the names and personal information of emergency contacts.
Designed to "facilitate easy global sharing of PNR data," the CRS-GDS companies "function both as data warehouses and data aggregators, and have a relationship to travel data analogous to that of credit bureaus to financial data.".[3]A canceled or completed trip does not erase the record since "copies of the PNRs are ‘purged’ from live to archival storage systems, and can be retained indefinitely by CRSs, airlines, and travel agencies."[4]Further, CRS-GDS companies maintain web sites that allow almost unrestricted access to PNR data – often, the information is accessible by just the reservation number printed on the ticket.
Additionally, "[t]hrough billing, meeting, and discount eligibility codes, PNRs contain detailed information on patterns of association between travelers. PNRs can contain religious meal preferences and special service requests that describe details of physical and medical conditions (e.g., "Uses wheelchair, can control bowels and bladder") – categories of information that have special protected status in the European Union and some other countries as“sensitive” personal data.”[5][6]Despite the sensitive character of the information they contain, PNRs are generally not recognized as deserving the same privacy protection afforded to medical and financial records. Instead, they are treated as a form of commercial transaction data.[5]
On 16 January 2004, theArticle 29 Working Partyreleased theirOpinion 1/2004 (WP85)on the level of PNR protection ensured in Australia for the transmission of Passenger Name Record data from airlines.
Customs applies a general policy of non-retention for these data. For those 0.05% to 0.1% of passengers who are referred to Customs for further evaluation, the airline PNR data are temporarily retained, but not stored, pending resolution of the border evaluation. After resolution, their PNR data are erased from the PC of the Customs PAU officer concerned and are not entered into Australian databases.
In 2010 the European Commission'sDirectorate-General for Justice, Freedom and Securitywas split in two. The resulting bodies were theDirectorate-General for Justice (European Commission)and theDirectorate-General for Home Affairs (European Commission).
On 4 May 2011,Stefano Manservisi, Director-General at theDirectorate-General for Home Affairs (European Commission)wrote to theEuropean Data Protection Supervisor(EDPS) with regards to a PNR sharing agreement with Australia,[7]a close ally of the US and signatory to theUKUSA Agreementonsignals intelligence.
The EDPS responded on 5 May inLetter 0420 D845:[7]
I am writing to you in reply to your letter of 4 May concerning the two draft Proposals for Council Decisions on (i) the conclusion and (ii) the signature of the Agreement between the European Union and Australia on the processing and transfer of Passenger Name Record (PNR) data by air carriers to the Australian Customs and Border Protection Service.We understand that the consultation of the EDPS takes place in the context of a fast track procedure. However,we regret that the time available for us to analyse the Proposal is reduced to a single day. Such a deadline precludes the EDPS from being able to exercise its competences in an appropriate way, even in the context of a file which we have been closely following since 2007.
TheArticle 29 Working PartydocumentOpinion 1/2005 on the level of protection ensured in Canada for the transmission of Passenger Name Record and Advance Passenger Information from airlines (WP 103), 19 January 2005, offers information on the nature of PNR agreements withCanada.Archived2014-11-27 at theWayback Machine.
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https://en.wikipedia.org/wiki/Passenger_name_record
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Apassportis an officialtravel documentissued by a government that certifies a person'sidentityand nationality for international travel.[1]A passport allows its bearer to enter and temporarily reside in a foreign country, access local aid and protection, and obtainconsular assistancefrom their government. In addition to facilitating travel, passports are a key mechanism forborder securityandregulating migration; they may also serve asofficial identificationfor various domestic purposes.
State-issued travel documents have existed in some form since antiquity; the modern passport was universally adopted and standardized in 1920.[2]The passport takes the form of a booklet bearing the official name andemblemof the issuing government and containing the biographical information of the individual, including their full name, photograph, place and date of birth, and signature. A passport does not create any rights in the country being visited nor impose any obligation on the issuing country; rather, it provides certification to foreign government officials of the holder's identity and right to travel, with pages available for insertingentry and exit stampsandtravel visas—endorsements that allow the individual to enter and temporarily reside in a country for a period of time and under certain conditions.
Since 1998, many countries have transitioned tobiometric passports, which contain an embeddedmicrochipto facilitateauthenticationand safeguard againstcounterfeiting.[3]As of July 2024, over 150 jurisdictions issue such "e-passports";[4]previously issued non-biometric passports usually remain valid until expiration.
Eligibility for a passport varies by jurisdiction, althoughcitizenshipis a common prerequisite. However, a passport may be issued to individuals who do not have the status or full rights of citizenship, such asAmericanorBritish nationals. Likewise, certain classes of individuals, such asdiplomatsand government officials, may be issued special passports that provide certain rights and privileges, such asimmunity from arrest or prosecution.[3]
While passports are typically issued by national governments, certain subnational entities are authorised to issue passports tocitizensresiding within their borders.[a]Additionally, othertypes of official documentsmay serve a similar role to passports but are subject to different eligibility requirements, purposes, or restrictions.
Etymologicalsources[example needed]show that the term "passport" may derive from a document required by some medieval Italian states in order for an individual to pass through the physical harbor (Italianpassa porto, "to pass the harbor") or gate (Italianpassa porte, "to pass the gates") of a walled city or jurisdiction.[5][6]Such documents were issued by local authorities to foreign travellers—as opposed to local citizens, as is the modern practice—and generally contained a list of towns and cities the document holder was permitted to enter or pass through. On the whole, documents were not required for travel to seaports, which were consideredopen trading points, but documents were required to pass harbor controls and travel inland from seaports.[7]The transition from private to state control over movement was an essential aspect of the transition fromfeudalismtocapitalism. Communal obligations to providepoor reliefwere an important source of the desire for controls on movement.[8]:10
One of the earliest known references to paperwork that served an analogous role to a passport is found in theHebrew Bible.Nehemiah2:7–9, dating from approximately 450 BC, states thatNehemiah, an official serving KingArtaxerxes I of Persia, asked permission to travel toJudea; the king granted leave and gave him a letter "to the governors beyond the river" requesting safe passage for him as he traveled through their lands.[9]
The ancient Indian political textArthashastra(third century BCE) mentions passes issued at the rate of onemashaper pass to enter and exit the country, and describes the duties of theMudrādhyakṣa(lit.'Superintendent of Seals') who must issue sealed passes before a person could enter or leave the countryside.[10]
Passports were an important part of the Chinese bureaucracy as early as theWestern Han(202 BC – 9 AD), if not in theQin dynasty. They required such details as age, height, and bodily features.[11]These passports (傳;zhuan) determined a person's ability to move throughout imperial counties and through points of control. Even children needed passports, but those of one year or less who were in their mother's care may not have needed them.[11]
In the medievalIslamic Caliphate, a form of passport was thebara'a, areceiptfor taxes paid. Only people who paid theirzakah(forMuslims) orjizya(fordhimmis) taxes were permitted to travel to different regions of the Caliphate; thus, thebara'areceipt was a "basic passport".[12]
In the12th century, theRepublic of Genoaissued a document calledBulletta, which was issued to the nationals of the Republic who were traveling to the ports of the emporiums and the ports of the Genoese colonies overseas, as well as to foreigners who entered them.
KingHenry V of Englandis credited with having invented what some consider the first British passport in the modern sense, as a means of helping his subjects prove who they were in foreign lands. The earliest reference to these documents is found in a1414 Act of Parliament.[13][14]In 1540, granting travel documents in England became a role of thePrivy Council of England, and it was around this time that the term "passport" was used. In 1794, issuing British passports became the job of the Office of theSecretary of State.[13]In theHoly Roman Empire, the 1548 ImperialDiet of Augsburgrequired the public to hold imperial documents for travel, at the risk of permanent exile.[15]
In 1791,Louis XVImasqueraded as a valet during hisFlight to Varennesas passports for the nobility typically included a number of persons listed by their function but without further description.[8]:31–32
A Pass-Card Treaty of October 18, 1850 among German states standardized information including issuing state, name, status, residence, and description of bearer. Tramping journeymen and jobseekers of all kinds were not to receive pass-cards.[8]:92–93
A rapid expansion ofrailway infrastructureand wealth in Europe beginning in the mid-nineteenth century led to large increases in the volume of international travel and a consequent unique dilution of the passport system for approximately thirty years prior toWorld War I. The speed of trains, as well as the number of passengers that crossed multiple borders, made enforcement of passport laws difficult. The general reaction was the relaxation of passport requirements.[16]In the later part of the nineteenth century and up to World War I, passports were not required, on the whole, for travel within Europe, and crossing a border was a relatively straightforward procedure. Consequently, comparatively few people held passports.
During World War I, European governments introduced border passport requirements for security reasons, and to control the emigration of people with useful skills. These controls remained in place after the war, becoming a standard, though controversial, procedure. British tourists of the 1920s complained, especially about attached photographs and physical descriptions, which they considered led to a "nasty dehumanisation".[17]TheBritish Nationality and Status of Aliens Actwas passed in 1914, clearly defining the notions ofcitizenshipand creating a booklet form of the passport.
In 1920, theLeague of Nationsheld a conference on passports, theParis Conference on Passports & Customs Formalities and Through Tickets.[18]Passport guidelines and a general booklet design resulted from the conference,[19]which was followed up by conferences in 1926 and 1927.[20]TheLeague of NationsissuedNansen passportstostatelessrefugeesfrom 1922 to 1938.[21]
While the United Nations held a travel conference in 1963, no passport guidelines resulted from it. Passport standardization came about in 1980, under the auspices of theICAO. ICAO standards include those formachine-readable passports.[22]Such passports have an area where some of the information otherwise written in textual form is written as strings of alphanumeric characters, printed in a manner suitable foroptical character recognition. This enables border controllers and other law enforcement agents to process these passports more quickly, without having to input the information manually into a computer. ICAO publishes Doc 9303Machine Readable Travel Documents, the technical standard for machine-readable passports.[23]A more recent standard is forbiometric passports. These containbiometricsto authenticate the identity of travellers. The passport's critical information is stored on a smallRFIDcomputer chip, much like information stored onsmartcards. Like some smartcards, the passport booklet design calls for an embedded contactless chip that is able to holddigital signaturedata to ensure the integrity of the passport and the biometric data.
Historically, legal authority to issue passports is founded on the exercise of each country's executive discretion. Certain legal tenets follow, namely: first, passports are issued in the name of the state; second, no person has a legal right to be issued a passport; third, each country's government, in exercising its executive discretion, has complete and unfettered discretion to refuse to issue or to revoke a passport; and fourth, that the latter discretion is not subject to judicial review. However, legal scholars including A.J. Arkelian have argued that evolutions in both the constitutional law of democratic countries and the international law applicable to all countries now render those historical tenets both obsolete and unlawful.[24][25]
Governments around the world issue a variety of passports for different purposes. The most common variety are ordinary passports issued to individualcitizensand othernationals. In the past, certain countries issued collective passports[b]or family passports.[c]Today, passports are typically issued to individual travellers rather than groups. Aside from ordinary passports issued to citizens by national governments, there are a variety of other types of passports by governments in specific circumstances.
While individuals are typically only permitted to hold one passport, certain governments permit citizens to hold more than one ordinary passport.[d]Individuals may also simultaneously hold an ordinary passport and an official or diplomatic passport.
Emergency passports (also called temporary passports) are issued to persons with urgent need to travel who do not have passports, e.g. someone abroad whose passport has been lost or stolen who needs to travel home within a few days, someone whose passport expires abroad, or someone who urgently needs to travel abroad who does not have a passport with sufficient validity. These passports are intended for very short durations, e.g. to allow immediate one-way travel back to the home country.Laissez-passerare also used for this purpose.[28]Uniquely, the United Kingdom issues emergency passports to citizens of certainCommonwealth stateswho lose their passports in non-Commonwealth countries where their home state does not maintain a diplomatic or consular mission.
Pursuant to theVienna Convention on Diplomatic Relations,Vienna Convention on Consular Relations, and theimmunity afforded to officials of a foreign stateundercustomary international law, diplomats and other individuals travelling on government business are entitled to reduced scrutiny atborder checkpointswhen travelling overseas. Consequently, such individuals are typically issued special passports indicating their status. These passports come in three distinct varieties:
Unlike most countries, the United Kingdom and the Republic of China issue various categories of passports to individuals without the right of abode in their territory. In the United Kingdom's case, these passports are typically issued to individuals connected with a former British colony while, in the ROC's case, these passports are the result of the legal distinction between ROC nationals with and without residence in the area it administers.[f]In both cases, holders of such passports are able to obtain residence on an equal footing with foreigners by applying forindefinite leave to remain(UK) or aresident certificate(ROC).
ARepublic of China citizenwho does not havehousehold registration(Chinese:戶籍;pinyin:hùjí) in the area administered by the ROC[f]is classified as a National Without Household Registration (NWOHR;Chinese:無戶籍國民) and is subject to immigration controls when clearing ROC border controls, does not have automatic residence rights, and cannot vote inTaiwanese elections. However, they are exempt fromconscription. Most individuals with this status are children born overseas to ROC citizens who do hold household registration. Additionally, because the ROC observes the principle ofjus sanguinis, members of theoverseas Chinesecommunity are also regarded as citizens.[32]During theCold War, both the ROC and PRC governments actively sought the support of overseas Chinese communities in their attempts to secure the position as the legitimate sole government of China. The ROC also encouraged overseas Chinese businessmen to settle in Taiwan to facilitate economic development and regulations concerning evidence of ROC nationality by descent were particularly lax during the period, allowing many overseas Chinese the right to settle in Taiwan.[33]About 60,000 NWOHRs currently holdTaiwanese passportswith this status.[34]
The United Kingdom issues several similar but distinct passports which correspond to the country's several categories of nationality. FullBritish citizensare issued a standardBritish passport. British citizens resident in theCrown Dependenciesmay hold variants of the British passport which confirm theirIsle of Man,Jersey, orGuernseyidentity. Many of the other categories of nationality do not grant bearers right of abode in the United Kingdom itself.
British National (Overseas)passports are issued to individuals connected to Hong Kong prior to its return to China.British Overseas Citizenpassports are primarily issued to individuals who did not acquire the citizenship of the colony they were connected to when it obtained independence (or their stateless descendants). British Overseas Citizen passports are also issued to certain categories of Malaysian nationals in Penang and Malacca, and individuals connected to Cyprus as a result of the legislation granting independence to those former British colonies.British Protected Personpassports are issued to otherwise stateless people connected to a former Britishprotectorate.British subjectpassports are issued to otherwise stateless individuals connected toBritish Indiaor to certain categories of Irish citizens (though, in the latter case, they do convey right of abode).
Additionally, individuals connected to aBritish overseas territoryare accordedBritish Overseas Territories citizenshipand may hold passports issued by the governments of their respective territory. All overseas territory citizens are also now eligible for full British citizenship. Each territory maintainsits own criteriafor determining whom it grants right of abode. Consequently, individuals holding BOTC passports are not necessarily entitled to enter or reside in the territory that issued their passport. Most countries distinguish between BOTC and other classes of British nationality for border control purposes. For instance, only Bermudian passport holders with an endorsement stating that they possess right of abode or belonger status in Bermuda are entitled to enter America without an electronic travel authorisation.[35]
Border control policies in many jurisdictions distinguish between holders of passports with and without right of abode, including NWOHRs and holders of the various British passports that do not confer right of abode upon the bearer. Certain jurisdictions may additionally distinguish between holders of such British passports with and withoutindefinite leave to remainin the United Kingdom. NWOHRs do not, for instance, have access to theVisa Waiver Program, or to visa free access to the Schengen Area or Japan. Other countries, such as India which allows all Chinese nationals to apply foreVisas, do not make such a distinction. Notably, while Singapore does permit visa free entry to all categories of British passport holders, it reduces length of stay for British nationals without right of abode in the United Kingdom, but does not distinguish between ROC passport holders with and without household registration.
Until 31 January 2021, holders of British National (Overseas) passports were able to use their UK passports for immigration clearance in Hong Kong[36]and to seek consular protection fromoverseas Chinese diplomatic missions. This was a unique arrangement as it involved a passport issued by one state conferring right of abode (or, more preciselyright to land) in and consular protection from another state. Since that date, the Chinese and Hong Kong governments have prohibited the use of BN(O) passports as travel documents or proof of identity and it; much like British Overseas Citizen, British Protected Person, or ROC NWOHR passports; is not associated with right of abode in any territory. BN(O)s who do not possess Chinese (or any other) nationality are required to use aDocument of Identity for Visa Purposesfor travel.[36]This restriction disproportionally affects ease of travel forpermanent residents of Indian, Pakistani, and Nepali ethnicity,[37]who were not granted Chinese nationality in 1997. As an additional consequence, Hongkongers seeking early pre-retirement withdrawals from theMandatory Provident Fundpension scheme may not use BN(O) passports for identity verification.[38]
Similarly, non-citizensin Latviaandin Estoniaare individuals, primarily of Russian or Ukrainian ethnicity, who are not citizens of Latvia or Estonia, but who have settled during theSoviet occupation, and thus have the right to a special non-citizen passport issued by the government as well as some other specific rights. Approximately two thirds of them areethnic Russians, followed by ethnic Belarusians, ethnic Ukrainians, ethnic Poles and ethnic Lithuanians.[39][40]According to theUN Special Rapporteur, the citizenship and naturalization laws in Latvia "are seen by the Russian community as discriminatory practices".[41]PerRussian visa policy, holders of theEstonian alien's passportor the Latvian non-citizen passport are entitled to visa free entry to Russia, in contrast to Estonian and Latvian citizens who must obtain an electronic visa.
ThePeople's Republic of China(PRC) authorises itsSpecial Administrative RegionsofHong KongandMacauto issue passports to their permanent residents withChinese nationalityunder the "one country, two systems" arrangement. Visa policies imposed by foreign authorities on Hong Kong and Macau permanent residents holding such passports are different from those holding ordinary passports of the People's Republic of China. AHong Kong Special Administrative Region passport(HKSAR passport) andMacau Special Administrative Region passport(MSAR passport) gain visa-free access to many more countries than ordinaryPRC passports.[42]
On 1 July 2011, theMinistry of Foreign Affairs of the People's Republic of Chinalaunched a trial issuance of e-passports for individuals conducting public affairs work overseas on behalf of the Chinese government.[43][44]The face, fingerprints, and otherbiometricfeatures of the passport holder isdigitizedand stored in pre-installed contactlesssmart chip,[45][46]along with "the passport owner's name, sex and personal photo as well as the passport's term of validity and [the] digital certificate of the chip".[47]Ordinary biometric passports were introduced by theMinistry of Public Securityon 15 May 2012.[48]As of January 2015, all new passports issued by China are biometric e-passports, and non-biometric passports are no longer issued.[47]
In 2012, over 38 million Chinese citizens held ordinary passports, comprising only 2.86 percent of the total population at the time.[49]In 2014, China issued 16 million passports, ranking first in the world, surpassing the United States (14 million) and India (10 million).[50]The number of ordinary passports in circulation rose to 120 million by October 2016, which was approximately 8.7 percent of the population.[51]As of April 2017 to date, China had issued over 100 million biometric ordinary passports.[52]
The three constituent countries of theDanish Realmhave a common nationality.Denmark properis a member of theEuropean Union, butGreenlandandFaroe Islandsare not. Danish citizens residing in Greenland or Faroe Islands can choose between holding aDanish EU passportand a Greenlandic or Faroese non-EU Danish passport.[53]
As of 21 September 2022, Danish citizens had visa-free or visa on arrival access to 188 countries and territories, thus ranking the Danish passport fifth in the world (tied with the passports ofAustria,the Netherlands, andSweden) according to theHenley Passport Index.[54]According to theWorld Tourism Organization2016 report, the Danish passport is first in the world (tied with Finland, Germany, Italy, Luxembourg, Singapore, and the United Kingdom) in terms of travel freedom, with the mobility index of 160 (out of 215 with no visa weighted by 1, visa on arrival weighted by 0.7, eVisa by 0.5 and traditional visa weighted by 0).[55]
Under Serbian law, people born or otherwise legally settled in Kosovo[g]are considered Serbian nationals and as such they are entitled to a Serbian passport.[56]However, these passports are not issued directly by theSerbian Ministry of Internal Affairsbut by the SerbianCoordination Directorate for Kosovo and Metohijainstead.[57]These particular passports do not allow the holder to enter theSchengen Areawithout a visa.[58][59]
As of August 2023, Serbian citizens had visa-free or visa on arrival access to 138 countries and territories, ranking the Serbian passport 38th overall in terms of travel freedom according to theHenley Passport Index.[60][61]The Serbian passport is one of the 5 passports with the most improved rating globally since 2006, in terms of the number of countries that its holders may visit without a visa.[62][63][64]
Although all U.S. citizens are also U.S. nationals, the reverse is not true. As specified in8 U.S.C.§ 1408, a person whose only connection to the United States is through birth in an outlying possession (which is defined in8 U.S.C.§ 1101asAmerican SamoaandSwains Island, the latter of which is administered as part of American Samoa), or through descent from a person so born, acquires U.S. nationality but not the citizenship. This was formerly the case in a few other current or formerU.S. overseas possessions, i.e. thePanama Canal ZoneandTrust Territory of the Pacific Islands.[65]Thepassportissued to non-citizen nationals contains the endorsement code 9 which states: "THE BEARER IS A UNITED STATES NATIONAL AND NOT A UNITED STATES CITIZEN." on the annotations page.[66]Non-citizen nationals may reside and work in the United States without restrictions, and may apply for citizenship under the same rules as resident aliens. Like resident aliens, they arenot presently allowed by any U.S. state to vote in federal or state elections.
Several entities without a sovereign territory issue documents described as passports, most notablyIroquois League,[67][68]theAboriginal Provisional Governmentin Australia and theSovereign Military Order of Malta.[69]Such documents are not necessarily accepted for entry into a country.
Each country sets its own conditions for the issue of passports.[71]Under the law of most countries, passports are government property, and may be limited or revoked at any time, usually on specified grounds, and possibly subject to judicial review.[72]In many countries, surrender of one's passport is a condition of grantingbailin lieu of imprisonment for a pending criminal trial due to the risk of the person leaving the country.[73]When passport holders apply for a new passport (commonly, due to expiration of the previous passport, insufficient validity for entry to some countries or lack of blank pages), they may be required to surrender the old passport for invalidation. In some circumstances an expired passport is not required to be surrendered or invalidated (for example, if it contains an unexpired visa).
Requirements for passport applicants vary significantly from country to country, with some states imposing stricter measures than others. For example,Pakistanrequires applicants to be interviewed before aPakistani passportwill be granted.[74]When applying for a passport or a national ID card, all Pakistanis are required to sign an oath declaringMirza Ghulam Ahmadto be an impostor prophet and allAhmadisto be non-Muslims.[75]In contrast, individuals holdingBritish National (Overseas)status are legally entitled to hold a passport in that capacity.
Countries withconscriptionornational servicerequirements may impose restrictions on passport applicants who have not yet completed their military obligations. For example, inFinland, male citizens aged 18–30 years must prove that they have completed, or are exempt from,their obligatory military serviceto be granted anunrestricted passport; otherwise a passport is issued valid only until the end of their 28th year, to ensure that they return to carry out military service.[76]Other countries with obligatory military service, such asSouth KoreaandSyria, have similar requirements, e.g.South Korean passportandSyrian passport.[77]
Passports have a limited validity, usually between 5 and 10 years. Many countries require passports to be valid for a minimum of six months beyond the planned date of departure, as well as having at least two to four blank pages.[78]It is recommended that a passport be valid for at least six months from the departure date as many airlines deny boarding to passengers whose passport has a shorter expiry date, even if the destination country does not have such a requirement for incoming visitors.
There is an increasing trend for adult passports to be valid for ten years, such as aUnited Kingdom passport,United States Passport,New Zealand Passport(after 30 November 2015)[79]orAustralian passport.
Some countries issue passports that valid for longer than 10 years, which ICAO does not recommend due to the security concerns and even some countries including all member states of theEuropean Uniondo not accept passports older than 10 years.
Passport booklets from almost all countries around the world display thenational coat of armsof the issuing country on the front cover. The United Nations keeps a record of national coats of arms, but displaying a coat of arms is not an internationally recognised requirement for a passport.
There are several groups of countries that have, by mutual agreement, adopted common designs for their passports:
Passports sometimes contain a message, usually near the front, requesting that the passport's bearer be allowed to pass freely, and further requesting that, in the event of need, the bearer be granted assistance. The message is sometimes made in the name of the government or the head of state, and may be written in more than one language, depending on the language policies of the issuing authority.
In 1920, an international conference on passports and through tickets held by theLeague of Nationsrecommended that passports be issued in theFrench language, historically the language of diplomacy, and one other language.[86]Currently, theICAOrecommends that passports be issued in English, French, and Spanish; or in the national language of the issuing country and in either English, French, or Spanish.[87]Many European countries use their national language, along with English and French.
Some additional language combinations are:
A passport is merely an identity document that is widely recognised for international travel purposes, and the possession of a passport does not in itself entitle a traveller to enter any country other than the country that issued it, and sometimes not even then, as with holders of theBritish Overseas citizenpassport. Many countries normally require visitors to obtain a visa. Each country has different requirements or conditions for the grant of visas, such as for the visitor not being likely to become a public charge for financial, health, family, or other reasons, and the holder not having been convicted of a crime or considered likely to commit one.[89][90]Where a country does not recognise another, or is in dispute with it, entry may be prohibited to holders of passports of the other party to the dispute, and sometimes to others who have, for example, visited the other country; examples are listed below. A country that issues a passport may also restrict its validity or use in specified circumstances, such as use for travel to certain countries for political, security, or health reasons.
Many nations implement border controls restricting the entry of people of certain nationalities or who have visited certain countries. For instance, Georgia refuses entry to holders of passports issued by the Republic of China.[91]Similarly, since April 2017, nationals of Bangladesh, Pakistan, Sudan, Syria, Yemen, and Iran have been banned from entering the parts of eastern Libya under the control of theTobruk government.[91][92][93]The Pakistani passports explicitly mention that these passports are valid in all countries except Israel. The majority ofArabcountries, as well as Iran and Malaysia, ban Israeli citizens;[91]however, exceptional entry to Malaysia is possible with approval from theMinistry of Home Affairs.[94]Certain countries may also restrict entry to those with Israeli stamps or visas in their passports. As a result of tension over the formerRepublic of Artsakhdispute, Azerbaijan currently forbids entry to Armenian citizens as well as to individuals with proof of travel to Artsakh.
Between September 2017 and January 2021, theUnited States of Americadid not issue new visas to nationals of Iran, North Korea, Libya, Somalia, Syria, or Yemen pursuant torestrictionsimposed by theTrump administration,[95]which were subsequently repealed by theBiden administrationon 20 January 2021.[96]While in force, the restrictions were conditional and could be lifted if the countries affected meet the required security standards specified by the Trump administration, anddual citizensof these countries could still enter if they presented a passport from a non-designated country.
One method by which to rank the value of a passport is to calculate its mobility score (MS). The mobility score of a passport is the number of countries that allow the holder of that passport to enter for general tourism visa-free, visa-on-arrival, eTA, or eVisa issued within 3 days. As of 2023, the strongest passport in the world is the Singaporean passport.[97]
However, another way to determine passport mobility score is the number of countries it allows holders to live and work in. For example, by this measure, the Irish passport would be most powerful because it allows the holder to live in all European Union/European Economic Area countries, as well as Switzerland and the United Kingdom, as the Irish passport is the only European Union passport now that still allows its users the right to live/work in the United Kingdom.[citation needed]
1British Overseas Territories.2These countries span the conventional boundary between Europe and Asia.3Partially recognized.4Unincorporated territoryof the United States.5Part of the Kingdom of Denmark.6Egyptspans the boundary between Africa and Asia.
1British Overseas Territories.2Azerbaijan,Georgia,Turkey,Kazakhstan,Russiaand the partially recognised republics ofAbkhaziaandSouth Ossetiaeach span the conventional boundary between Europe and Asia.3Cyprus,Armenia, and the partially recognised republic ofNorthern Cyprusare entirely in Western Asia but have socio-political connections with Europe.4Egyptspans the boundary between Africa and Asia.5Partially recognized.
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Abiometric passport(also known as anelectronic passport,e-passportor adigital passport) is apassportthat has an embedded electronicmicroprocessorchip, which containsbiometricinformation that can be used to authenticate the identity of the passport holder. It usescontactless smart cardtechnology, including a microprocessor chip (computer chip) and antenna (for both power to the chip and communication) embedded in the front or back cover, or centre page, of the passport. The passport's critical information is printed on the data page of the passport, repeated on themachine readable linesand stored in the chip.Public key infrastructure(PKI) is used to authenticate the data stored electronically in the passport chip, making it expensive and difficult to forge when all security mechanisms are fully and correctly implemented.
Most countries are issuing biometric passports to their citizens.Malaysiawas the first country to issuebiometric passportsin 1998.[1]By the end of 2008, 60 countries were issuing such passports,[2]which increased to over 150 by mid-2019.[3]
The currently standardised biometrics used for this type of identification system arefacial recognition,fingerprint recognition, andiris recognition. These were adopted after assessment of several different kinds of biometrics includingretinal scan. Document and chip characteristics are documented in theInternational Civil Aviation Organization's (ICAO) Doc 9303 (ICAO 9303).[4]The ICAO defines the biometric file formats and communication protocols to be used in passports. Only the digital image (usually inJPEGorJPEG 2000format) of each biometric feature is actually stored in the chip. The comparison of biometric features is performed outside the passport chip by electronic border control systems (e-borders). To store biometric data on the contactless chip, it includes a minimum of 32 kilobytes ofEEPROMstorage memory, and runs on an interface in accordance with theISO/IEC 14443international standard, amongst others. These standards intend interoperability between different countries and different manufacturers of passport books.
Somenational identity cards, such as those fromAlbania,Brazil, theNetherlands, andSaudi Arabiaare fully ICAO 9303 compliant biometrictravel documents. However others, such as theUnited States passport card, are not.[5]
Biometric passports have protection mechanisms to avoid and/or detect attacks:
To assure interoperability and functionality of the security mechanisms listed above, ICAO andGermanFederal Office for Information Security(BSI) have specified several test cases. These test specifications are updated with every new protocol and are covering details starting from the paper used and ending in the chip that is included.[9]
Since the introduction of biometric passports, several attacks have been presented and demonstrated.
Privacyproponents in many countries question and protest the lack of information about exactly what the passports' chip will contain, and whether they affectcivil liberties. The main problem they point out is that data on the passports can be transferred with wirelessRFIDtechnology, which can become a major vulnerability. Although this could allowID-check computers to obtain a person's information without a physical connection, it may also allow anyone with the necessary equipment to perform the same task. If the personal information and passport numbers on the chip are notencrypted, the information might wind up in the wrong hands.
On 15 December 2006, theBBCpublished an article[26]on the British ePassport, citing the above stories and adding that:
and adding that the Future of Identity in the Information Society (FIDIS) network's research team (a body of IT security experts funded by the European Union) has "also come out against the ePassport scheme... [stating that] European governments have forced a document on its people that dramatically decreases security and increases the risk of identity theft."[27]
Most security measures are designed against untrusted citizens (the "provers"), but the scientific security community recently also addressed the threats from untrustworthy verifiers, such as corrupt governmental organizations, or nations using poorly implemented, unsecure electronic systems.[28]New cryptographic solutions such asprivate biometricsare being proposed to mitigate threats of mass theft of identity. These are under scientific study, but not yet implemented in biometric passports.
It was planned that, except for Denmark andIreland,EU passportswould have digital imaging andfingerprintscan biometrics placed on their RFID chips.[116]This combination ofbiometricsaims to create an unrivaled level of security and protection against fraudulent identification papers[vague]. Technical specifications for the new passports have been established by the European Commission.[117]The specifications are binding for theSchengen agreementparties, i.e. the EU countries, except Ireland, and the fourEuropean Free Trade Associationcountries—Iceland, Liechtenstein,[118][119]Norway and Switzerland.[120]These countries are obliged to implement machine readable facial images in the passports by 28 August 2006, and fingerprints by 26 June 2009.[121]TheEuropean Data Protection Supervisorhas stated that the current legal framework fails to "address all the possible and relevant issues triggered by the inherent imperfections of biometric systems".[122]
Irish biometric passports only used a digital image and not fingerprinting. German passports printed after 1 November 2007 contain two fingerprints, one from each hand, in addition to a digital photograph. Romanian passports will also contain two fingerprints, one from each hand. The Netherlands also takes fingerprints and was[123]the only EU member that had plans to store these fingerprints centrally.[124]According to EU requirements, only nations that are signatories to theSchengen acquisare required to add fingerprint biometrics.[125]
In the EU nations, passport prices will be:
In the EFTA, passport prices will be:
The ICAO standard sets a 35x45 mm image with adequate resolution with the following requirements:
Though some countries like USA use a 2x2 inch photo format (51x51 mm), they usually crop it to be closer to 35:45 in ratio when issuing a passport.
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Amachine-readable passport(MRP) is amachine-readabletravel document(MRTD) with the data on the identity page encoded inoptical character recognitionformat. Many countries began to issue machine-readable travel documents in the 1980s. Most travelpassportsworldwide are MRPs. TheInternational Civil Aviation Organization(ICAO) requires all ICAO member states to issue only MRPs as of April 1, 2010, and all non-MRP passports must expire by November 24, 2015.[1]
Machine-readable passports are standardized by theICAODocument 9303(endorsed by theInternational Organization for Standardizationand theInternational Electrotechnical Commissionas ISO/IEC 7501-1) and have a specialmachine-readable zone(MRZ), which is usually at the bottom of the identity page at the beginning of a passport. The ICAO 9303 describes three types of documents corresponding to theISO/IEC 7810sizes:
The fixed format allows specification of document type, name, document number, nationality, date of birth, sex, and document expiration date. All these fields are required on a passport. There is room for optional, often country-dependent, supplementary information. There are also two sizes of machine-readable visas similarly defined.
Computers with a camera and suitable software can directly read the information on machine-readable passports. This enables faster processing of arriving passengers by immigration officials, and greater accuracy than manually-read passports, as well as faster data entry, more data to be read and better data matching against immigration databases and watchlists.
Apart from optically readable information, many passports contain anRFIDchip which enables computers to read a higher amount of information, for example a photo of the bearer. These passports are calledbiometric passportsand are also described by ICAO 9303.
Passport booklets have an identity page containing the identity data. This page is in theID-3size of 125 × 88 mm (4.92 × 3.46 in).
The data of the machine-readable zone consists of two rows of 44 characters each. The only characters used are A–Z, 0–9 and the filler character<.
In the name field, spaces, hyphens and other punctuation are represented by<, exceptapostrophes, which are skipped. If the names are too long, names are abbreviated to their most significant parts. In that case, the last position must contain an alphabetic character to indicate possible truncation, and if there is a given name, the two fillers and at least one character of it must be included.
Smaller documents such as identity and passport cards are usually in theID-1size, which is 85.6 × 54.0 mm (3.37 × 2.13 in), the same size as credit cards. The data of the machine-readable zone in a TD1 size card consists of three rows of 30 characters each. The only characters used are A–Z, 0–9 and the filler character<.
Some official travel documents are in the largerID-2size, 105.0 × 74.0 (4.13 × 2.91 in). They have a layout of the MRZ with two rows of 36 characters each, similar to the TD3 format, but with 31 characters for the name, 7 for the personal number and one less check digit. Yet some official travel documents are in the booklet format with a TD3 identity page.
The format of the first row for ID-1 (credit card size) documents is:
The format of the second row is:
1:United States Passport Cards, as of 2011, use this field for the application number that produced the card.[citation needed]
The format of the third row is:
The format of the first row for ID-2[3](medium size) documents is:
The format of the second row is:
The ICAO Document 9303 part 7 describes machine-readable visas. They come in two different formats:
The format of the first row of the machine-readable zone is:
The format of the second row is:
The ICAO document 9303 part 3 describes specifications common to all Machine Readable Travel Documents.
The dimensions of the effective reading zone (ERZ) is standardized at 17.0 mm (0.67 in) in height with a margin of 3 mm at the document edges and 3.2 mm at the edge against the visual readable part. This is in order to allow use of a single machine reader.
The nationality codes shall contain theISO 3166-1 alpha-3code with modifications for all formats. The check digit calculation method is also the same for all formats.
Some values that are different from ISO 3166-1 alpha-3 are used for the issuing country and nationality field:[4]
Other values, which do not have broad acceptance internationally, include:
Uruguay currently issues passports with the country of birth code in the place of the citizenship code, affecting naturalised citizens as their passports return "error", causing significant travel challenges for passport holders. This is due to a combination of the official Spanish translation of 9303 using "nacionalidad" rather than "ciudadania" to reflect the English original of citizenship - notable in part 3 section 7.1 which specifically addresses this potential error. In October 2023, the high level technical team ofTAG/TRIPS4addressed the Uruguay case and it is proposed the translation is adjusted and the update is communicated to Uruguayan authorities. Uruguayan authorities have committed to reviewing their policy on the understanding citizenship should be used, which overcomes the challenge of domestic definitions ofnationalitycurrently differing fromcitizenship.[citation needed]
The check digit calculation is as follows: each position is assigned a value; for the digits 0 to 9 this is the value of the digits, for the letters A to Z this is 10 to 35, for the filler<this is 0. The value of each position is then multiplied by its weight; the weight of the first position is 7, of the second it is 3, and of the third it is 1, and after that the weights repeat 7, 3, 1, and so on. All values are added together and the remainder of the final value divided by 10 is the check digit.
Due to technical limits, characters inside the Machine Readable Zone (MRZ) need to be restricted to the 10Arabic numerals, the 26 capital Latin letters A through Z, and the filler character <.
Apostrophesand similar punctuation marks have to be omitted, buthyphensand spaces should be replaced by an opening angle bracket.
Section 6 of the 9303 part 3 document specifies transliteration of letters outside the A–Z range. It recommends that diacritical marks onLatinletters A-Z are simply omitted (ç →C, ď →D, ê →E, ñ →Netc.), but it allows the following transliterations:
The following transliterations are mandatory:
In Germany, Austria, Switzerland and Scandinavia it is standard to use the Å→AA, Ä or Æ→AE, Ö or Ø→OE, Ü→UE, and ß→SS mappings, soMüllerbecomes MUELLER, Gößmann becomes GOESSMANN, andHämäläinenbecomes HAEMAELAEINEN. ð, ñ and ü occur in Iceland and Spain, but they write them as D, N and U.
Austrian passports may (but do not always) contain a trilingual (in German, English, and French) explanation of the Germanumlautsand ß.
Russian visas (and Russian internal passports since 2011) have a different transliteration of Cyrillic into the machine-readable zone. As an example, the letter "ч" is usually transcribed as "ch" in Russian travel documents, however, Russian visas and internal passports use "3" in the machine-readable zone instead. Another example is "Алексей" (Cyrillic version) → "ALEKSEQ" (machine readable version in an internal document). This makes it easier to transliterate the name back to Cyrillic.
For airline tickets, visas and more, the advice is to only use the first name written in the passport. This is a problem for people who use their second name (as defined by the order in the passport) as their main name in daily speech. It is common, for example in Scandinavia, that the second or even third name is the one defined for daily usage: for example, the actorHugh Laurie, whose full name is James Hugh Calum Laurie. Swedish travel agents usually book people using the first and daily name if the first one is not their main name, despite advice to use only the first name. If this is too long, the spelling in the MRZ could be used.
For people using a variant of their first name in daily speech, for example the former US presidentBill Clintonwhose full name is William Jefferson Clinton, the advice is to spell their name as in the passport.
In Scandinavian legislation, a middle name is a name placed between the given name and surname, and is usually a family name. Such names are written as an extra surname in passports. People have been stranded at airports since they entered this extra family name in the "middle name" field in airline booking forms, which in English speaking tradition is a given name.
Chinese,Japanese,KoreanandHungarian namesmight pose a challenge too, since the family name is normally written first. Tickets should use given name and surname as indicated in passports.
This name issue is also an issue for post-Brexit EU women under the Brexit settled status (they have two family names, a birth and marriage name, but only the birth name was used by the passport MRZ and therefore used in the settlement application, although they have been using the married name in UK population register).[further explanation needed][6]
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A nationalidentity documentis an identity card with a photo, usable as an identity card at least inside the country, and which is issued by an official national authority. Identity cards can be issued voluntarily or may be compulsory to possess as a resident or citizen.[1]
Driving licencesand other cards issued bystate or regional governmentsindicating certain permissions are not counted here as national identity cards. So for example, by this criterion, theUnited States driver's licenseis excluded, as these are issued by local (state) governments.
Generally, most countries in the world issue identity cards, with less than 10 countries worldwide not issuing them, mostly confined to theanglosphere,microstatesandunrecognised states.[1]Many states issue voluntary identity cards to citizens as a convenience. As of 1996, identity cards were compulsory in over 100 countries.[2]In these countries, the meaning of compulsory varies.[2]
In theEuropean Union, anEU/EEA national identity cardcan be used to travel freely within theEU/EEAin lieu of a passport.[3]Similarly, in South America, citizens may use an identity card to travel betweenMERCOSURstates.[4]In many other areas of the world, simplified travel arrangements are in place for neighbouring countries, allowing the use of identity cards for travel.
The term "compulsory" may have different meanings and implications in different countries.Possessionof a card may only become compulsory at a certain age. There may be a penalty for notcarryinga card or identification such as adriving licence. In some cases a person may be detained until identity is proven. This facilitates police identification of fugitives. In some countries, police need a reason to ask for identification, such as suspicion of a crime or security risk, while in others, they can do so without stating a reason. Random checks are rare, except inpolice states. Normally there is an age limit, such as 18, after which possession is mandatory, even if minors aged 15–17 may need a card in order to prove that they are under 18.
The card's front has the bearer's picture (with an electronic stamp on it) and right thumb print. It also includes either the bearer's signature or – if the bearer is illiterate – the phrase "cannot sign" (não assina) The verso has the unique number assigned the bearer (registro geralor RG), the bearer's full name,filiation, birthplace (locality and federation unit), birth date, andCPF number. It may include some additional information. It is officially 102 × 68 mm,[13]but lamination tends to make it slightly larger than theISO/IEC 7810 ID-2standard of 105 × 74 mm, so it is a tight fit in most wallets. A driver's licence has only recently been given the same legal status as the national identity card. In most situations, only a few other documents can be substituted for a national identity card: for example, identification documents issued by national councils of professionals.
As of 2020, a new Electronic Identity Document is being issued which must be renewed every 10 years. This new document is available both physically, as a card, and electronically, through a mobile application[25]
In Greece, there are many everyday things one cannot do without an ID. In fact, according to an older law, the Police ID is the only legal identity document and no one has a right to ask for more identity documents. Since the 1980s all legal services in Greece must be done with this ID. It is possible to travel within the EU using a Greek national ID card, although it may cause delays at border controls because those cards do not have machine-readable zones. Carrying any ID is not de jure compulsory. However, during routine police checks, if a citizen is found without an ID, the police officer may take them to the nearest police station for further investigation, thus rendering always carrying the ID card de facto compulsory.
The Guatemalan constitution requires personal identification via documentation, person rooting or the government. If the person cannot be identified, they may be sent to a judge until identification is provided.[37]
Police officers have an absolute right to require every person aged 15 or above on public premises to produce their HKID or valid passport for inspection; failure to produce such photo ID constitutes an offence in law. The reason for setting up police random checks is due to the end of theTouch Base Policyon 24 October 1980, which meant that all illegal immigrants fromChinathat failed to present a validHong Kong Identity Cardat random checks would subsequently be sent back toMainland China.
The Directorate General of National Security of Morocco announced it will issue a newer version of the national electronic identity card (NEIC) from 2020. The NEIC isbiometricand provides citizens of a birth certificate, residence certificate, extract of birth and citizenship certificates.
North Korea is probably the country which imposes the strongest fines for citizens not carrying ID cards. For travel, North Koreans need both an identity card, and a "travel pass", with specified destination and written permission. Sometimes citizens may be punished with time in a labour camp for not carrying their cards, however this is often only a short sentence and people are usually released upon presentation of the card at a later date. Although much is not known about the properties of the card, it is probably plastic and similar in size to most European ID cards.
Between 2004 and 2008, all records were transferred to an electronic Korean-language central database. Obtaining a driving license in North Korea is not usual – except in the case of professional drivers, mechanics, and assistants – since few citizens own cars. Only government officials are issuedpassportsbecause the state restricts citizens travel. North Koreans working abroad are issued contracts between North Korea and the host country to allow for travel, and government officers often accompany and supervise workers.
The Philippine Identification System (PhilSys) ID also known as the Philippine identity card is issued to all Filipino citizens and resident aliens in the Philippines. The pilot implementation began in selected regions in 2018 and full implementation began in 2019.[73]The national ID card is not compulsory and will harmonize existing government-initiated identification cards issued including theUnified Multi-Purpose IDissued to members of theSocial Security System,Government Service Insurance System,Philippine Health Insurance Corporation, andHome Development Mutual Fund(Pag-IBIG Fund).[74]This will also replace theAlien Certificate of Registration (ACR) Cardfor foreign residents and expatriates who are living in the Philippines permanently.
Because it is sometimes necessary to produce a national identity card, many South African permanent residents carry their card at all times.
All citizens must submit and save their 10 fingerprints to the criminal database operated by National Police Agency and right thumb fingerprint to Ministry of Interior and Safety at the time of ID card application.
國民身份證
Documents for uruguayan citizens are in blue and documents for legal residents are in yellow with inscription "EXTRANJERO".
It is required for many things such as credit card transactions, age verification, etc.
These are countries where official authorities issue identity cards to those who request them, but where it is not illegal to be without an official identity document. For some services, identification is needed, but documents such as passports or identity cards issued by banks or driving licences can be used. In countries where national identity cards are fully voluntary, they are often not so commonly used, because many already have a passport and a driving licence, so a third identity document is often considered superfluous.
This national digital ID system also offers real-time online and offline authentication to supporteKYC. It is consent-basedbiometric-backed identification for alllegal residentsof Ethiopia (non-citizens and minors are also eligible).[121][122]
While police officers and some other officials have a right to demand to see one of those documents, the law does not state that one is obliged to submit the document immediately. Fines may only be applied if an identity card or passport is not possessed at all, if the document is expired or if one explicitly refuses to show ID to the police. If one is unable to produce an ID card or passport (or any other form of credible identification) during a police control, one can (in theory) be brought to the next police post and detained for a maximum of 12 hours, or until positive identification is possible. However, this measure is only applied if the police have reasonable grounds to believe the person detained has committed an offence.[127]
The British Overseas Territory of Gibraltar has a voluntary ID card system for citizens, valid in the UK and EU/European Free Trade Associationmember countries.
Police has the legal power to stop people on streets at random and ask for ID card. If the person has no proof for identification one can be detained for maximum 24 hours.
The US uses theSocial Security numberas the de facto national ID number of the country.
It is unclear if it is compulsory or not.
These are countries where official authorities do not issue any identity cards. When identification is needed, e.g. passports, driving licences, bank cards etc. can be used, along with manual verification such as utility bills and bank statements.[167]Most countries that are not listed at all in this page have no national ID card.
In 1985, there was a failed proposal to create anAustralia Card. In 2007, there was another failed proposal to create a non-compulsoryAccess Cardthat would act as a gateway to The Department of Human Services.
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Adigital wallet, also known as ane-walletormobile wallet, is anelectronic device,online service, orsoftware programthat allows one party to makeelectronic transactionswith another party barteringdigital currencyunits forgoods and services. This can include purchasing items eitheronlineor at thepoint of salein abrick and mortarstore, using eithermobile payment(on asmartphoneor othermobile device) or (for online buying only) using alaptopor otherpersonal computer. Money can be deposited in the digital wallet prior to any transactions or, in other cases, an individual's bank account can be linked to the digital wallet. Users might also have theirdriver's license,health card,loyalty card(s)and other ID documents stored within the wallet. The credentials can be passed to a merchant's terminal wirelessly vianear field communication(NFC).
Increasingly, digital wallets are being made not just for basic financial transactions but to also authenticate the holder's credentials. For example, a digital wallet could verify the age of the buyer to the store while purchasing alcohol. The system has already gained popularity in Japan, where digital wallets are known as "wallet mobiles".[1]In addition, a few US states have adapted digital driver's license and state IDs to be added to digital wallet in lieu of the physical card and it can be used at selected TSA checkpoints at airports, banking or enterprise.[2]
Acryptocurrency walletis a digital wallet where private keys are stored forcryptocurrencieslikebitcoin.
A digital wallet has both a software and information component. Secure and fair electronic payment systems are an important issue.[3]The software provides security and encryption for the personal information and for the actual transaction. Typically, digital wallets are stored on the client side and are easily self-maintained and fully compatible with moste-commercewebsites. A server-side digital wallet, also known as a thin wallet, is one that an organization creates for and about its members and maintains on itsservers. Server-side digital wallets are gaining popularity among major retailers due to the security, efficiency, and added utility it provides to the end-user, which increases their satisfaction of their overall purchase.[4]The information component is basically a database of user-input information. This information consists of a user's shipping address, billing address, payment methods (including credit card numbers, expiry dates, and security numbers), and other information.
Digital wallets are composed of both digital wallet devices and digital wallet systems. There are dedicated digital wallet devices such as the biometric wallet byDunhill,[5]a physical device that holds cash and cards along with aBluetoothmobile connection. Presently there are further explorations for smartphones with NFC digital wallet capabilities, such as smartphones utilizingGoogle'sAndroidandApple'siOSoperating systems to power wallets such asGoogle PayandApple Pay.[citation needed]
Digital wallet systems enable the widespread use of digital wallet transactions among various retail vendors in the form ofmobile paymentssystems and digital wallet applications. TheM-PESAmobile payments system andmicrofinancingservice has widespread use inKenyaandTanzania,[6]while theMasterCard PayPassapplication has been adopted by a number of vendors in the U.S. and worldwide.[7]
Digital wallets are being used more frequently among Asian countries as well. One in every five consumers in Asia are now using a digital wallet, representing a twofold increase from two years ago. AMasterCardmobile shopping survey among 8500 adults, aged 18–64 across 14 markets, showed that 45% of users in China, 36.7% of users in India and 23.3% of users in Singapore are the biggest adopters of digital wallets. The survey was conducted between October and December 2015. Further analysis showed that 48.5% of consumers in these regions made purchases using smartphones. Indian consumers are leading the way with 76.4% using a smartphone to make a purchase, which is a drastic increase of 29.3% from the previous year. This has inspired companies like Reliance and Amazon India to come out with their own digital wallet.Flipkarthas already introduced its own digital wallet.[8]
Consumers are not required to fill out order forms on each site when they purchase an item because the information has already been stored and is automatically updated and entered into the order fields across merchant sites when using a digital wallet. Consumers also benefit when using digital wallets because their information is encrypted or protected by a private software code; merchants benefit by receiving a combination of protection against fraud, faster receipt of payment, decreased transaction costs, and decreased theft loss.
Digital wallets are available to consumers free of charge, and they're fairly easy to obtain. For example, when a consumer makes a purchase at a merchant site that's set up to handle server-side digital wallets, they type their name, payment and shipping information into the merchant's own form. At the end of the purchase, the consumer is asked to sign up for a wallet of their choice by entering a user name and password for future purchases. Users can also acquire wallets at a wallet vendor's site.
Most, if not all digital wallets offer advanced security features e.g.biometric authenticationand encryption, this protects the financial information of the users thus preventing fraud.[9]
With the acquisition ofiDEAL,European Payments Initiative(EPI) Company has announced that it will create an all European digital wallet.[10][11][12][13]
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https://en.wikipedia.org/wiki/Digital_wallet
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Amobile driving licence(alsomobile driver's licenseormDL) is amobile appthat replaces a physicaldriver's license. AnInternational Organization for Standardization(ISO) standard for the mobile driving licence (ISO/IEC 18013-5) was approved on August 18, 2021 and published on 30 September 2021.[1]
In November 2020, Denmark publicly released a digital/mobile driving licence using a proprietary app implementation using a QR code, also not conforming to the ISO/IEC 18013-5 standard. Similar to Iceland's implementation, it is fully equivalent to physical IDs, however only valid in Denmark.[2]
Icelandwas the second country in Europe to introduce a digital/mobile driver's licence in July 2020.Icelandic driving licenceholders can request a digital version of their licence online by using theirelectronic ID(Icelandic: rafræn skilríki) and is issued as a.pkpass fileloaded into theWallet apponiPhoneor a third-party app onAndroid. Digital driving licences display the same information as a physical licence, along with a barcode (renewed regularly by the server, acting as verification). Commercial establishments (e.g. for proof of age) can use the island.is app to verify barcodes. The licences are equally valid as official ID, even for voting, however only within Iceland. The implementation does not conform to the ISO/IEC 18013-5 standard.[3]As of August 2022, 60% of driver's licences have been issued in digital/mobile form.[4]
The first instance of an electronic driver's license was deployed in Mexico as early as 2007, using theGemaltosmart-card platform. In 2016, the U.S. National Institute of Standards and Technology (NIST) partnered with Gemalto to pilot the "digital driver's license" inWashington D.C.,Idaho,Colorado,MarylandandWyoming.[5]
On 1 October 2019,Norwaybecame the first country inEuropeto introduce a digital driver's license. A holder of aNorwegian driver's licensecan request a digital version of their physical driver's license after downloading theappFørerkortfrom their preferredapp marketplace. The applicant must verify their identity withBankIDupon logging in on the app for the first time, which will then retrieve information from the national database for driving licenses. After this procedure, the digital driver's license will display the exact same information as on the physical driver's license. The app only allows one phone with a digital driver's license per user.
If the holder has recently passed their driving exam or upgraded to a new category, and hence awaiting to receive their physical driving license, the app will display a temporary driving permit. When the physical driver's license has been produced, the app is able to display the holder's digital driver's license, regardless if the holder has received their physical driving license bymailyet or not. If a holder's driver's license has been revoked or suspended, this will information will be displayed in the app as long as the holder has not gotten their driver's license back.
Upon atraffic stopby thepoliceor coming in contact with thePublic Roads Administration, the digital driver's license is valid as a proof of identification. Although the driver's license comes with a barcode, which can be scanned by either government authorities, commercial establishments or even private persons to verify the details, it is not considered as a proof of identification in most places. The digital driver's license is not valid outside of Norway.[6]
The first mDL that claims compliance with ISO/IEC 18013-5 isLouisiana's, developed in part by Envoc, a software firm inBaton Rouge, whose president claimed that most drivers under 40 won't go back home if they forget their physical laminated license, "but if they forget their phone, they always turn around."[7]Ontarioin 2020, in response tothe COVID-19 pandemic, announced a "Digital Identity Program," including a mobile driver's license.[8]
Colorado was the first state to deploy a production version of a digital license, primarily based onQR codesstored in adigital wallet, which it claims is accepted bypolice officers throughout the state.[9]After going through the standard process at the stateDepartment of Motor Vehicles, volunteers installed the "DigiDL" app on their phones and then downloaded the license. Volunteers tested the digital driver's license in stores, theColorado Lotteryclaim center, and anart fair.[10]
Smartphoneoperating systemsare adapting to the new standard. For example,Android's JetPack suite comes with specific support for ISO 18013–5 from version API 24.[11][12]In March 2022,Appleintroduced support for mobile IDs conforming to ISO 18013-5 inApple Wallet, through a proprietary enrollment process which is implemented in partnership with governments.[13]ArizonaandGeorgiabecame the first two states to announce that IDs were supported on Apple Wallet, starting with versioniOS 15.4. On March 23, 2022, Arizona officially launched their program which includes the first TSA checkpoint to support Apple’s mobile driver’s license, Phoenix Sky Harbor International Airport.[14][15]
Safety organization Fime has a product to help test an app's conformance with the ISO/IEC 18013-5 standard.[16]TheKantara Initiativecreated a "Privacy & Identity Protection in Mobile Driving License Ecosystems Discussion Group" to issue a report on the need for conformance specifications around identity and privacy.[17]
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Inmathematics, aDrinfeld module(orelliptic module) is roughly a special kind ofmoduleover a ring of functions on a curve over afinite field, generalizing theCarlitz module. Loosely speaking, they provide a function field analogue ofcomplex multiplicationtheory. Ashtuka(also calledF-sheaforchtouca) is a sort of generalization of a Drinfeld module, consisting roughly of avector bundleover a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.
Drinfeld modules were introduced byDrinfeld(1974), who used them to prove theLanglands conjecturesfor GL2of analgebraic function fieldin some special cases. He later invented shtukas and used shtukas of rank 2 to prove
the remaining cases of the Langlands conjectures for GL2.Laurent Lafforgueproved the Langlands conjectures for GLnof a function field by studying themoduli stackof shtukas of rankn.
"Shtuka" is a Russian word штука meaning "a single copy", which comes from the German noun “Stück”, meaning “piece, item, or unit". In Russian, the word "shtuka" is also used in slang for a thing with known properties, but having no name in a speaker's mind.
We letL{\displaystyle L}be a field of characteristicp>0{\displaystyle p>0}. The ringL{τ}{\displaystyle L\{\tau \}}is defined to be the ring ofnoncommutative(or twisted)polynomialsa0+a1τ+a2τ2+⋯{\displaystyle a_{0}+a_{1}\tau +a_{2}\tau ^{2}+\cdots }overL{\displaystyle L}, with the multiplication given by
The elementτ{\displaystyle \tau }can be thought of as aFrobenius element: in fact,L{\displaystyle L}is a left module overL{τ}{\displaystyle L\{\tau \}}, with elements ofL{\displaystyle L}acting as multiplication andτ{\displaystyle \tau }acting as the Frobenius endomorphism ofL{\displaystyle L}. The ringL{τ}{\displaystyle L\{\tau \}}can also be thought of as the ring of all (absolutely) additive polynomials
inL[x]{\displaystyle L[x]}, where a polynomialf{\displaystyle f}is calledadditiveiff(x+y)=f(x)+f(y){\displaystyle f(x+y)=f(x)+f(y)}(as elements ofL[x,y]{\displaystyle L[x,y]}). The ring of additive polynomials is generated as an algebra overL{\displaystyle L}by the polynomialτ=xp{\displaystyle \tau =x^{p}}. The multiplication in the ring of additive polynomials is given by composition of polynomials, not by multiplication of commutative polynomials, and is not commutative.
LetFbe an algebraic function field with a finite field of constants and fix aplace∞{\displaystyle \infty }ofF. DefineAto be the ring of elements inFthat are regular at every place except possibly∞{\displaystyle \infty }. In particular,Ais aDedekind domainand it isdiscreteinF(with the topology induced by∞{\displaystyle \infty }). For example, we may takeAto be the polynomial ringFq[t]{\displaystyle F_{q}[t]}. LetLbe a field equipped with a ring homomorphismι:A→L{\displaystyle \iota :A\to L}.
The condition that the image ofAis not inLis a non-degeneracy condition, put in to eliminate trivial cases, while the condition thatd∘ϕ=ι{\displaystyle d\circ \phi =\iota }gives the impression that a Drinfeld module is simply a deformation of the mapι{\displaystyle \iota }.
AsL{τ} can be thought of as endomorphisms of the additive group ofL, a DrinfeldA-module can be regarded as an action ofAon the additive group ofL, or in other words as anA-module whose underlying additive group is the additive group ofL.
Suppose thatXis a curve over the finite fieldFp.
A (right)shtukaof rankrover ascheme(or stack)Uis given by the following data:
whose cokernels are supported on certain graphs of morphisms fromUtoX(called the zero and pole of the shtuka, and usually denoted by 0 and ∞), and are locally free of rank 1 on their supports. Here (Fr×1)*Eis the pullback ofEby the Frobenius endomorphism ofU.
Aleft shtukais defined in the same way except that the direction of the morphisms is reversed. If the pole and zero of the shtuka are disjoint then left shtukas and right shtukas are essentially the same.
By varyingU, we get analgebraic stackShtukarof shtukas of rankr, a "universal" shtuka overShtukar×Xand a morphism (∞,0) fromShtukartoX×Xwhich is smooth and of relative dimension 2r− 2. The stackShtukaris not of finite type forr> 1.
Drinfeld modules are in some sense special kinds of shtukas. (This is not at all obvious from the definitions.) More precisely, Drinfeld showed how to construct a shtuka from a Drinfeld module.
See Drinfeld, V. G.Commutative subrings of certain noncommutative rings.Funkcional. Anal. i Prilovzen. 11 (1977), no. 1, 11–14, 96. for details.
The Langlands conjectures for function fields state (very roughly) that there is a bijection between cuspidal automorphic representations ofGLnand certain representations of a Galois group. Drinfeld used Drinfeld modules to prove some special cases of the Langlands conjectures, and later proved the full Langlands conjectures forGL2by generalizing Drinfeld modules to shtukas.
The "hard" part of proving these conjectures is to construct Galois representations with certain properties, and Drinfeld constructed the necessary Galois representations by finding them inside thel-adic cohomology of certain moduli spaces of rank 2 shtukas.
Drinfeld suggested that moduli spaces of shtukas of rankrcould be used in a similar way to prove the Langlands conjectures forGLr; the formidable technical problems involved in carrying out this program were solved by Lafforgue after many years of effort.
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Inalgebra, anadditive map,Z{\displaystyle Z}-linear maporadditive functionis afunctionf{\displaystyle f}that preserves the addition operation:[1]f(x+y)=f(x)+f(y){\displaystyle f(x+y)=f(x)+f(y)}for every pair of elementsx{\displaystyle x}andy{\displaystyle y}in thedomainoff.{\displaystyle f.}For example, anylinear mapis additive. When the domain is thereal numbers, this isCauchy's functional equation. For a specific case of this definition, seeadditive polynomial.
More formally, an additive map is aZ{\displaystyle \mathbb {Z} }-module homomorphism. Since anabelian groupis aZ{\displaystyle \mathbb {Z} }-module, it may be defined as agroup homomorphismbetween abelian groups.
A mapV×W→X{\displaystyle V\times W\to X}that is additive in each of two arguments separately is called abi-additive mapor aZ{\displaystyle \mathbb {Z} }-bilinear map.[2]
Typical examples include maps betweenrings,vector spaces, ormodulesthat preserve theadditive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring.
Iff{\displaystyle f}andg{\displaystyle g}are additive maps, then the mapf+g{\displaystyle f+g}(definedpointwise) is additive.
Definition of scalar multiplication by an integer
Suppose thatX{\displaystyle X}is an additive group with identity element0{\displaystyle 0}and that the inverse ofx∈X{\displaystyle x\in X}is denoted by−x.{\displaystyle -x.}For anyx∈X{\displaystyle x\in X}and integern∈Z,{\displaystyle n\in \mathbb {Z} ,}let:nx:={0whenn=0,x+⋯+x(nsummands)whenn>0,(−x)+⋯+(−x)(|n|summands)whenn<0,{\displaystyle nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0,\\&&&x&&+\cdots +&&x&&~~~~{\text{(}}n&&{\text{ summands) }}&&~{\text{ when }}n>0,\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{(}}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0,\\\end{alignedat}}\right.}Thus(−1)x=−x{\displaystyle (-1)x=-x}and it can be shown that for all integersm,n∈Z{\displaystyle m,n\in \mathbb {Z} }and allx∈X,{\displaystyle x\in X,}(m+n)x=mx+nx{\displaystyle (m+n)x=mx+nx}and−(nx)=(−n)x=n(−x).{\displaystyle -(nx)=(-n)x=n(-x).}This definition of scalar multiplication makes the cyclic subgroupZx{\displaystyle \mathbb {Z} x}ofX{\displaystyle X}into aleftZ{\displaystyle \mathbb {Z} }-module; ifX{\displaystyle X}is commutative, then it also makesX{\displaystyle X}into a leftZ{\displaystyle \mathbb {Z} }-module.
Homogeneity over the integers
Iff:X→Y{\displaystyle f:X\to Y}is an additive map between additive groups thenf(0)=0{\displaystyle f(0)=0}and for allx∈X,{\displaystyle x\in X,}f(−x)=−f(x){\displaystyle f(-x)=-f(x)}(where negation denotes the additive inverse) and[proof 1]f(nx)=nf(x)for alln∈Z.{\displaystyle f(nx)=nf(x)\quad {\text{ for all }}n\in \mathbb {Z} .}Consequently,f(x−y)=f(x)−f(y){\displaystyle f(x-y)=f(x)-f(y)}for allx,y∈X{\displaystyle x,y\in X}(where by definition,x−y:=x+(−y){\displaystyle x-y:=x+(-y)}).
In other words, every additive map ishomogeneous over the integers. Consequently, every additive map betweenabelian groupsis ahomomorphism ofZ{\displaystyle \mathbb {Z} }-modules.
Homomorphism ofQ{\displaystyle \mathbb {Q} }-modules
If the additiveabelian groupsX{\displaystyle X}andY{\displaystyle Y}are also aunitalmodulesover the rationalsQ{\displaystyle \mathbb {Q} }(such as real or complexvector spaces) then an additive mapf:X→Y{\displaystyle f:X\to Y}satisfies:[proof 2]f(qx)=qf(x)for allq∈Qandx∈X.{\displaystyle f(qx)=qf(x)\quad {\text{ for all }}q\in \mathbb {Q} {\text{ and }}x\in X.}In other words, every additive map ishomogeneous over the rational numbers. Consequently, every additive maps between unitalQ{\displaystyle \mathbb {Q} }-modulesis ahomomorphism ofQ{\displaystyle \mathbb {Q} }-modules.
Despite being homogeneous overQ,{\displaystyle \mathbb {Q} ,}as described in the article onCauchy's functional equation, even whenX=Y=R,{\displaystyle X=Y=\mathbb {R} ,}it is nevertheless still possible for the additive functionf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }tonotbehomogeneous over the real numbers; said differently, there exist additive mapsf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }that arenotof the formf(x)=s0x{\displaystyle f(x)=s_{0}x}for some constants0∈R.{\displaystyle s_{0}\in \mathbb {R} .}In particular, there exist additive maps that are notlinear maps.
Proofs
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In themathematicalfield ofknot theory, theJones polynomialis aknot polynomialdiscovered byVaughan Jonesin 1984.[1][2]Specifically, it is aninvariantof an orientedknotorlinkwhich assigns to each oriented knot or link aLaurent polynomialin the variablet1/2{\displaystyle t^{1/2}}withintegercoefficients.[3]
Suppose we have anoriented linkL{\displaystyle L}, given as aknot diagram. We will define the Jones polynomialV(L){\displaystyle V(L)}by usingLouis Kauffman'sbracket polynomial, which we denote by⟨⟩{\displaystyle \langle ~\rangle }. Here the bracket polynomial is aLaurent polynomialin the variableA{\displaystyle A}with integer coefficients.
First, we define the auxiliary polynomial (also known as the normalized bracket polynomial)
wherew(L){\displaystyle w(L)}denotes thewritheofL{\displaystyle L}in its given diagram. The writhe of a diagram is the number of positive crossings (L+{\displaystyle L_{+}}in the figure below) minus the number of negative crossings (L−{\displaystyle L_{-}}). The writhe is not a knot invariant.
X(L){\displaystyle X(L)}is a knot invariant since it is invariant under changes of the diagram ofL{\displaystyle L}by the threeReidemeister moves. Invariance under type II and III Reidemeister moves follows from invariance of the bracket under those moves. The bracket polynomial is known to change by a factor of−A±3{\displaystyle -A^{\pm 3}}under a type I Reidemeister move. The definition of theX{\displaystyle X}polynomial given above is designed to nullify this change, since the writhe changes appropriately by+1{\displaystyle +1}or−1{\displaystyle -1}under type I moves.
Now make the substitutionA=t−1/4{\displaystyle A=t^{-1/4}}inX(L){\displaystyle X(L)}to get the Jones polynomialV(L){\displaystyle V(L)}. This results in a Laurent polynomial with integer coefficients in the variablet1/2{\displaystyle t^{1/2}}.
This construction of the Jones polynomial fortanglesis a simple generalization of theKauffman bracketof a link. The construction was developed byVladimir Turaevand published in 1990.[4]
Letk{\displaystyle k}be a non-negative integer andSk{\displaystyle S_{k}}denote the set of all isotopic types of tangle diagrams, with2k{\displaystyle 2k}ends, having no crossing points and no closed components (smoothings). Turaev's construction makes use of the previous construction for the Kauffman bracket and associates to each2k{\displaystyle 2k}-end oriented tangle an element of the freeR{\displaystyle \mathrm {R} }-moduleR[Sk]{\displaystyle \mathrm {R} [S_{k}]}, whereR{\displaystyle \mathrm {R} }is theringofLaurent polynomialswith integer coefficients in the variablet1/2{\displaystyle t^{1/2}}.
Jones' original formulation of his polynomial came from his study of operator algebras. In Jones' approach, it resulted from a kind of "trace" of a particular braid representation into an algebra which originally arose while studying certain models, e.g. thePotts model, instatistical mechanics.
Let a linkLbe given. Atheorem of Alexanderstates that it is the trace closure of a braid, say withnstrands. Now define a representationρ{\displaystyle \rho }of thebraid grouponnstrands,Bn, into theTemperley–Lieb algebraTLn{\displaystyle \operatorname {TL} _{n}}with coefficients inZ[A,A−1]{\displaystyle \mathbb {Z} [A,A^{-1}]}andδ=−A2−A−2{\displaystyle \delta =-A^{2}-A^{-2}}. The standard braid generatorσi{\displaystyle \sigma _{i}}is sent toA⋅ei+A−1⋅1{\displaystyle A\cdot e_{i}+A^{-1}\cdot 1}, where1,e1,…,en−1{\displaystyle 1,e_{1},\dots ,e_{n-1}}are the standard generators of the Temperley–Lieb algebra. It can be checked easily that this defines a representation.
Take the braid wordσ{\displaystyle \sigma }obtained previously fromL{\displaystyle L}and computeδn−1trρ(σ){\displaystyle \delta ^{n-1}\operatorname {tr} \rho (\sigma )}wheretr{\displaystyle \operatorname {tr} }is theMarkov trace. This gives⟨L⟩{\displaystyle \langle L\rangle }, where⟨{\displaystyle \langle }⟩{\displaystyle \rangle }is the bracket polynomial. This can be seen by considering, asLouis Kauffmandid, the Temperley–Lieb algebra as a particular diagram algebra.
An advantage of this approach is that one can pick similar representations into other algebras, such as theR-matrix representations, leading to "generalized Jones invariants".
The Jones polynomial is characterized by taking the value 1 on any diagram of the unknot and satisfies the followingskein relation:
whereL+{\displaystyle L_{+}},L−{\displaystyle L_{-}}, andL0{\displaystyle L_{0}}are three oriented link diagrams that are identical except in one small region where they differ by the crossing changes or smoothing shown in the figure below:
The definition of the Jones polynomial by the bracket makes it simple to show that for a knotK{\displaystyle K}, the Jones polynomial of its mirror image is given by substitution oft−1{\displaystyle t^{-1}}fort{\displaystyle t}inV(K){\displaystyle V(K)}. Thus, anamphicheiral knot, a knot equivalent to its mirror image, haspalindromicentries in its Jones polynomial. See the article onskein relationfor an example of a computation using these relations.
Another remarkable property of this invariant states that the Jones polynomial of an alternating link is analternating polynomial. This property was proved byMorwen Thistlethwaite[5]in 1987. Another proof of this last property is due toHernando Burgos-Soto, who also gave an extension of the property to tangles.[6]
The Jones polynomial is not a complete invariant. There exist an infinite number of non-equivalent knots that have the same Jones polynomial. An example of two distinct knots having the same Jones polynomial can be found in the book by Murasugi.[7]
For a positive integerN{\displaystyle N}, theN{\displaystyle N}-colored Jones polynomialVN(L,t){\displaystyle V_{N}(L,t)}is a generalisation of the Jones polynomial. It is theReshetikhin–Turaev invariantassociated with the(N+1){\displaystyle (N+1)}-irreducible representation of thequantum groupUq(sl2){\displaystyle U_{q}({\mathfrak {sl}}_{2})}. In this scheme, the Jones polynomial is the 1-colored Jones polynomial, the Reshetikhin-Turaev invariant associated to the standard representation (irreducible and two-dimensional) ofUq(sl2){\displaystyle U_{q}({\mathfrak {sl}}_{2})}. One thinks of the strands of a link as being "colored" by a representation, hence the name.
More generally, given a linkL{\displaystyle L}ofk{\displaystyle k}components and representationsV1,…,Vk{\displaystyle V_{1},\ldots ,V_{k}}ofUq(sl2){\displaystyle U_{q}({\mathfrak {sl}}_{2})}, the(V1,…,Vk){\displaystyle (V_{1},\ldots ,V_{k})}-colored Jones polynomialVV1,…,Vk(L,t){\displaystyle V_{V_{1},\ldots ,V_{k}}(L,t)}is theReshetikhin–Turaev invariantassociated toV1,…,Vk{\displaystyle V_{1},\ldots ,V_{k}}(here we assume the components are ordered). Given two representationsV{\displaystyle V}andW{\displaystyle W}, colored Jones polynomials satisfy the following two properties:[8]
These properties are deduced from the fact that colored Jones polynomials are Reshetikhin-Turaev invariants.
LetK{\displaystyle K}be a knot. Recall that by viewing a diagram ofK{\displaystyle K}as an element of the Temperley-Lieb algebra thanks to the Kauffman bracket, one recovers the Jones polynomial ofK{\displaystyle K}. Similarly, theN{\displaystyle N}-colored Jones polynomial ofK{\displaystyle K}can be given a combinatorial description using theJones-Wenzl idempotents, as follows:
The resulting element ofQ(t){\displaystyle \mathbb {Q} (t)}is theN{\displaystyle N}-colored Jones polynomial. See appendix H of[9]for further details.
As first shown byEdward Witten,[10]the Jones polynomial of a given knotγ{\displaystyle \gamma }can be obtained by consideringChern–Simons theoryon the three-sphere withgauge groupSU(2){\displaystyle \mathrm {SU} (2)}, and computing thevacuum expectation valueof aWilson loopWF(γ){\displaystyle W_{F}(\gamma )}, associated toγ{\displaystyle \gamma }, and thefundamental representationF{\displaystyle F}ofSU(2){\displaystyle \mathrm {SU} (2)}.
By substitutingeh{\displaystyle e^{h}}for the variablet{\displaystyle t}of the Jones polynomial and expanding it as the series of h each of the coefficients turn to be theVassiliev invariantof the knotK{\displaystyle K}. In order to unify the Vassiliev invariants (or, finite type invariants),Maxim Kontsevichconstructed theKontsevich integral. The value of the Kontsevich integral, which is the infinite sum of 1, 3-valuedchord diagrams, named the Jacobi chord diagrams, reproduces the Jones polynomial along with thesl2{\displaystyle {\mathfrak {sl}}_{2}}weight system studied byDror Bar-Natan.
By numerical examinations on some hyperbolic knots,Rinat Kashaevdiscovered that substituting then-th root of unityinto the parameter of thecolored Jones polynomialcorresponding to then-dimensional representation, and limiting it asngrows to infinity, the limit value would give thehyperbolic volumeof theknot complement. (SeeVolume conjecture.)
In 2000Mikhail Khovanovconstructed a certain chain complex for knots and links and showed that the homology induced from it is a knot invariant (seeKhovanov homology). The Jones polynomial is described as theEuler characteristicfor this homology.
It is anopen questionwhether there is a nontrivial knot with Jones polynomial equal to that of theunknot. It is known that there are nontriviallinkswith Jones polynomial equal to that of the correspondingunlinksby the work ofMorwen Thistlethwaite.[11]It was shown by Kronheimer and Mrowka that there is no nontrivial knot with Khovanov homology equal to that of the unknot.[12]
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Indiscrete geometry, anopaque setis a system of curves or other set in theplanethat blocks alllines of sightacross apolygon, circle, or other shape. Opaque sets have also been calledbarriers,beam detectors,opaque covers, or (in cases where they have the form of aforestofline segmentsor other curves)opaque forests. Opaque sets were introduced byStefan Mazurkiewiczin 1916,[1]and the problem of minimizing their total length was posed byFrederick Bagemihlin 1959.[2]
For instance, visibility through aunit squarecan be blocked by its four boundary edges, with length 4, but a shorter opaque forest blocks visibility across the square with length2+126≈2.639{\displaystyle {\sqrt {2}}+{\tfrac {1}{2}}{\sqrt {6}}\approx 2.639}. It is unproven whether this is the shortest possible opaque set for the square, and for most other shapes this problem similarly remains unsolved. The shortest opaque set for any boundedconvex setin the plane has length at most theperimeterof the set, and at least half the perimeter. For the square, a slightly stronger lower bound than half the perimeter is known. Another convex set whose opaque sets are commonly studied is theunit circle, for which the shortestconnectedopaque set has length2+π{\displaystyle 2+\pi }. Without the assumption of connectivity, the shortest opaque set for the circle has length at leastπ{\displaystyle \pi }and at most4.7998{\displaystyle 4.7998}.
Several publishedalgorithmsclaiming to find the shortest opaque set for aconvex polygonwere later shown to be incorrect. Nevertheless, it is possible to find an opaque set with a guaranteedapproximation ratioinlinear time, or to compute the subset of the plane whose visibility is blocked by a given system of line segments inpolynomial time.
Every setS{\displaystyle S}in the plane blocks the visibility through a superset ofS{\displaystyle S}, itscoverageC{\displaystyle C}.C{\displaystyle C}consists of points for which all lines through the point intersectS{\displaystyle S}. If a given setK{\displaystyle K}forms a subset of the coverage ofS{\displaystyle S}, thenS{\displaystyle S}is said to be anopaque set,barrier,beam detector, oropaque coverforK{\displaystyle K}. If, additionally,S{\displaystyle S}has a special form, consisting of finitely manyline segmentswhose union forms aforest, it is called anopaque forest. There are many possible opaque sets for any given setK{\displaystyle K}, includingK{\displaystyle K}itself, and many possible opaque forests. For opaque forests, or more generally for systems ofrectifiable curves, their length can be measured in the standard way. For more general point sets, one-dimensionalHausdorff measurecan be used, and agrees with the standard length in the cases of line segments and rectifiable curves.[3]
Most research on this problem assumes that the given setK{\displaystyle K}is aconvex set. When it is not convex but merely aconnected set, it can be replaced by itsconvex hullwithout changing its opaque sets. Some variants of the problem restrict the opaque set to lie entirely inside or entirely outsideK{\displaystyle K}. In this case, it is called aninterior barrieror anexterior barrier, respectively. When this is not specified, the barrier is assumed to have no constraints on its location. Versions of the problem in which the opaque set must be connected or form a single curve have also been considered. It is not known whether everyconvex setP{\displaystyle P}has a shortest opaque set, or whether instead the lengths of its opaque sets might approach aninfimumwithout ever reaching it.[3]Every opaque set forP{\displaystyle P}can be approximated arbitrarily closely in length by an opaque forest,[4]and it has been conjectured that everyconvex polygonhas an opaque forest as its shortest opaque set, but this has not been proven.[3]
When the region to be covered is aconvex set, the length of its shortest opaque set must be at least half its perimeter and at most its perimeter. For some regions, additional improvements to these bounds can be made.
IfK{\displaystyle K}is a bounded convex set to be covered, then itsboundary∂K{\displaystyle \partial K}forms an opaque set whose length is the perimeter|∂K|{\displaystyle |\partial K|}. Therefore, the shortest possible length of an opaque set is at most the perimeter. For setsK{\displaystyle K}that are strictly convex, meaning that there are no line segments on the boundary, and for interior barriers, this bound is tight. Every point on the boundary must be contained in the opaque set, because every boundary point has atangent linethrough it that cannot be blocked by any other points.[5]The same reasoning shows that for interior barriers ofconvex polygons, allverticesmust be included. Therefore, theminimum Steiner treeof the vertices is the shortestconnectedopaque set, and thetraveling salesperson pathof the vertices is the shortestsingle-curveopaque set.[4]However, for interior barriers of non-polygonal convex sets that are not strictly convex, or for barriers that are not required to be connected, other opaque sets may be shorter; for instance, it is always possible to omit the longest line segment of the boundary. In these cases, the perimeter or Steiner tree length provide anupper boundon the length of an opaque set.[3][4]
There are several proofs that an opaque set for any convex setK{\displaystyle K}must have total length at least|∂K|/2{\displaystyle |\partial K|/2}, half the perimeter. One of the simplest involves theCrofton formula, according to which the length of any curve is proportional to its expected number of intersection points with a random line from an appropriateprobability distributionon lines. It is convenient to simplify the problem by approximatingK{\displaystyle K}by a strictly convex superset, which can be chosen to have perimeter arbitrarily close to the original set. Then, except for the tangent lines toK{\displaystyle K}(which form a vanishing fraction of all lines), a line that intersectsK{\displaystyle K}crosses its boundary twice. Therefore, if a random line intersectsK{\displaystyle K}with probabilityp{\displaystyle p}, the expected number of boundary crossings is2p{\displaystyle 2p}. But each line that intersectsK{\displaystyle K}intersects its opaque set, so the expected number of intersections with the opaque set is at leastp{\displaystyle p}, which is at least half that forK{\displaystyle K}. By the Crofton formula, the lengths of the boundary and barrier have the same proportion as these expected numbers.[6]
This lower bound of|∂K|/2{\displaystyle |\partial K|/2}on the length of an opaque set cannot be improved to have a larger constant factor than 1/2, because there exist examples of convex sets that have opaque sets whose length is close to this lower bound. In particular, for very long thin rectangles, one long side and two short sides form a barrier, with total length that can be made arbitrarily close to half the perimeter. Therefore, among lower bounds that consider only the perimeter of the coverage region, the bound of|∂K|/2{\displaystyle |\partial K|/2}is best possible.[6]However, getting closer to|∂K|/2{\displaystyle |\partial K|/2}in this way involves considering a sequence of shapes rather than just a single shape, because for any convex setK{\displaystyle K}that is not a triangle, there exists aδ{\displaystyle \delta }such that all opaque sets have length at least|∂K|/2+δ{\displaystyle |\partial K|/2+\delta }.[7]
For atriangle, as for any convex polygon, the shortest connected opaque set is its minimum Steiner tree.[8]In the case of a triangle, this tree can be described explicitly: if the widest angle of the triangle is2π/3{\displaystyle 2\pi /3}(120°) or more, it uses the two shortest edges of the triangle, and otherwise it consists of three line segments from the vertices to theFermat pointof the triangle.[9]However, without assuming connectivity, the optimality of the Steiner tree has not been demonstrated. Izumi has proven a small improvement to the perimeter-halving lower bound for theequilateral triangle.[10]
For aunit square, the perimeter is 4, the perimeter minus the longest edge is 3, and the length of the minimum Steiner tree is1+3≈2.732{\displaystyle 1+{\sqrt {3}}\approx 2.732}. However, a shorter, disconnected opaque forest is known, with length2+126≈2.639{\displaystyle {\sqrt {2}}+{\tfrac {1}{2}}{\sqrt {6}}\approx 2.639}. It consists of the minimum Steiner tree of three of the square's vertices, together with a line segment connecting the fourth vertex to the center.Ross Honsbergercredits its discovery to Maurice Poirier, a Canadian schoolteacher,[11]but it was already described in 1962 and 1964 by Jones.[12][13]It is known to be optimal among forests with only two components,[5][14]and has been conjectured to be the best possible more generally, but this remains unproven.[7]The perimeter-halving lower bound of 2 for the square, already proven by Jones,[12][13]can be improved slightly, to2.00002{\displaystyle 2.00002}, for any barrier that consists of at most countably manyrectifiable curves,[7]improving similar previous bounds that constrained the barrier to be placed only near to the given square.[6]
The case of theunit circlewas described in a 1995Scientific Americancolumn byIan Stewart, with a solution of length2+π{\displaystyle 2+\pi },[15]optimal for a single curve or connected barrier[8][16][17]but not for an opaque forest with multiple curves.Vance FaberandJan Mycielskicredit this single-curve solution toMenachem Magidorin 1974.[8]By 1980, E. Makai had already provided a better three-component solution, with length approximately4.7998{\displaystyle 4.7998},[18]rediscovered by John Day in a followup to Stewart's column.[19]The unknown length of the optimal solution has been called thebeam detection constant.[20]
Two published algorithms claim to generate the optimal opaque forest for arbitrary polygons, based on the idea that the optimal solution has a special structure: a Steiner tree for one triangle in atriangulation of the polygon, and a segment in each remaining triangle from one vertex to the opposite side, of length equal to the height of the triangle. This structure matches the conjectured structure of the optimal solution for a square. Although the optimal triangulation for a solution of this form is not part of the input to these algorithms, it can be found by the algorithms inpolynomial timeusingdynamic programming.[21][22]However, these algorithms do not correctly solve the problem for all polygons, because some polygons have shorter solutions with a different structure than the ones they find. In particular, for a long thin rectangle, the minimum Steiner tree of all four vertices is shorter than the triangulation-based solution that these algorithms find.[23]No known algorithm has been guaranteed to find a correct solution to the problem, regardless of its running time.[3]
Despite this setback, the shortest single-curve barrier of a convex polygon, which is the traveling salesperson path of its vertices, can be computed exactly inpolynomial timefor convex polygons by adynamic programmingalgorithm, in models of computation for whichsums of radicalscan be computed exactly.[4]There has also been more successful study ofapproximation algorithmsfor the problem, and for determining the coverage of a given barrier.
By the general bounds for opaque forest length in terms of perimeter, the perimeter of a convex set approximates its shortest opaque forest to within a factor of two in length. In two papers, Dumitrescu, Jiang, Pach, and Tóth provide severallinear-timeapproximation algorithms for the shortest opaque set for convex polygons, with betterapproximation ratiosthan two:
Additionally, because the shortest connected interior barrier of a convex polygon is given by the minimum Steiner tree, it has apolynomial-time approximation scheme.[4]
The region covered by a given forest can be determined as follows:
If the input consists ofn{\displaystyle n}line segments formingm{\displaystyle m}connected components, then each of then{\displaystyle n}setsCp{\displaystyle C_{p}}consists of at most2m{\displaystyle 2m}wedges. It follows that the combinatorial complexity of the coverage region, and the time to construct it, isO(m2n2){\displaystyle O(m^{2}n^{2})}as expressed inbig O notation.[25]
Although optimal in the worst case for inputs whose coverage region has combinatorial complexity matching this bound, this algorithm can be improved heuristically in practice by a preprocessing phase that merges overlapping pairs of hulls until all remaining hulls are disjoint, in timeO(nlog2n){\displaystyle O(n\log ^{2}n)}. If this reduces the input to a single hull, the more expensive sweeping and intersecting algorithm need not be run: in this case the hull is the coverage region.[26]
Mazurkiewicz (1916)showed that it is possible for an opaque set to avoid containing any nontrivial curves and still have finite total length.[1]A simplified construction ofBagemihl (1959), shown in the figure, produces an example for the unit square. The construction begins with line segments that form an opaque set with an additional property: the segments of negative slope block all lines of non-negative slope, while the segments of positive slope block all lines of non-positive slope. In the figure, the initial segments with this property are four disjoint segments along the diagonals of the square. Then, it repeatedly subdivides these segments while maintaining this property. At each level of the construction, each line segment is split by a small gap near its midpoint into two line segments, with slope of the same sign, that together block all lines of the opposite sign that were blocked by the original line segment. Thelimit setof this construction is aCantor spacethat, like all intermediate stages of the construction, is an opaque set for the square. With quickly decreasing gap sizes, the construction produces a set whoseHausdorff dimensionis one, and whose one-dimensionalHausdorff measure(a notion of length suitable for such sets) is finite.[2]
Thedistance setsof the boundary of a square, or of the four-segment shortest known opaque set for the square, both contain all distances in the interval from 0 to2{\displaystyle {\sqrt {2}}}. However, by using similar fractal constructions, it is also possible to find fractal opaque sets whose distance sets omit infinitely many of the distances in this interval, or that (assuming thecontinuum hypothesis) form aset of measure zero.[2]
Opaque sets were originally studied byStefan Mazurkiewiczin 1916.[1]Other early works on opaque sets include the papers ofH. M. Sen Guptaand N. C. Basu Mazumdar in 1955,[27]and byFrederick Bagemihlin 1959,[2]but these are primarily about the distance sets and topological properties of barriers rather than about minimizing their length. In a postscript to his paper, Bagemihl asked for the minimum length of an interior barrier for the square,[2]and subsequent work has largely focused on versions of the problem involving length minimization. They have been repeatedly posed, with multiple colorful formulations: digging a trench of as short a length as possible to find a straight buried telephone cable,[8]trying to find a nearby straight road while lost in a forest,[17]swimming to a straight shoreline while lost at sea,[4]efficiently painting walls to render a glass house opaque,[28]etc.
The problem has also been generalized to sets that block allgeodesicson aRiemannian manifold,[29][30]or that block lines through sets in higher-dimensions. In three dimensions, the corresponding question asks for a collection of surfaces of minimum total area that blocks all visibility across a solid. However, for some solids, such as a ball, it is not clear whether such a collection exists, or whether instead the area has aninfimumthat cannot be attained.[8][31]
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Music theoryanalyzes thepitch, timing, and structure of music. It uses mathematics to studyelements of musicsuch astempo,chord progression,form, andmeter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications ofset theory,abstract algebraandnumber theory.
While music theory has noaxiomaticfoundation in modern mathematics, the basis of musicalsoundcan be described mathematically (usingacoustics) and exhibits "a remarkable array of number properties".[1]
Though ancient Chinese, Indians, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,[2]thePythagoreans(in particularPhilolausandArchytas)[3]of ancient Greece were the first researchers known to have investigated the expression ofmusical scalesin terms of numericalratios,[4]particularly the ratios of small integers. Their central doctrine was that "all nature consists ofharmonyarising out of numbers".[5]
From the time ofPlato, harmony was considered a fundamental branch ofphysics, now known asmusical acoustics. EarlyIndianandChinesetheorists show similar approaches: all sought to show that the mathematical laws ofharmonicsandrhythmswere fundamental not only to our understanding of the world but to human well-being.[6]Confucius, like Pythagoras, regarded the small numbers 1,2,3, and 4 as the source of all perfection.[7]
Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement ofpulserepetition,accent,phraseand duration – music would not be possible.[8]Modern musical use of terms likemeterandmeasurealso reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time andperiodicitythat is fundamental to physics.[citation needed]
The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).[9]
Musical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, practicing economy and making use of repetition and order.[10]The common types of form known asbinaryandternary("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.[11][12]
Amusical scaleis a discrete set ofpitchesused in making or describing music. The most important scale in the Western tradition is thediatonic scalebut many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally theoctave. Theoctaveof any pitch refers to a frequency exactly twice that of the given pitch.
Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times, and so on, of the fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and so on of the fundamental are called suboctaves. There is no case in musical harmony where, if a given pitch be considered accordant, that its octaves are considered otherwise. Therefore, any note and its octaves will generally be found similarly named in musical systems (e.g. all will be calleddohorAorSa, as the case may be).
When expressed as a frequency bandwidth an octaveA2–A3spans from 110 Hz to 220 Hz (span=110 Hz). The next octave will span from 220 Hz to 440 Hz (span=220 Hz). The third octave spans from 440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave spans twice the frequency range of the previous octave.
Because we are often interested in the relations orratiosbetween the pitches (known asintervals) rather than the precise pitches themselves in describing a scale, it is usual to refer to all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written1/1), generally a note which functions as thetonicof the scale. For interval size comparison,centsare often used.
There are two main families of tuning systems:equal temperamentandjust tuning. Equal temperament scales are built by dividing an octave into intervals which are equal on alogarithmic scale, which results in perfectly evenly divided scales, but with ratios of frequencies which areirrational numbers. Just scales are built by multiplying frequencies byrational numbers, which results in simple ratios between frequencies, but with scale divisions that are uneven.
One major difference between equal temperament tunings and just tunings is differences inacoustical beatwhen two notes are sounded together, which affects the subjective experience ofconsonance and dissonance. Both of these systems, and the vast majority of music in general, have scales that repeat on the interval of everyoctave, which is defined as frequency ratio of 2:1. In other words, every time the frequency is doubled, the given scale repeats.
Below areOgg Vorbisfiles demonstrating the difference between just intonation and equal temperament. You might need to play the samples several times before you can detect the difference.
5-limit tuning, the most common form ofjust intonation, is a system of tuning using tones that areregular numberharmonicsof a singlefundamental frequency. This was one of the scalesJohannes Keplerpresented in hisHarmonices Mundi(1619) in connection with planetary motion. The same scale was given in transposed form by Scottish mathematician and musical theorist, Alexander Malcolm, in 1721 in his 'Treatise of Musick: Speculative, Practical and Historical',[13]and by theoristJose Wuerschmidtin the 20th century. A form of it is used in the music of northern India.
American composerTerry Rileyalso made use of the inverted form of it in his "Harp of New Albion". Just intonation gives superior results when there is little or nochord progression: voices and other instruments gravitate to just intonation whenever possible. However, it gives two different whole tone intervals (9:8 and 10:9) because a fixed tuned instrument, such as a piano, cannot change key.[14]To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the tonic frequency. For instance, with a tonic ofA4(A natural above middle C), the frequency is 440Hz, and a justly tuned fifth above it (E5) is simply 440×(3:2) = 660 Hz.
Pythagorean tuningis tuning based only on the perfect consonances, the (perfect) octave, perfect fifth, and perfect fourth. Thus the major third is considered not a third but a ditone, literally "two tones", and is (9:8)2= 81:64, rather than the independent and harmonic just 5:4 = 80:64 directly below. A whole tone is a secondary interval, being derived from two perfect fifths minus an octave, (3:2)2/2 = 9:8.
The just major third, 5:4 and minor third, 6:5, are asyntonic comma, 81:80, apart from their Pythagorean equivalents 81:64 and 32:27 respectively. According toCarlDahlhaus (1990, p. 187), "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals."
Westerncommon practice musicusually cannot be played in just intonation but requires a systematically tempered scale. The tempering can involve either the irregularities ofwell temperamentor be constructed as aregular temperament, either some form ofequal temperamentor some other regular meantone, but in all cases will involve the fundamental features ofmeantone temperament. For example, the root of chordii, if tuned to a fifth above the dominant, would be a major whole tone (9:8) above the tonic. If tuned a just minor third (6:5) below a just subdominant degree of 4:3, however, the interval from the tonic would equal a minor whole tone (10:9). Meantone temperament reduces the difference between 9:8 and 10:9. Their ratio, (9:8)/(10:9) = 81:80, is treated as a unison. The interval 81:80, called thesyntonic commaor comma of Didymus, is the key comma of meantone temperament.
Inequal temperament, the octave is divided into equal parts on the logarithmic scale. While it is possible to construct equal temperament scale with any number of notes (for example, the 24-toneArab tone system), the most common number is 12, which makes up the equal-temperamentchromatic scale. In western music, a division into twelve intervals is commonly assumed unless it is specified otherwise.
For the chromatic scale, the octave is divided into twelve equal parts, each semitone (half-step) is an interval of thetwelfth root of twoso that twelve of these equal half steps add up to exactly an octave. With fretted instruments it is very useful to use equal temperament so that the frets align evenly across the strings. In the European music tradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such asmusical keyboards. Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world.
Equally tempered scales have been used and instruments built using various other numbers of equal intervals. The19 equal temperament, first proposed and used byGuillaume Costeleyin the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal 12-semitone equal temperament at the cost of a flatter fifth. The overall effect is one of greater consonance.Twenty-four equal temperament, with twenty-four equally spaced tones, is widespread in the pedagogy andnotationofArabic music. However, in theory and practice, the intonation of Arabic music conforms torational ratios, as opposed to theirrational ratiosof equally tempered systems.[15]
While any analog to the equally temperedquarter toneis entirely absent from Arabic intonation systems, analogs to a three-quarter tone, orneutral second, frequently occur. These neutral seconds, however, vary slightly in their ratios dependent onmaqam, as well as geography. Indeed, Arabic music historianHabib Hassan Toumahas written that "the breadth of deviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music. To temper the scale by dividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristic elements of this musical culture."[15]
53 equal temperamentarises from the near equality of 53perfect fifthswith 31 octaves, and was noted byJing FangandNicholas Mercator.
Musical set theory uses the language of mathematicalset theoryin an elementary way to organize musical objects and describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such astranspositionandinversion, one can discover deep structures in the music. Operations such as transposition and inversion are calledisometriesbecause they preserve the intervals between tones in a set.
Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. For example, the pitch classes in an equally tempered octave form anabelian groupwith 12 elements. It is possible to describejust intonationin terms of afree abelian group.[16][17]
Transformational theoryis a branch of music theory developed byDavid Lewin. The theory allows for great generality because it emphasizes transformations between musical objects, rather than the musical objects themselves.
Theorists have also proposed musical applications of more sophisticated algebraic concepts. The theory of regular temperaments has been extensively developed with a wide range of sophisticated mathematics, for example by associating each regular temperament with a rational point on aGrassmannian.
Thechromatic scalehas a free and transitive action of thecyclic groupZ/12Z{\displaystyle \mathbb {Z} /12\mathbb {Z} }, with the action being defined viatranspositionof notes. So the chromatic scale can be thought of as atorsorfor the group.
Some composers have incorporated thegolden ratioandFibonacci numbersinto their work.[18][19]
ThemathematicianandmusicologistGuerino Mazzolahas usedcategory theory(topos theory) for a basis of music theory, which includes usingtopologyas a basis for a theory ofrhythmandmotives, anddifferential geometryas a basis for a theory ofmusical phrasing,tempo, andintonation.[20]
Music portal
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Cache poisoningrefers to acomputer security vulnerabilitywhere invalid entries can be placed into acache, which are then assumed to be valid when later used.[1]Two common varieties areDNS cache poisoning[2]andARP cache poisoning.Web cache poisoninginvolves the poisoning ofweb caches[3](which has led to security issues in programming languages, including all Python versions at the time in 2021, and expedited security updates[4]). Attacks on other, more specific, caches also exist.[5][6][7]
Thiscomputer securityarticle is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Cache_poisoning
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DNS spoofing, also referred to asDNS cache poisoning, is a form of computer securityhackingin which corruptDomain Name Systemdata is introduced into theDNS resolver'scache, causing thename serverto return an incorrect result record, e.g. anIP address. This results intraffic being divertedto any computer that the attacker chooses. Put simply, a hacker makes the device think it is connecting to the chosen website, when in reality, it is redirected to a different website by altering theIP addressassociated with thedomain namein the DNS server.[1]
ADomain Name System servertranslates a human-readabledomain name(such asexample.com) into a numericalIP addressthat is used toroutecommunications betweennodes.[2]Normally if the server does not know a requested translation it will ask another server, and the process continuesrecursively. To increase performance, a server will typically remember (cache) these translations for a certain amount of time. This means if it receives another request for the same translation, it can reply without needing to ask any other servers, until that cache expires.
When a DNS server has received a false translation and caches it for performance optimization, it is consideredpoisoned, and it supplies the false data to clients. If a DNS server is poisoned, it may return an incorrect IP address, diverting traffic to another computer (often an attacker's).[3]
Normally, a networked computer uses a DNS server provided by an Internet service provider (ISP) or the computer user's organization. DNS servers are used in an organization's network to improve resolution response performance by caching previously obtained query results. Poisoning attacks on a single DNS server can affect the users serviced directly by the compromised server or those serviced indirectly by its downstream server(s) if applicable.[4]
To perform acache poisoningattack, the attackerexploitsflaws in the DNS software. A server should correctly validate DNS responses to ensure that they are from an authoritative source (for example by usingDNSSEC); otherwise the server might end up caching the incorrect entries locally and serve them to other users that make the same request.
This attack can be used to redirect users from a website to another site of the attacker's choosing. For example, anattackerspoofsthe IP address DNS entries for a target website on a given DNS server and replaces them with the IP address of a server under their control. The attacker then creates files on the server under their control with names matching those on the target server. These files usually containmaliciouscontent, such ascomputer wormsorviruses. A user whose computer has referenced the poisoned DNS server gets tricked into accepting content coming from a non-authentic server and unknowingly downloads the malicious content. This technique can also be used forphishingattacks, where a fake version of a genuine website is created to gather personal details such as bank and credit/debit card details.
The vulnerability of systems to DNS cache poisoning goes beyond its immediate effects as it can open users up to further risks such asphishing,malwareinjections,denial of service, and website hijacking due to system vulnerabilities. Various methods, ranging from the use ofsocial engineeringtactics to the exploitation of weaknesses present in the DNS server software, can lead to these attacks.[5]
In the following variants, the entries for the serverns.target.examplewould be poisoned and redirected to the attacker's name server at IP addressw.x.y.z. These attacks assume that the name server fortarget.exampleisns.target.example.
To accomplish the attacks, the attacker must force the target DNS server to make a request for a domain controlled by one of the attacker's nameservers.[citation needed]
The first variant of DNS cache poisoning involves redirecting the name server of the attacker's domain to the name server of the target domain, then assigning that name server an IP address specified by the attacker.
DNS server's request: what are the address records forsubdomain.attacker.example?
Attacker's response:
A vulnerable server would cache the additional A-record (IP address) forns.target.example, allowing the attacker to resolve queries to the entiretarget.exampledomain.
The second variant of DNS cache poisoning involves redirecting the nameserver of another domain unrelated to the original request to an IP address specified by the attacker.[citation needed]
DNS server's request: what are the address records forsubdomain.attacker.example?
Attacker's response:
A vulnerable server would cache the unrelated authority information fortarget.example's NS-record (nameserverentry), allowing the attacker to resolve queries to the entiretarget.exampledomain.
Many cache poisoning attacks against DNS servers can be prevented by being less trusting of the information passed to them by other DNS servers, and ignoring any DNS records passed back that are not directly relevant to the query. For example, versions ofBIND9.5.0-P1[6]and above perform these checks.[7]Source port randomization for DNS requests, combined with the use of cryptographically secure random numbers for selecting both the source port and the 16-bitcryptographic nonce, can greatly reduce the probability of successful DNS race attacks.[citation needed]
However, when routers,firewalls,proxies, and other gateway devices performnetwork address translation(NAT), or more specifically, port address translation (PAT), they may rewrite source ports in order to track connection state. When modifying source ports, PAT devices may remove source port randomness implemented bynameserversand stub resolvers.[citation needed][8]
Secure DNS (DNSSEC) uses cryptographic digital signatures signed with a trustedpublic key certificateto determine the authenticity of data. DNSSEC can counter cache poisoning attacks. In 2010 DNSSEC was implemented in the Internet root zone servers,[9]but needs to be deployed on alltop level domainservers as well. The DNSSEC readiness of these is shown in thelist of Internet top-level domains. As of 2020, all of the original TLDs support DNSSEC, as do country code TLDs of most large countries, but many country-code TLDs still do not.[10]
This kind of attack can be mitigated at thetransport layerorapplication layerby performing end-to-end validation once a connection is established. A common example of this is the use ofTransport Layer Securityanddigital signatures. For example, by usingHTTPS(the secure version ofHTTP), users may check whether the server's digital certificate is valid and belongs to a website's expected owner.[11]Similarly, thesecure shellremote login program checks digital certificates at endpoints (if known) before proceeding with the session. For applications that download updates automatically, the application can embed a copy of the signing certificate locally and validate the signature stored in the software update against the embedded certificate.[12]
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Incomputer networking,IP address spoofingorIP spoofingis the creation ofInternet Protocol(IP)packetswith a false sourceIP address, for the purpose of impersonating another computing system.[1]
The basic protocol for sending data over the Internet network and many othercomputer networksis theInternet Protocol(IP). The protocol specifies that each IP packet must have aheaderwhich contains (among other things) the IP address of the sender of the packet. The source IP address is normally the address that the packet was sent from, but the sender's address in the header can be altered, so that to the recipient it appears that the packet came from another source.
The protocol requires the receiving computer to send back a response to the source IP address therefore spoofing is mainly used when the sender can anticipate the network response or does not care about the response.
The source IP address provides only limited information about the sender. It may provide general information on the region, city and town when on the packet was sent. It does not provide information on the identity of the sender or the computer being used.
IP address spoofing involving the use of a trusted IP address can be used by network intruders to overcome network security measures, such asauthenticationbased on IP addresses. This type of attack is most effective where trust relationships exist between machines. For example, it is common on some corporate networks to have internal systems trust each other, so that users can log in without a username or password provided they are connecting from another machine on the internal network – which would require them already being logged in. By spoofing a connection from a trusted machine, an attacker on the same network may be able to access the target machine without authentication.
IP address spoofing is most frequently used indenial-of-service attacks,[2]where the objective is to flood the target with an overwhelming volume of traffic, and the attacker does not care about receiving responses to the attack packets. Packets with spoofed IP addresses are more difficult to filter since each spoofed packet appears to come from a different address, and they hide the true source of the attack. Denial of service attacks that use spoofing typically randomly choose addresses from the entire IP address space, though more sophisticated spoofing mechanisms might avoid non-routable addresses or unused portions of the IP address space. The proliferation of largebotnetsmakes spoofing less important in denial of service attacks, but attackers typically have spoofing available as a tool, if they want to use it, so defenses against denial-of-service attacks that rely on the validity of the source IP address in attack packets might have trouble with spoofed packets.
In DDoS attacks, the attacker may decide to spoof the IP source address to randomly generated addresses, so the victim machine cannot distinguish between the spoofed packets and legitimate packets. The replies would then be sent to random addresses that do not end up anywhere in particular. Such packages-to-nowhere are called thebackscatter, and there arenetwork telescopesmonitoring backscatter to measure the statistical intensity of DDoS attacks on the internet over time.[3]
The use of packets with a false source IP address is not always evidence of malicious intent. For example, in performance testing of websites, hundreds or even thousands of "vusers" (virtual users) may be created, each executing a test script against the website under test, in order to simulate what will happen when the system goes "live" and a large number of users log in simultaneously.[citation needed]
Since each user will normally have its own IP address, commercial testing products (such asHP LoadRunner,WebLOAD, and others) can use IP spoofing, allowing each user its own "return address" as well.[citation needed]
IP spoofing is also used in some server-side load balancing. It lets the load balancer spray incoming traffic, but not need to be in the return path from the servers to the client. This saves a networking hop through switches and the load balancer as well as outbound message processing load on the load balancer. Output usually has more packets and bytes, so the savings are significant.[4][5]
Configuration and services that are vulnerable to IP spoofing:
Packet filteringis one defense against IPspoofing attacks. The gateway to a network usually performsingress filtering, which is blocking of packets from outside the network with a source address inside the network. This prevents an outside attacker spoofing the address of an internal machine. Ideally, the gateway would also performegress filteringon outgoing packets, which is blocking of packets from inside the network with a source address that is not inside. This prevents an attacker within the network performing filtering from launching IP spoofing attacks against external machines. Anintrusion detection system(IDS) is a common use of packet filtering, which has been used to secure the environments for sharing data over network and host-based IDS approaches.[6]
It is also recommended to design network protocols and services so that they do not rely on the source IP address for authentication.
Someupper layer protocolshave their own defense against IP spoofing attacks. For example,Transmission Control Protocol(TCP) uses sequence numbers negotiated with the remote machine to ensure that arriving packets are part of an established connection. Since the attacker normally cannot see any reply packets, the sequence number must be guessed in order to hijack the connection. The poor implementation in many older operating systems and network devices, however, means that TCP sequence numbers can be predicted.
The termspoofingis also sometimes used to refer toheader forgery, the insertion of false or misleading information ine-mailornetnewsheaders. Falsified headers are used to mislead the recipient, or network applications, as to the origin of a message. This is a common technique ofspammersandsporgers, who wish to conceal the origin of their messages to avoid being tracked.
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https://en.wikipedia.org/wiki/IP_address_spoofing
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MAC spoofingis a technique for changing a factory-assignedMedia Access Control (MAC) addressof anetwork interfaceon anetworkeddevice. The MAC address that is hard-coded on anetwork interface controller(NIC) cannot be changed. However, manydriversallow the MAC address to be changed. Additionally, there are tools which can make an operating system believe that the NIC has the MAC address of a user's choosing. The process of masking a MAC address is known as MAC spoofing. Essentially, MAC spoofing entails changing a computer's identity, for any reason.[1]
Changing the assigned MAC address may allow the user to bypassaccess control listsonserversorrouters, either hiding a computer on a network or allowing it to impersonate another network device. It may also allow the user to bypass MAC address blacklisting to regain access to a Wi-Fi network.However, MAC spoofing does not work when trying to bypass parental controls if automatic MAC filtering is turned on.[citation needed]MAC spoofing is done for legitimate and illicit purposes alike.[2][dead link]
ManyISPsregister the client's MAC address for service and billing services.[3]Since MAC addresses are unique and hard-coded onnetwork interface controller (NIC)cards,[1]when the client wants to connect a new device or change an existing one, the ISP will detect different MAC addresses and might not grant Internet access to those new devices. This can be circumvented easily by MAC spoofing, with the client only needing to spoof the new device's MAC address so it appears to be the MAC address that was registered by the ISP.[3]In this case, the client spoofs their MAC address to gain Internet access from multiple devices. While this is generally a legitimate case, MAC spoofing of new devices can be considered illegal if the ISP's user agreement prevents the user from connecting more than one device to their service. Moreover, the client is not the only person who can spoof their MAC address to gain access to the ISP.Computer crackerscan gain unauthorized access to the ISP via the same technique. This allows them to gain access to unauthorized services, while being difficult to identify and track as they are using the client's identity. This action is considered an illegitimate and illegal use of MAC spoofing.[4]
This also applies tocustomer-premises equipment, such ascableandDSL modems. If leased to the customer on a monthly basis, the equipment has a hard-coded MAC address known to the provider's distribution networks, allowing service to be established as long as the customer is not in billing arrears. In cases where the provider allows customers to provide their own equipment (and thus avoid the monthly leasing fee on their bill), the provider sometimes requires that the customer provide the MAC address of their equipment before service is established.
Somesoftwarecan only be installed and run on systems with pre-defined MAC addresses as stated in the softwareend-user license agreement, and users have to comply with this requirement in order to gain access to the software. If the user has to install different hardware due to malfunction of the original device or if there is a problem with the user's NIC card, then the software will not recognize the new hardware. However, this problem can be solved using MAC spoofing. The user has to spoof the new MAC address so that it appears to be the address that was in use when the software was registered.[citation needed]Legal issues might arise if the software is run on multiple devices at once by using MAC spoofing. At the same time, the user can access software for which they have not secured a license. Contacting the software vendor might be the safest route to take if there is a hardware problem preventing access to the software.
Some softwares may also performMAC filteringin an attempt to ensure unauthorized users cannot gain access to certain networks which would otherwise be freely accessible with the software. Such cases can be considered illegitimate or illegal activity and legal action may be taken.[5]
If a user chooses to spoof their MAC address in order to protect their privacy,[citation needed]this is called identity masking. As an example motivation, on Wi-Fi network connections a MAC address is not encrypted. Even the secureIEEE 802.11i-2004(WPA) encryption method does not prevent Wi-Fi networks from sending out MAC addresses.[citation needed]Hence, in order to avoid being tracked, the user might choose to spoof the device's MAC address. However, computer crackers use the same technique to bypass access control methods such as MAC filtering, without revealing their identity. MAC filtering prevents access to a network if the MAC address of the device attempting to connect does not match any addresses marked as allowed, which is used by some networks. Computer crackers can use MAC spoofing to gain access to networks utilising MAC filtering if any of the allowed MAC addresses are known to them, possibly with the intent of causing damage, while appearing to be one of the legitimate users of the network. As a result, the real offender may go undetected by law enforcement.[citation needed]
To prevent third parties from using MAC addresses to track devices, Android, Linux, iOS, macOS, and Windows[6]have implemented MAC address randomization. In June 2014, Apple announced that future versions of iOS would randomize MAC addresses for all WiFi connections. TheLinux kernelhas supported MAC address randomization during network scans since March 2015,[7]but drivers need to be updated to use this feature.[8]Windows has supported it since the release ofWindows 10[6]in July 2015.
Although MAC address spoofing is not illegal, its practice has caused controversy in some cases. In the 2012 indictment againstAaron Swartz, an Internethacktivistwho was accused of illegally accessing files from theJSTORdigital library, prosecutors claimed that because he had spoofed his MAC address, this showed purposeful intent to commit criminal acts.[5]In June 2014, Apple announced that future versions of their iOS platform would randomize MAC addresses for all WiFi connections, making it more difficult for internet service providers to track user activities and identities, which resurrected moral and legal arguments surrounding the practice of MAC spoofing among several blogs and newspapers.[9]
MAC address spoofing is limited to the localbroadcast domain. UnlikeIP address spoofing, where senders spoof their IP address in order to cause the receiver to send the response elsewhere, in MAC address spoofing the response is usually received by the spoofing party if MAC filtering is not turned on making the spoofer able to impersonate a new device.
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https://en.wikipedia.org/wiki/MAC_spoofing
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Proxy ARPis a technique by which aproxy serveron a given network answers theAddress Resolution Protocol(ARP) queries for anIP addressthat is not on that network. The proxy is aware of the location of the traffic's destination and offers its ownMAC addressas the (ostensibly final) destination.[1]The traffic directed to the proxy address is then typically routed by the proxy to the intended destination via another interface or via atunnel.
The process, which results in the proxy server responding with its own MAC address to an ARP request for a different IP address for proxying purposes, is sometimes referred to aspublishing.
Below are some typical uses for proxy ARP:
Disadvantage of proxy ARP include scalability as ARP resolution by a proxy is required for every device routed in this manner, and reliability as no fallback mechanism is present, and masquerading can be confusing in some environments.
Proxy ARP can create DoS attacks on networks if misconfigured. For example, a misconfigured router with proxy ARP has the ability to receive packets destined for other hosts (as it gives its own MAC address in response to ARP requests for other hosts/routers), but may not have the ability to correctly forward these packets on to their final destination, thus blackholing the traffic.
Proxy ARP can hide device misconfigurations, such as a missing or incorrectdefault gateway.
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https://en.wikipedia.org/wiki/Proxy_ARP
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Lower Saxony[a]is aGerman state(Land) innorthwestern Germany. It is the second-largest state by land area, with 47,614 km2(18,384 sq mi), and fourth-largest in population (8 million in 2021) among the 16Länderof theFederal Republic of Germany. In rural areas,Northern Low SaxonandSaterland Frisianare still spoken, though by declining numbers of people.
Lower Saxony borders on (from north and clockwise) theNorth Sea, the states ofSchleswig-Holstein,Hamburg,Mecklenburg-Vorpommern,Brandenburg,Saxony-Anhalt,Thuringia,HesseandNorth Rhine-Westphalia, and theNetherlands. Furthermore, thestate of Bremenforms twoenclaveswithin Lower Saxony, one being the city ofBremen, the other itsseaport,Bremerhaven(which is asemi-exclave, as it has a coastline). Lower Saxony thus borders more neighbours than any other singleBundesland. The state's largest cities are the state capitalHanover,Braunschweig(Brunswick),Oldenburg,Osnabrück,Wolfsburg,Göttingen,Salzgitter,Hildesheim, mainly situated in its central and southern parts, except Oldenburg.
Lower Saxony is the onlyBundeslandthat encompasses both maritime and mountainous areas. The northwestern area of the state, on the coast of the North Sea, is calledEast Frisiaand the sevenEast Frisian Islandsoffshore are popular with tourists. In the extreme west of Lower Saxony is theEmsland, an economically emerging but rather sparsely populated area, once dominated by inaccessible swamps. The northern half of Lower Saxony, also known as theNorth German Plain, is almost invariably flat except for the gentle hills around the Bremengeestland. Towards the south and southwest lie the northern parts of theCentral Uplands: theWeser Uplandsand theHarzMountains. Between these two lie theLower Saxon Hills, a range of low ridges.
The region in the northeast, theLüneburg Heath(Lüneburger Heide), is the largest heathland area of Germany. In the Middle Ages, the town of Lüneburg was wealthy due to salt-mining and the salt trade. To the north theElbevalley separates Lower Saxony from Hamburg, Schleswig-Holstein,Mecklenburg-Vorpommern, and Brandenburg. The left banks of the Elbe downstream Hamburg are known as theAltes Land(Old Country). Due to its gentle local climate and fertile soil, it is the state's largest area of fruit farming, its chief produce beingapples.
Most of the state's territory was part of the historicKingdom of Hanover, and the state of Lower Saxony has adopted the coat of arms and other symbols of the former kingdom. It was created by the merger of theState of Hanoverwith three smaller states on 1 November 1946.
Lower Saxony has a natural boundary in the north in the North Sea and thelowerand middle reaches of theRiver Elbe, although parts of the city of Hamburg lie south of the Elbe. The state and city of Bremen is anenclaveentirely surrounded by Lower Saxony. TheBremen/Oldenburg Metropolitan Regionis a cooperative body for the enclave area. To the southeast, the state border runs through the Harz, low mountains that are part of the GermanCentral Uplands. The northeast and west of the state, which form roughly three-quarters of its land area, belong to the North German Plain, while the south is in theLower Saxon Hills, including theWeser Uplands,Leine Uplands,Schaumburg Land,Brunswick Land,Untereichsfeld,Elm, andLappwald. In the northeast of Lower Saxony is the Lüneburg Heath. The heath is dominated by the poor, sandy soils of thegeest, while in the central-east and southeast in theloessbördezone, productive soils with high natural fertility occur. Under these conditions—withloamandsand-containing soils—the land is well-developedagriculturally. In the west lie theCounty of Bentheim,Osnabrück Land,Emsland,Oldenburg Land,Ammerland,Oldenburg Münsterland, and on the coastEast Frisia.
The state is dominated by several large northwards-flowing rivers, including theEms,Weser,Aller, and the Elbe.
The highest point in Lower Saxony is theWurmberg(971 metres or 3,186 feet) in the Harz. Most of thesignificant hills and mountainsare found in the southeastern part of the state. The lowest point in the state, at about 2.5 metres or 8 feet 2 inches below sea level, is a depression nearFreepsumin East Frisia.
The state's economy, population, and infrastructure are centred on the cities and towns of Hanover, Stadthagen, Celle, Braunschweig,Wolfsburg, Hildesheim, and Salzgitter. Together with Göttingen in southern Lower Saxony, they form the core of theHannover–Braunschweig–Göttingen–Wolfsburg Metropolitan Region.
Lower Saxony has clear regional divisions that manifest themselves geographically, as well as historically and culturally. In the regions that used to be independent, especially the heartlands of the former states ofBrunswick,Hanover,OldenburgandSchaumburg-Lippe, a marked local regional awareness exists. By contrast, the areas surrounding the Hanseatic cities of Bremen and Hamburg are much more oriented towards those centres.
Sometimes, overlaps and transition areas happen between the various regions of Lower Saxony. Several of the regions listed here are part of other, larger regions, that are also included in the list.
Just under 20% of the land area of Lower Saxony is designated as nature parks, i.e.:Dümmer,Elbhöhen-Wendland,Elm-Lappwald,Harz,Lüneburger Heide,Münden,Terra.vita,Solling-Vogler,Lake Steinhude,Südheide,Weser Uplands,Wildeshausen Geest,Bourtanger Moor-Bargerveen.[4]
Lower Saxony falls climatically into thenorth temperate zoneof central Europe that is affected by prevailingWesterliesand is located in a transition zone between themaritime climateofWestern Europeand thecontinental climateofEastern Europe. This transition is clearly noticeable within the state: while the northwest experiences an Atlantic (North Sea coastal) to Sub-Atlantic climate, with comparatively low variations in temperature during the course of the year and a surplus water budget, the climate towards the southeast is increasingly affected by the Continent. This is clearly shown by greater temperature variations between the summer and winter halves of the year and in lower and more variable amounts of precipitation across the year. This sub-continental effect is most sharply seen in the Wendland, in the Weser Uplands (Hamelin to Göttingen) and in the area of Helmstedt. The highest levels of precipitation are experienced in the Harz because the Lower Saxon part forms thewindward sideof this mountain range against whichorographic rainfalls. The average annual temperature is 8 °C (46 °F); 7.5 °C (45.5 °F) in theAltes Landand 8.5 °C (47.3 °F) in thedistrict of Cloppenburg.
Lower Saxony is divided into 37 districts (Landkreiseor simplyKreise):
Furthermore, there are eight urban districts and two cities with special status:
Between 1946 and 2004, the state's districts and independent towns were grouped into eight regions, with a different status for two regions (Verwaltungsbezirke), comprising the formerly free states of Brunswick and Oldenburg. In 1978 these regions were merged into four governorates (Regierungsbezirke). In 2005 theBezirksregierungen(regional governments) were again split into separate bodies.
1946–1978:
1978–2004:
On 1 January 2005 the four administrative regions or governorates (Regierungsbezirke), into which Lower Saxony had been hitherto divided, were dissolved.[5]These were the governorates of Braunschweig, Hanover, Lüneburg and Weser-Ems.
The largest towns in Lower Saxony as of 31 December 2017:[6]
The name ofSaxonyderives from that of theGermanicconfederation of tribes called theSaxons. Before the late medieval period, there was a singleDuchy of Saxony. The term "Lower Saxony" was used after the dissolution of the stem duchy in the late 13th century to distinguish the parts of the former duchy ruled by theHouse of Welffrom theElectorate of Saxonyon one hand, and from theDuchy of Westphaliaon the other.
The name and coat of arms of the present state go back to theGermanic tribe of Saxons. During theMigration Periodsome of the Saxon peoples left their homeland inHolsteinabout the 3rd century and pushed southwards over theElbe, where they expanded into the sparsely populated regions in the rest of the lowlands, in present-day Northwest Germany and the northeastern part of what is now theNetherlands. From about the 7th century the Saxons had occupied a settlement area that roughly corresponds to the present state of Lower Saxony, ofWestphaliaand a number of areas to the east, for example, in what is now west and north Saxony-Anhalt. The land of the Saxons was divided into about 60Gaue. TheFrisianshad not moved into this region; for centuries they preserved their independence in the most northwesterly region of the present-day Lower Saxon territory. The original language of the folk in the area of Old Saxony wasWest Low German, one of the varieties of language in the Low German dialect group.
The establishment of permanent boundaries between what later became Lower Saxony and Westphalia began in the 12th century. In 1260, in a treaty between theArchbishopric of Cologneand theDuchy of Brunswick-Lüneburgthe lands claimed by the two territories were separated from each other.[7]The border ran along the Weser to a point north of Nienburg. The northern part of the Weser-Ems region was placed under the rule of Brunswick-Lüneburg.
The wordNiedersachsenwas first used before 1300 in a Dutch rhyming chronicle (Reimchronik). From the 14th century it referred to theDuchy of Saxe-Lauenburg(as opposed toSaxe-Wittenberg).[8]On the creation of theimperial circlesin 1500, aLower Saxon Circlewas distinguished from aLower Rhenish–Westphalian Circle. The latter included the following territories that, in whole or in part, belong today to the state of Lower Saxony: theBishopric of Osnabrück, theBishopric of Münster, theCounty of Bentheim, theCounty of Hoya, the Principality ofEast Frisia, thePrincipality of Verden, theCounty of Diepholz, theCounty of Oldenburg, theCounty of Schaumburgand theCounty of Spiegelberg. At the same time a distinction was made with the eastern part of the old Saxon lands from thecentral Germanprincipalities later calledUpper Saxonyfor dynastic reasons.[9]
The close historical links between the domains of the Lower Saxon Circle now in modern Lower Saxony survived for centuries especially from a dynastic point of view. The majority of historic territories whose land now lies within Lower Saxony were sub-principalities of the medieval, Welf estates of theDuchy of Brunswick-Lüneburg. All the Welf princes called themselves dukes "of Brunswick and Lüneburg" despite often ruling parts of a duchy that was forever being divided and reunited as various Welf lines multiplied or died out.
Two major principalities survived east of the Weser after the Napoleonic Wars: theKingdom of Hanoverand theDuchy of Brunswick(after 1866 Hanover became aPrussian province; after 1919 Brunswick became a free state). Historically a close tie existed between the royal house of Hanover (Electorate of Hanover) and theUnited Kingdom of Great Britain and Irelandas a result of theirpersonal unionin the 18th century (the personal union was dissolved whenVictoriabecame the Queen of the United Kingdom in 1837 because Hanover did not allow female rulers).
West of the RiverHuntea "de-Westphalianising process" began in 1815.[10]After theCongress of Viennathe territories of the later administrative regions (Regierungsbezirke) ofOsnabrückandAurichtransferred to the Kingdom of Hanover. TheGrand Duchy of Oldenburgand thePrincipality of Schaumburg-Lipperetained state autonomy. Nevertheless, the entire Weser-Ems region (including the city ofBremen) were grouped in 1920 into a Lower Saxon Constituency Association (Wahlkreisverband IX (Niedersachsen)). This indicates that at that time the western administrations of thePrussianProvince of Hanover and the state ofOldenburgwere perceived as being "Lower Saxon".
The forerunners of today's state of Lower Saxony were lands that were geographically and, to some extent, institutionally interrelated from very early on. TheCounty of Schaumburg(not to be confused with the Principality of Schaumburg-Lippe) around the towns ofRintelnandHessisch Oldendorfdid indeed belong to the Prussian province ofHesse-Nassauuntil 1932, a province that also included large parts of the present state of Hesse, including the cities ofKassel,WiesbadenandFrankfurt am Main; but in 1932 the County of Schaumburg became part of the Prussian Province of Hanover.
When theNazi Partyseized powerin 1933, they quickly transformed Germany into a highly centralised state and divided the entireThird ReichintoGauewhich largely superseded (but did not outright replace) Germany's traditional federal system. Nevertheless, some changes to the old state and provincial borders were made in 1937, notably including the city ofCuxhavenbeing fully integrated into the Prussian Province of Hanover under theGreater Hamburg Act. The effect of this Nazi-era change was that in 1946, after the Third Reich had collapsed and when state of Lower Saxony was founded, only four states needed to be merged. With the exception of Bremen and the areas that were ceded to theSoviet Occupation Zonein 1945, all those areas allocated to the new state of Lower Saxony in 1946, had already been merged into the "Constituency Association of Lower Saxony" in 1920.
In a lecture on 14 September 2007, Dietmar von Reeken described the emergence of a "Lower Saxony consciousness" in the 19th century, the geographical basis of which was used to invent a territorial construct: the resultinglocal heritagesocieties (Heimatvereine) and their associated magazines routinely used the terms "Lower Saxony" or "Lower Saxon" in their names. At the end of the 1920s in the context of discussions about a reform of the Reich, and promoted by the expanding local heritage movement (Heimatbewegung), a 25-year conflict started between "Lower Saxony" and "Westphalia". The supporters of this dispute were administrative officials and politicians, but regionally focussed scientists of various disciplines were supposed to have fuelled the arguments. In the 1930s, a real Lower Saxony did not yet exist, but there were a plethora of institutions that would have called themselves "Lower Saxon". The motives and arguments in the disputes between "Lower Saxony" and "Westphalia" were very similar on both sides: economic interests, political aims, cultural interests and historical aspects.[11]
After theSecond World Warmost of Northwest Germany lay within theBritish Zone of Occupation. On 23 August 1946, theBritish Military GovernmentissuedOrdinance No. 46"Concerning the dissolution of the provinces of the former state ofPrussiain the British Zone and their reconstitution as independent states", which initially established theState of Hanoveron the territory of the former Prussian Province of Hanover. Its minister president,Hinrich Wilhelm Kopf, had already suggested in June 1945 the formation of a state of Lower Saxony, that was to include the largest possible region in the middle of the British Zone. In addition to the regions that actually became Lower Saxony subsequently, Kopf asked, in a memorandum dated April 1946, for the inclusion of the former Prussian district ofMinden-Ravensberg(i.e. the Westphalian city ofBielefeldas well as the Westphalian districts ofMinden,Lübbecke,Bielefeld,HerfordandHalle), thedistrict of Tecklenburgand the state ofLippe.[12]Kopf's plan was ultimately based on a draft for the reform of the German Empire from the late 1920s by Georg Schnath and Kurt Brüning. The strongWelfconnotations of this draft, according to Thomas Vogtherr, did not simplify the development of a Lower Saxon identity after 1946.[13]
An alternative model, proposed by politicians in Oldenburg and Brunswick, envisaged the foundation of the independent state of "Weser-Ems", that would be formed from the state of Oldenburg, the Hanseatic City of Bremen and the administrative regions of Aurich and Osnabrück. Several representatives of the state of Oldenburg even demanded the inclusion of the Hanoverian districts ofDiepholz,Syke,Osterholz-ScharmbeckandWesermündein the proposed state of "Weser-Ems". Likewise an enlarged State of Brunswick was proposed in the southeast to include theRegierungsbezirkofHildesheimand thedistrict of Gifhorn. Had this plan come to fruition, the territory of the present Lower Saxony would have consisted of three states of roughly equal size.
The district council ofVechtaprotested on 12 June 1946 against being incorporated into the metropolitan area of Hanover (Großraum Hannover). If the State of Oldenburg was to be dissolved, Vechta District would much rather be included in theWestphalianregion.[14]Particularly in the districts where there was a politicalCatholicismthe notion was widespread, thatOldenburg Münsterlandand theRegierungsbezirkof Osnabrück should be part of a newly formed State of Westphalia.[15]
Since the foundation of the states ofNorth Rhine-WestphaliaandHanoveron 23 August 1946 the northern and eastern border of North Rhine-Westphalia has largely been identical with that of the PrussianProvince of Westphalia. Only theFree State of Lippewas not incorporated into North Rhine-Westphalia until January 1947. With that the majority of the regions left of the Upper Weser became North Rhine-Westphalian.
In the end, at the meeting of the Zone Advisory Board on 20 September 1946, Kopf's proposal with regard to the division of the British occupation zone into three large states proved to be capable of gaining a majority.[16]Because this division of their occupation zone into relatively large states also met the interests of the British, on 8 November 1946 Regulation No. 55 of theBritishmilitary governmentwas issued, by which the State of Lower Saxony with its capitalHanoverwere founded, backdated to 1 November 1946. The state was formed by a merger of theFree States of Brunswick,of Oldenburgand ofSchaumburg-Lippewith the previously formed State of Hanover. But there were exceptions:
The demands of Dutch politicians that the Netherlands should be given the German regions east of the Dutch-German border aswar reparations, were roundly rejected at the London Conference of 26 March 1949. In fact only about 1.3 km2(0.50 sq mi) of west Lower Saxony was transferred to the Netherlands, in 1949.
→ see main articleDutch annexation of German territory after World War II
The firstLower Saxon parliamentorLandtagmet on 9 December 1946. It was not elected; rather it was established by the British Occupation Administration (a so-called "appointed parliament"). That same day the parliament elected theSocial Democrat,Hinrich Wilhelm Kopf, the former Hanoverian president (Regierungspräsident) as their first minister-president. Kopf led a five-party coalition, whose basic task was to rebuild a state afflicted by the war's rigours. Kopf's cabinet had to organise an improvement of food supplies and the reconstruction of the cities and towns destroyed by Allied air raids during the war years. Hinrich Wilhelm Kopf remained – interrupted by the time in office ofHeinrich Hellwege(1955–1959) – as the head of government in Lower Saxony until 1961.
The greatest problem facing the first state government in the immediate post-war years was the challenge of integrating hundreds of thousands ofrefugeesfrom Germany's former territories in the east (such asSilesiaandEast Prussia), which had been annexed byPolandand theSoviet Union. Lower Saxony was at the western end of the direct escape route from East Prussia and had the longest border with the Soviet Zone. On 3 October 1950 Lower Saxony took over the sponsorship of the very large number of refugees fromSilesia. In 1950 there was still a shortage of 730,000 homes according to official figures.
During the period when Germany was divided, the Lower Saxonborder crossing at Helmstedtfound itself on the main transport artery toWest Berlinand, from 1945 to 1990 was the busiest European border crossing point.
Of economic significance for the state was theVolkswagenconcern, that restarted the production of civilian vehicles in 1945, initially under British management, and in 1949 transferred into the ownership of the newly founded country ofWest Germanyand state of Lower Saxony. Overall, Lower Saxony, with its large tracts of rural countryside and few urban centres, was one of the industrially weaker regions of the federal republic for a long time. In 1960, 20% of the working population worked on the land. In the rest of the federal territory the figure was just 14%. Even in economically prosperous times the jobless totals in Lower Saxony are constantly higher than the federal average.
In 1961Georg Diederichstook office as the minister president of Lower Saxony as the successor to Hinrich Wilhelm Kopf. He was replaced in 1970 byAlfred Kubel. The arguments about theGorleben Nuclear Waste Repository, that began during the time in office of minister presidentErnst Albrecht(1976–1990), have played an important role in state and federal politics since the end of the 1970s.
In 1990Gerhard Schröderentered the office of minister-president. On 1 June 1993, the new Lower Saxon constitution entered force, replacing the "Provisional Lower Saxon Constitution" of 1951. It enablesreferendumsandplebiscitesand establishesenvironmental protectionas a fundamental state principle.
The former HanoverianAmt Neuhauswith its parishes of Dellien, Haar, Kaarßen, Neuhaus (Elbe), Stapel, Sückau,Sumteand Tripkau as well as the villages of Neu Bleckede, Neu Wendischthun and Stiepelse in the parish of Teldau and the historic Hanoverian region in the forest district of Bohldamm in the parish of Garlitz transferred with effect from 30 June 1993 fromMecklenburg-Vorpommernto Lower Saxony (Lüneburg district). From these parishes the new municipality of Amt Neuhaus was created on 1 October 1993.
In 1998Gerhard Glogowskisucceeded Gerhard Schröder who became Federal Chancellor. Because he had been linked with various scandals in his home city of Brunswick, he resigned in 1999 and was replaced bySigmar Gabriel.
From 2003 to his election as Federal President in 2010Christian Wulffwas minister president in Lower Saxony. TheOsnabrückerheaded a CDU-led coalition with the FDP as does his successor,David McAllister. After the elections on 20 January 2013 McAllister wasdeselected.[17]
At the end of 2014, there were almost 571,000 non-German citizens in Lower Saxony.[20]At the end of 2023, there were almost 1,085,315 non-German citizens in Lower Saxony.[21]
The highest share of migrants to Germans in Lower Saxony is in the urban agglomeration ofBraunschweig(44.8%) which consists the most populated parts ofSalzgittere.gLebenstedtorThiede, the townWolfenbüttel, parts of the districtsLandkreis Peine,Landkreis Gifhorn,Landkreis HelmstedtandLandkreis Wolfenbüttel. 46.4% ofSalzgitterhave a foreign background, which has the highest percentile of people from different countries in Lower Saxony.[22][23]
The following table illustrates the largest minority groups in Lower Saxony:
[24]
The 2011 census stated that a majority of the population were Christians (71.93%); 51.48% of the total population were members of the Protestant Church in Germany, 18.34% were Catholics, 2.11% were members of other Christian denominations, 2.27% were members of other religions. 25.8% have no denomination.[25]Even though there is a high level of official belonging to a Christian denomination, the people – especially in the cities – are highly secular in behaviour.
As of 2020, theProtestant Church in Germanywas the faith of 41.1% of the population.[26]It is organised in the fiveLandeskirchennamedEvangelical Lutheran State Church in Brunswick(comprising the formerFree State of Brunswick),Evangelical Lutheran Church of Hanover(comprising the formerProvince of Hanover),Evangelical Lutheran Church in Oldenburg(comprising the formerFree State of Oldenburg),Evangelical Lutheran Church of Schaumburg-Lippe(comprising the formerFree State of Schaumburg-Lippe), andEvangelical Reformed Church(covering all the state).
Together, these member churches of theProtestant Church in Germanygather a substantial part of the Protestant population in Germany.
TheCatholic Churchwas the faith of 16.3% of the population in 2020.[26]It is organised in the three dioceses ofOsnabrück(western part of the state),Münster(comprising the former Free State of Oldenburg) andHildesheim(northern and eastern part of the state). The Catholic faith is mainly concentrated to the regions of Oldenburger Münsterland, the region of Osnabrück, the region of Hildesheim and in the Western Eichsfeld.
42.6% of the Low Saxons were irreligious or adhere to other religions.[26]Judaism,IslamandBuddhismare minority faiths.
Thegross domestic product(GDP) of the state was 229.5 billion euros in 2018, accounting for 8.7% of German economic output. GDP per capita adjusted for purchasing power was 33,700 euros or 112% of the EU27 average in the same year. The GDP per employee was 100% of the EU average.[27]
Agriculture, strongly weighted towards the livestock sector, has always been a very important economic factor in the state. The north and northwest of Lower Saxony are mainly made up of coarse sandy soil that makes crop farming difficult and therefore grassland and cattle farming are more prevalent in those areas. Lower Saxony is home, in 2017, to one in five of Germany'scattle, one in three of the country'spigs, and 50% of itshens.[28]Wheat,potatoes,rye, andoatsare among the state's present-dayarablecrops. Towards the south and southeast, extensiveloesslayers in the soil left behind by the lastice ageallow high-yield crop farming. One of the principal crops there issugar beet. Consequently, the Land has a big food industry, mainly organised in small and medium-sized enterprises (SME). Big players areDeutsches MilchkontorandPHW Group(biggest German poultry farmer and producer).
Mining has also been an important source of income in Lower Saxony for centuries.Silver orebecame a foundation of notable economic prosperity in the Harz Mountains as early as the 12th century, whileiron miningin the Salzgitter area andsalt miningin various areas of the state became another important economic backbone. Although overall yields are comparatively low, Lower Saxony is also an important supplier of crude oil in the European Union. Mineral products still mined today includeiron.
Radioactive wasteis frequently transported in the area to the city ofSalzgitter, for thedeep geological repositorySchacht Konradand betweenSchacht Asse IIin theWolfenbütteldistrict andLindwedelandHöfer.
Manufacturingis another large part of the regional economy. Despite decades of gradual downsizing and restructuring, the carmakerVolkswagenwith its five production plants within the state's borders still remains the single biggest private-sector employer, its world headquarters inWolfsburg. Due to theVolkswagen Law, which has recently been ruled illegal by theEuropean Union's high court, the state of Lower Saxony is still the second-largest shareholder, owning 20.3% of the company.[29]Thanks to the importance of car manufacturing in Lower Saxony, a thriving supply industry is centred around its regional focal points. Other mainstays of the Lower Saxon industrial sector include aviation (the region of Stade is called CFK-Valley), shipbuilding (such asMeyer Werft),biotechnology, andsteel. Medicine plays a major role; Hanover and Göttingen have two large University Medical Schools and hospitals, andOtto Bockin Duderstadt is the largest producer ofprostheticsand associated componentry in the world.
The service sector has gained importance following the demise of manufacturing in the 1970s and 1980s. Important branches today are the tourism industry withTUI AGin Hanover, one of Europe's largest travel companies, as well astradeandtelecommunications. Hanover is one of Germany's main hubs for insurance and financial-services companies, for exampleTalanxandHannover Re.
In October 2018, the Lower Saxony unemployment rate stood at 5.0% and was marginally higher than the national average.[30]
Lower Saxony has fourWorld Heritage Sites.
Since 1948, politics in the state have been dominated by the centre-rightChristian Democratic Union(CDU) and the centre-leftSocial Democratic Party. Lower Saxony was one of the origins of the German environmentalist movement in reaction to the state government's support for underground nuclear waste disposal. This led to the formation of the German Green Party in 1980. Germany's multi-party system generally requires coalitions to be made because parties usually do not have outright majorities.
The Minister-President heads the state government, acting as a head of state (even if the federated states have the status of a state do not establish the office of a head of state but merge the functions with the head of the executive branch) as well as the government leader. They are elected by theLandtag of Lower Saxony.
The former Minister-President,Christian Wulff, led a coalition of his CDU with theFree Democratic Partybetween 2003 and 2010. Inthe 2008 election, the ruling CDU held on to its position as the leading party in the state, despite losing votes and seats. The CDU's coalition with the Free Democratic Party retained its majority although it was cut from 29 to 10. The election also saw the entry into the state parliament for the first time of the leftistThe Leftparty. On 1 July 2010David McAllisterwas elected Minister-President.
After thestate election on 20 January 2013,Stephan Weilof the Social Democrats was elected as the new Minister-President.[32]He governed incoalitionwith theGreens. After thestate election in September 2017, Stephan Weil remained Minister-President, with the SDP incoalitionwith theCDU. After the2022 Election, Weil was once again elected as Minister-President with an SDP-Greens coalition.[33]
The state of Lower Saxony was formed after World War II by merging the former states of Hanover, Oldenburg, Brunswick and Schaumburg-Lippe. Hanover, a former kingdom, is by far the largest of these contributors by area and population and has been a province of Prussia since 1866. The city of Hanover is the largest and capital city of Lower Saxony.
The constitution states that Lower Saxony be a free, republican, democratic,[34]social and environmentally sustainable state inside the Federal Republic of Germany; universal human rights, peace and justice are preassigned guidelines of society, and the human rights and civil liberties proclaimed by the constitution of the Federal Republic are genuine constituents of the constitution of Lower Saxony. Each citizen is entitled to education and there is universal compulsory school attendance.
All government authority is to be sanctioned by the will of the people, which expresses itself via elections and plebiscites. The legislative assembly is a unicameral parliament elected for terms of five years. The composition of the parliament obeys the principle of proportional representation of the participating political parties, but it is also ensured that each constituency delegates one directly elected representative. If a party wins more constituency delegates than their statewide share among the parties would determine, it can keep all these constituency delegates.
The states of the Federal Republic of Germany, and so Lower Saxony, have legislative responsibility and power mainly reduced to the policy fields of the school system, higher education, culture and media and police, while other policy fields like economic and social policies, foreign policy are a prerogative of the federal government. Hence the probably most important function of the federal states is their representation in the Federal Council (Bundesrat), where their approval on many crucial federal policy fields, including the tax system, is required for laws to become enacted.
Thecoat of armsshows a whitehorse(Saxon Steed) against a red background, which is an old symbol of the Saxon people. Legend has it that the horse was a symbol of the Saxon leaderWidukind, albeit a black horse against a yellow background. The colours changed after the Christian baptism of Widukind. White and red are colours (besides black and gold) of theHoly Roman EmpiresymbolisingChristas the saviour, who is still shown with a red cross against a white background.
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Elizabethan victory
Earl of WestmorlandEarl of NorthumberlandCountess of WestmorlandCountess of Northumberland (1538-1596)
TheRising of the Northof 1569, also called theRevolt of the Northern Earls,Northern Rebellionor theRebellion of the Earls, was an unsuccessful attempt byCatholicnobles fromNorthern Englandto depose QueenElizabeth I of Englandand replace her withMary, Queen of Scots.[1]
Elizabeth Isucceeded her half-sisterMary Ias queen of England in 1558. Elizabeth's accession was disputed due to the questioned legitimacy of the marriage of her parents (Henry VIIIandAnne Boleyn), and Elizabeth's own questioned legitimacy due to theAct of Succession 1536. Under Henry VIII and his advisorThomas Cromwell, power was gradually shifted from regional institutions to royal control. This course was encouraged by Elizabeth's counsellors such asWilliam Ceciland a policy of centralization was the approach favoured by Elizabeth herself at least in regards to the northern border region.
Opponents of Elizabeth looked toMary, Queen of Scots, the descendant of Henry VIII's sisterMargaret. The claims were initially put forward by Mary's father-in-law, KingHenry II of France, and Mary upheld them after her return to Scotland in 1561.
ManyEnglish Catholics, then a significant portion of the population, supported Mary's claim as a way to restore Roman Catholicism. This position was especially strong inNorthern England, where several powerful nobles were Roman Catholics; there had been similar risings against Henry VIII; thePilgrimage of Graceof 1536 andBigod's Rebellionof 1537. Supporters of Mary hoped for aid from France (among Scots) and possibly Spain (among English). Mary's position was strengthened by the birth of her son,James, in 1566 but weakened again when she was deposed in July 1567. Following this, she fled to England and at the time of the Rising was in the custody of theEarl of Shrewsbury, on Elizabeth's orders.
The rebellion was led byCharles Neville, 6th Earl of Westmorland, andThomas Percy, 7th Earl of Northumberland. Seven hundred soldiers assembled atBrancepeth Castle.[2]In November 1569 Westmorland and Northumberland occupiedDurham.Thomas Plumtree(see right) celebratedMassinDurham Cathedral.[3][4]From Durham, the rebels marched south toBramham Moor, while Elizabeth struggled to raise forces sufficient to confront them. But, hearing of a large force being raised by theEarl of Sussex, the rebels abandoned plans to besiegeYork, and capturedBarnard Castleinstead. They proceeded to Clifford Moor, but found little popular support. Sussex marched out from York on 13 December 1569 with 10,000 men against the rebels' 6,000,[5]and was followed by 12,000 men underBaron Clinton. The rebel earls retreated northward and finally dispersed their forces, fleeing into Scotland.
A questionable role in the rebellion was played byLeonard Dacre, an early sympathiser of Mary. At the outbreak of the rebellion, he travelled to Elizabeth's court at Windsor to claim the heritage of his young nephew, the 5th Baron Dacre. After the latter's untimely death in 1569, this had descended to his sisters, all married to sons ofThomas Howard, 4th Duke of Norfolk. Dacre returned to Northern England, ostensibly a faithful partisan of Elizabeth, but his intentions remain unclear.
After the retreat of the rebels, he seizedGreystoke Castleand fortified his ownNaworth Castle, where he gathered 3,000 Cumbrian troops and tried to keep up the appearance of good relations with the Queen. He held out against a siege of the royal army underBaron Hunsdonbut then attacked the retreating army atGelt River. Though Hunsdon was outnumbered, he charged Dacre's foot with his cavalry, killing 300–400 and capturing 200–300 men. Dacre escaped via Scotland toBrussels, where he died in exile.[6]
Some of the rebels escaped into Scotland.Regent Marwrote thatAgnes Gray, Lady Home, had been a busy worker to receive the rebels.[7]Two of the leaders, the Earls of Northumberland and Westmorland, had fled into Scotland. Northumberland was captured byJames Douglas, 4th Earl of Morton, and turned over to Elizabeth in 1572, who had him beheaded at York. After having been hidden atFerniehirst Castle, Westmorland escaped to Flanders, where he died impoverished. His family lost their ancestral homes and his wife,Jane Howard, also fled to the Continent. She lived the rest of her life under house arrest. Her brother, the Duke of Norfolk, was first imprisoned, then pardoned. He was imprisoned again following theRidolfi plotin 1571 and finally executed in 1572. Norfolk's treason charges included "comforting and relieving of the English rebels that stirred the Rebellion in the North since they have fled out of the realm."[8]Altogether, 600 supporters of Mary were executed, while many others fled into exile.
Queen Elizabeth declared martial law, exacting terrible retribution on the ordinary folk of the Yorkshire Dales, despite the lack of any popular support for the Earls' Rising, with her demand for at least 700 executions. The victims of this purge were, as a contemporary account said "wholly of the meanest sort of people", so that hardly a village escaped the sight of a public hanging.[9]
In 1570,Pope Pius Vhad tried to aid the rebellion by excommunicating Elizabeth and declaring her deposed in thepapal bullRegnans in Excelsis, but the document did not arrive until the rebellion had been suppressed. The bull gave Elizabeth more reason to view Catholics with suspicion. It inspired conspiracies to assassinate her, starting with the Ridolfi plot.
In 1587, Elizabeth broughtMary, Queen of Scots, to trial fortreason; she was convicted by the court and executed.
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TheRidolfi plotwas aCatholicplot in 1571 to overthrow QueenElizabeth Iof England and replace her withMary, Queen of Scots. The plot was hatched and planned byRoberto Ridolfi, an international banker who was able to travel between Brussels, Rome and Madrid to gather support without attracting too much suspicion.
Thomas Howard, 4th Duke of Norfolk, a Roman Catholic with a Protestant education, a second cousin of Queen Elizabeth's and the wealthiest landowner in the country, had been proposed as a possible husband for Mary since her imprisonment in 1568. This suited Norfolk, who had ambitions and felt Elizabeth persistently undervalued him.[1]In pursuit of his goals, he agreed to support theNorthern Rebellion, though he quickly lost his nerve. Norfolk was imprisoned in theTower of Londonfor nine months and only freed under house arrest when he confessed all and begged for mercy.[2]Pope Pius V, in his 1570papal bullRegnans in Excelsis, excommunicated the Protestant Elizabeth and permitted all faithful Catholics to do all they could to depose her. The majority of English Catholics ignored the bull,[3]but in response to it, Elizabeth became much harsher to Catholics and their sympathisers.[4]
Roberto Ridolfi, aFlorentinebanker and ardent Roman Catholic, had been involved in the planning of the Northern rebellion and had been plotting to overthrow Elizabeth as early as 1569.[5]With the failure of the rebellion, he concluded that foreign intervention was needed to restore Catholicism and bring Mary to the English throne, and so he began to contact potential conspirators. Mary's advisor,John Lesley, theBishop of Ross, gave his assent to the plot as the way to free Mary.[6]The plan was to have theDuke of Albainvade from the Netherlands with 10,000 men, foment a rebellion of the northern English nobility, murder Elizabeth, and marry Mary to Thomas Howard. Ridolfi optimistically estimated half of all English peers were Catholic and could muster in excess of 39,000 men.[7]Norfolk verbally assured Ridolfi that he was a Catholic, guaranteeing that was true even though the Duke had been raised as a Protestant as a child.[3][8]Both Mary and Norfolk, desperate to remedy their respective situations, agreed to the plot.[9]With their blessing, Ridolfi set off to the Continent to gain the support of Alba, Pius V, and King Philip II.
Ridolfi's co-conspirators, some of them mentioned above, played an important role in the plot to overthrow Elizabeth:
DonGuerau de Espés: Spain's ambassador to England, who was expelled after the discovery of his involvement. Elizabeth had raised her concerns about de Espés' behaviour withAnna of Austria.[10]
John Lesley: the Bishop of Ross, who was Mary Stuart's chief agent; arranged meetings and delivered letters for Mary during her house arrest.
Thomas Howard, 4th Duke of Norfolk, who was Queen Elizabeth I's second cousin. He was to marry Mary, Queen of Scots and together with her restore Catholic rule to the English and Scottish thrones. After the plot was discovered he was given a day-long trial that ended with his execution.
Mary, Queen of Scots: after it became clear that Elizabeth I was not going to restore her to the Scottish throne or return her to France, Mary plotted for her freedom. She wrote to Ridolfi denouncing the French and soliciting Spanish aid, while simultaneously professing friendship and loyalty to Elizabeth I and England. Giving her consent to the plot in March 1571, her role was to marry the Duke of Norfolk, with the plan that when the troops arrived in London she would be returned to the Scottish throne. However, when the plot was uncovered, her deep involvement in it altered Elizabeth's opinion of Mary; Elizabeth never spoke of restoring her to the throne again.
King Philip II, who welcomed Ridolfi to court and, with the council, discussed the plot's pros and cons. He supported overthrowing Elizabeth and later came to support the assassination. Philip, however, disapproved of the papal bull against Elizabeth because, according to Cyril Hamshere, he feared it would "prompt Elizabeth to take reprisals against Catholics."[11]
Fernando Álvarez de Toledo, 3rd Duke of Alba, who was the leader of the Spanish army stationed in the Netherlands and was to lead more than 10,000 men to Harwich or Portsmouth. His army was to invade England and make its way to London to establish Mary on the throne.
PopePius V, who made Ridolfi his papal agent in England in 1567, was not only aware of the plot but gave his written approval in a letter for Ridolfi to take to Philip II.
In 1571, Elizabeth's intelligence network was sending her information about a plot against her life. By gaining the confidence of Spain's ambassador to England,John Hawkinslearned the details of the conspiracy and notified the government so as to arrest the plotters. Elizabeth was also sent a private warning byCosimo I de' Medici, Grand Duke of Tuscany, who had learned of the plot against her.Charles Baillie, Ridolfi's messenger, was arrested on c.12 April 1571 atDoverfor carrying compromising letters, and by the use oftortureand prison informers such asWilliam Herle, he was forced to reveal the cipher of the messages he carried.
On 29 August 1571, Norfolk's secretaries William Barker and Robert Higford entrusted to Thomas Browne, aShrewsbury draper, what was purported to be a bag of silver coin for delivery to Laurence Bannister, one of Norfolk's officials in the north of England. Browne grew suspicious of the bag's weight, opened it, and discovered 600 pounds in gold from the French ambassador, destined for Scotland on Mary's behalf, and ciphered letters. Because he knew Norfolk was under suspicion, Browne reported his find toWilliam Cecil, 1st Baron Burghley, theSecretary of State. Higford and Barker were interrogated, the letters were partly deciphered, and a search for the cipher key at Howard House uncovered a ciphered letter from Mary Stuart hidden under a doormat.
Norfolk's servants were arrested and interrogated, and confessions were extracted from them by threats or application of torture. Sir Thomas Smith and Thomas Wilson were sent to confront Norfolk, who claimed the money was for his own private purposes. The deciphered letter, however, proved that he was lying. Unaware of his servants' confessions or the survival of letters which, contrary to his instructions, had not been burnt, he denied the charges against him. On 7 September, the queen's warrant for conveying him to theTower of Londonarrived. Thereupon, the duke admitted a degree of involvement in the transmission of money and correspondence to Mary's Scottish supporters. In January 1572, Norfolk was tried and convicted on three counts of high treason, and on 2 June he was beheaded onTower Hill.
Guerau de Spes, the Spanish ambassador, was expelled from the country in January 1571.[12]Still abroad when the plot was discovered, Ridolfi never returned to England; he became a Florentine senator in 1600.
Despite his plot's ultimate failure, Roberto Ridolfi's story is surprising and memorable. He had played the relatively minor role of banker but nevertheless found himself at the centre of a major plot to overthrow the English government. Ridolfi had been jailed in 1568 because of a rumour that he had distributed money to dissenting nobles associated with the Northern Rebellion. The Pope did, in fact, give him 12,000 crowns for that purpose, but Ridolfi was released in 1570 because no evidence could be found to incriminate him. Even after his arrest and release, Ridolfi remained a spy for the Pope.[13]Ridolfi's banking connections helped him become acquainted with the Duke of Norfolk, and he became a supporter of a marriage between Norfolk and Mary, Queen of Scots, who would, if the plot succeeded, rule England and reinstate Catholicism there.[13]
After Norfolk's release from prison in August 1570, Ridolfi "picked up the broken threads of Catholic intrigue".[11]Ridolfi was in an advantageous position to orchestrate a Catholic rebellion in England, since he was employed by the Pope, France, and Spain, and had ties to the Catholic contingent in England. He could use banking as an excuse to travel among these groups for the purpose of conspiring.[13]When he travelled to mainland Europe to inform King Phillip and the Pope of the plot, it is believed that he was still working for Elizabeth.[13]
The Duke of Alba, the Spanish Viceroy in the Netherlands who was to lead the attack on England, felt Ridolfi was too garrulous to be the leader of a conspiracy, but Spanish Ambassador DonGuerau de Spesdescribed Ridolfi as "A person of great truth and virtue and an intimate friend of mine."[14]Ridolfi's talkative nature did eventually cause him trouble, as he was not very discreet and trumpeted his plan all over Europe. His boasting was partially responsible for the plot failure, as he told it toCosimo I de' Medici, Grand Duke of Tuscanywho immediately informed Elizabeth of the plot.[13]
Ridolfi escaped execution, unlike some of his co-conspirators, and lived until 1612.[15]
According to historian Cyril Hamshere, retrospective critics of the conspiracy cited a number of reasons why the Ridolfi Plot would have been doomed to fail even if it had not been discovered prematurely. For one, the small number of Spanish soldiers (between 6000 and 10,000) would have been absurdly inadequate to the task of overthrowing the English government. Additionally, the vagueness of the invasion point was a logistical shortcoming. The plan was to land at eitherHarwichorPortsmouth, but Ridolfi apparently did not know exactly where Harwich was. Also dubious was Ridolfi's reliance on the Duke of Norfolk, who was regarded as a bad leader and was not even a Catholic. This did not make him an ideal co-conspirator, but, according to Hamshere, "his main merit lay in his title: in 1571 he was the only Duke in England".[11].The weakness of this theory is that it discount the quality of Spanish Tercio considered among the best troops of their time plus one has only to imagine if the Spanish joined with theNorthern Rebellionthis movement happening again ; more than 10000 English Catholics plus 10,000 Spanish elite troops could have proven a deadly combination for Elizabeth Regime .
Norfolk's Protestantism was but one irony of the Ridolfi Plot: Norfolk and Mary, Queen of Scots had each been married three times before their proposed marriage to each other. Pope Pius was, apparently, willing to grant Mary an annulment of her marriage to her imprisoned husband,[14]but the notion of two thrice wed royals leading England back to Catholicism is somewhat problematic, nonetheless.[16]
The Ridolfi Plot was covered inMary Queen of Scots(1971), starringVanessa Redgraveas Mary andGlenda Jacksonas Elizabeth.
An altered and fictionalised version of the Ridolfi Plot was featured in the filmElizabeth(1998), starringCate Blanchettas Elizabeth.
Notes
Bibliography
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The 1583Throckmorton Plotwas one of a series of attempts byEnglish Roman Catholicsto deposeElizabeth I of Englandand replace her withMary, Queen of Scots, then held under house arrest in England. The alleged objective was to facilitate a Spanish invasion of England, assassinate Elizabeth, and put Mary on the English throne.
The plot is named after the key conspirator,Sir Francis Throckmorton, cousin ofBess Throckmorton,lady in waitingto Queen Elizabeth. Throckmorton was arrested in November 1583 and executed on 10 July 1584.[1]
The plot aimed to free Mary, Queen of Scots, under house arrest in England since 1568, make her queen in place of Elizabeth, and legally restoreRoman Catholicism.[2]This would be achieved by a Spanish-backed invasion of England, led by the FrenchDuke of Guise, supported by a simultaneous revolt of English Roman Catholics.[3]Guise would then marry Mary and become king.
It was typical of the amateurish and overly optimistic approach of many such attempts. Throckmorton was placed under surveillance almost as soon as he returned to England, and subsequently arrested and executed. The plot was never put into action.[4]
Francis Throckmorton (1554-1584) came from a prominent English Catholic family, his fatherJohn Throckmortonbeing a senior judge and witness toQueen Mary's will.[5]While travelling in Europe with his brother Thomas from 1580 to 1583, they visitedParisand met with Catholic exilesCharles PagetandThomas Morgan.[6]
After returning to London in 1583, Francis Throckmorton carried messages between Mary, Queen of Scots, Morgan, andBernardino de Mendoza,Philip II of Spain's ambassador in London. This correspondence was routed through the French embassy in London. Throckmorton also carried some letters written by Mary to the French ambassadorMichel de Castelnau. An agent within the French embassy atSalisbury CourtnearFleet Street, known as "Henry Fagot", notifiedFrancis Walsingham, Elizabeth'sSecretary of State.[7]
Throckmorton was taken into custody in November, along with incriminating documents, including lists of English Catholic supporters.[8]He was encoding a letter to Mary, Queen of Scots when he was arrested. After a few days, he was taken to theTower of London.[9]Another conspirator and letter carrier,George More, was also arrested and questioned, but released after making a deal with Walsingham.[10]
Shortly before his arrest, Throckmorton managed to send a casket of other documents to Mendoza; it has been suggested this was exactly what Walsingham wanted him to do. Throckmorton was a relatively minor player, whose significance was to confirm the extent of Spanish involvement in seeking to overthrow Elizabeth.[11]
Protected bydiplomatic immunity, Mendoza was expelled in January 1584.[1]He was the last Spanish ambassador to England during theElizabethan era.[12]Throckmorton was tortured with therack,[13]first on 16 November, to ensure he revealed as much information as possible. On 19 November, he confessed to giving the Spanish ambassador a list of suitable havens and ports on the English coast.[14]
Throckmorton was put on trial on 21 May 1584 and executed on 10 July.[15]His brother Thomas and many others managed to escape; some were imprisoned in the Tower of London, but Francis Throckmorton was the only one executed.[4][16]
Unsurprisingly, Mary denied any knowledge of the plot. She was able to claim that she was not the author of letters coded in cipher by her secretaries. More of these letters were rediscovered and deciphered in 2023, and seem to implicate her. In June 1583, she asked the French ambassador Michel de Castelnau to apologise to Throckmorton for not writing to him in her own hand, and observed the potential for "great danger". A few months later, as the conspiracy unravelled, she offered money from her French dowry income to the Guises to maintain their interest in her cause after the fall of theGowrie Regimein Scotland.[17]
Mary was placed under strict confinement atChartley HallinStaffordshire. A new and stricter custodianAmias Pauletwas appointed in January 1585.[18]Walsingham andLord Burghleydrew up theBond of Association, obliging all signatories to execute anyone who attempted to usurp the throne or to assassinate the Queen.[19]Mary herself was one of the signatories and it provided the basis forher executionfollowing the 1586Babington Plot.[20][21]
A servant of Mary, Queen of Scots,Jérôme Pasquier, was questioned byThomas Phelippesin September 1586. He confessed to writing a letter in cipher for Mary to send to the French ambassador Castelnau asking him to negotiate a pardon for Francis Throckmorton.[22]
Many participants in the Babington andGunpowder Plotswere related by blood or marriage to Francis Throckmorton, among themRobert CatesbyandFrancis Tresham. Bess Throckmorton (1565-1647) secretly marriedSir Walter Raleigh(1554-1618).
A ballad celebrating the discovery of the plot compared Elizabeth's escape to the survival ofShadrach, Meshach, and Abednegoin Nebuchadnezzar's fiery furnace.[23]
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Cryptography, the use of codes and ciphers, began thousands of years ago.[1]Until recent decades, it has been the story of what might be calledclassical cryptography— that is, of methods ofencryptionthat use pen and paper, or perhaps simple mechanical aids. In the early 20th century, the invention of complex mechanical and electromechanical machines, such as theEnigmarotor machine, provided more sophisticated and efficient means of encryption; and the subsequent introduction of electronics and computing has allowed elaborate schemes of still greater complexity, most of which are entirely unsuited to pen and paper.
The development ofcryptographyhas been paralleled by the development ofcryptanalysis— the "breaking" of codes andciphers. The discovery and application, early on, offrequency analysisto the reading of encrypted communications has, on occasion, altered the course of history. Thus theZimmermann Telegramtriggered the United States' entry into World War I; andAlliesreading ofNazi Germany's ciphers shortened World War II, in some evaluations by as much as two years.
Until the 1960s, secure cryptography was largely the preserve of governments. Two events have since brought it squarely into the public domain: the creation of a public encryption standard (DES), and the invention ofpublic-key cryptography.
The earliest known use of cryptography is found in non-standardhieroglyphscarved into the wall of a tomb from theOld Kingdom of Egyptcirca 1900 BC.[1]These are not thought to be serious attempts at secret communications, however, but rather to have been attempts at mystery, intrigue, or even amusement for literate onlookers.[2]
Someclay tabletsfrom Mesopotamia somewhat later are clearly meant to protect information—one dated near 1500 BC was found to encrypt a craftsman's recipe for pottery glaze, presumably commercially valuable.[3][4]Furthermore,Hebrewscholars made use of simple monoalphabeticsubstitution ciphers(such as theAtbash cipher) beginning perhaps around 600 to 500 BC.[5][6]
The Kama Sutra, estimated to have been composed in India between 400 BC to 300 AD,[7]lists 64 arts recommended for a better quality of life, including a reference toMlecchita vikalpa, "the art of understanding writing in cypher, and the writing of words in a peculiar way." This was recommended for private communication between lovers.[8][9]As the Kama Sutra only contains a general reference in a list and not a description, it is unclear what precisely it referred to at the time. Later commentaries on the Kama Sutra offer detailed instructions for substitution ciphers, but these were composed between the tenth and thirteenth centuries AD.[10]Parts of the EgyptiandemoticGreek Magical Papyriwere written in acypherscript.[11]
Theancient Greeksare said to have known of ciphers.[12]Thescytaletransposition cipherwas used by theSpartanmilitary,[6]but it is not definitively known whether the scytale was for encryption, authentication, or avoiding bad omens in speech.[13][14]Herodotustells us[15]of secret messages physically concealed beneath wax on wooden tablets or as a tattoo on a slave's head concealed by regrown hair, although these are not properly examples of cryptographyper seas the message, once known, is directly readable; this is known assteganography. Another Greek method was developed byPolybius(now called the "Polybius Square").[6]TheRomansknew something of cryptography (e.g., theCaesar cipherand its variations).[16]
David Kahnnotes inThe Codebreakersthat modern cryptology originated among theArabs, the first people to systematically document cryptanalytic methods.[17]Al-Khalil(717–786) wrote theBook of Cryptographic Messages, which contains the first use ofpermutations and combinationsto list all possibleArabicwords with and without vowels.[18]
The invention of thefrequency analysistechnique for breaking monoalphabeticsubstitution ciphers, byAl-Kindi, anArab mathematician,[19][20]sometime around AD 800, proved to be the single most significant cryptanalytic advance until World War II. Al-Kindi wrote a book on cryptography entitledRisalah fi Istikhraj al-Mu'amma(Manuscript for the Deciphering Cryptographic Messages), in which he described the first cryptanalytic techniques, including some forpolyalphabetic ciphers, cipher classification, Arabic phonetics and syntax, and most importantly, gave the first descriptions on frequency analysis.[21]He also covered methods of encipherments, cryptanalysis of certain encipherments, and statistical analysis of letters and letter combinations in Arabic.[22][23]An important contribution ofIbn Adlan(1187–1268) was onsample sizefor use of frequency analysis.[18]
In early medieval England between the years 800–1100, substitution ciphers were frequently used by scribes as a playful and clever way to encipher notes, solutions to riddles, and colophons. The ciphers tend to be fairly straightforward, but sometimes they deviate from an ordinary pattern, adding to their complexity, and possibly also to their sophistication.[24]This period saw vital and significant cryptographic experimentation in the West.
Ahmad al-Qalqashandi(AD 1355–1418) wrote theSubh al-a 'sha, a 14-volume encyclopedia which included a section on cryptology. This information was attributed toIbn al-Durayhimwho lived from AD 1312 to 1361, but whose writings on cryptography have been lost. The list of ciphers in this work included bothsubstitutionandtransposition, and for the first time, apolyalphabetic cipher[25]with multiple substitutions for eachplaintextletter (later called homophonic substitution). Also traced to Ibn al-Durayhim is an exposition on and a worked example of cryptanalysis, including the use of tables ofletter frequenciesand sets of letters which cannot occur together in one word.
The earliest example of the homophonicsubstitution cipheris the one used byDuke of Mantuain the early 1400s.[26]Homophonic cipher replaces each letter with multiple symbols depending on the letter frequency. The cipher is ahead of the time because it combines monoalphabetic and polyalphabetic features.
Essentially all ciphers remained vulnerable to the cryptanalytic technique of frequency analysis until the development of the polyalphabetic cipher, and many remained so thereafter. The polyalphabetic cipher was most clearly explained byLeon Battista Albertiaround AD 1467, for which he was called the "father of Western cryptology".[1]Johannes Trithemius, in his workPoligraphia, invented thetabula recta, a critical component of the Vigenère cipher. Trithemius also wrote theSteganographia.Giovan Battista Bellasoin 1553 first described the cipher that would become known in the 19th century as theVigenère cipher, misattributed toBlaise de Vigenère.[27]In Europe, cryptography became (secretly) more important as a consequence of political competition and religious revolution. For instance, in Europe during and after theRenaissance, citizens of the various Italian states—thePapal Statesand the Roman Catholic Church included—were responsible for rapid proliferation of cryptographic techniques, few of which reflect understanding (or even knowledge) of Alberti's polyalphabetic advance. "Advanced ciphers", even after Alberti, were not as advanced as their inventors/developers/users claimed (and probably even they themselves believed). They were frequently broken. This over-optimism may be inherent in cryptography, for it was then – and remains today – difficult in principle to know how vulnerable one's own system is. In the absence of knowledge, guesses and hopes are predictably common.
Cryptography,cryptanalysis, and secret-agent/courier betrayal featured in theBabington plotduring the reign of QueenElizabeth Iwhich led to the execution ofMary, Queen of Scots.Robert Hookesuggested in the chapterOf Dr. Dee's Book of Spirits, thatJohn Deemade use of Trithemian steganography, to conceal his communication with Queen Elizabeth I.[28]
The chief cryptographer of King Louis XIV of France wasAntoine Rossignol; he and his family created what is known as theGreat Cipherbecause it remained unsolved from its initial use until 1890, when French military cryptanalyst,Étienne Bazeriessolved it.[29]An encrypted message from the time of theMan in the Iron Mask(decrypted just prior to 1900 byÉtienne Bazeries) has shed some, regrettably non-definitive, light on the identity of that real, if legendary and unfortunate, prisoner.
Outside of Europe, after the Mongols brought about the end of theIslamic Golden Age, cryptography remained comparatively undeveloped.Cryptography in Japanseems not to have been used until about 1510, and advanced techniques were not known until after the opening of the country to the West beginning in the 1860s.
Although cryptography has a long and complex history, it wasn't until the 19th century that it developed anything more than ad hoc approaches to either encryption orcryptanalysis(the science of finding weaknesses in crypto systems). Examples of the latter includeCharles Babbage'sCrimean Warera work on mathematical cryptanalysis ofpolyalphabetic ciphers, redeveloped and published somewhat later by the PrussianFriedrich Kasiski. Understanding of cryptography at this time typically consisted of hard-won rules of thumb; see, for example,Auguste Kerckhoffs' cryptographic writings in the latter 19th century.Edgar Allan Poeused systematic methods to solve ciphers in the 1840s. In particular he placed a notice of his abilities in thePhiladelphiapaperAlexander's Weekly (Express) Messenger, inviting submissions of ciphers, most of which he proceeded to solve. His success created a public stir for some months.[30]He later wrote an essay on methods of cryptography which proved useful as an introduction for novice British cryptanalysts attempting to break German codes and ciphers during World War I, and a famous story,The Gold-Bug, in which cryptanalysis was a prominent element.
Cryptography, and its misuse, were involved in the execution ofMata Hariand inDreyfus' convictionand imprisonment, both in the early 20th century. Cryptographers were also involved in exposing the machinations which had led to the Dreyfus affair; Mata Hari, in contrast, was shot.
In World War I theAdmiralty'sRoom 40broke German naval codes and played an important role in several naval engagements during the war, notably in detecting major German sorties into theNorth Seathat led to the battles ofDogger BankandJutlandas the British fleet was sent out to intercept them. However, its most important contribution was probably indecryptingtheZimmermann Telegram, acablefrom the German Foreign Office sent via Washington to itsambassadorHeinrich von Eckardtin Mexico which played a major part in bringing the United States into the war.
In 1917,Gilbert Vernamproposed a teleprinter cipher in which a previously prepared key, kept on paper tape, is combined character by character with the plaintext message to produce the cyphertext. This led to the development of electromechanical devices as cipher machines, and to the only unbreakable cipher, theone time pad.
During the 1920s, Polish naval-officers assisted the Japanese military with code and cipher development.
Mathematical methods proliferated in the period prior to World War II (notably inWilliam F. Friedman's application of statistical techniques to cryptanalysis and cipher development and inMarian Rejewski's initial break into the German Army's version of theEnigmasystem in 1932).
By World War II, mechanical and electromechanicalcipher machineswere in wide use, although—where such machines were impractical—code booksand manual systems continued in use. Great advances were made in both cipher design andcryptanalysis, all in secrecy. Information about this period has begun to be declassified as the official British 50-year secrecy period has come to an end, as US archives have slowly opened, and as assorted memoirs and articles have appeared.
The Germans made heavy use, in several variants, of an electromechanicalrotor machineknown asEnigma.[31]MathematicianMarian Rejewski, at Poland'sCipher Bureau, in December 1932 deduced the detailed structure of the German Army Enigma, using mathematics and limited documentation supplied by CaptainGustave Bertrandof Frenchmilitary intelligenceacquired from a German clerk. This "was one of the great achievements of cryptology," according to historianDavid Kahn.[32]Rejewski and his mathematical Cipher Bureau colleagues,Jerzy RóżyckiandHenryk Zygalski, continued reading Enigma and keeping pace with the evolution of the German Army machine's components and encipherment procedures for some time. As the Poles' resources became strained by the changes being introduced by the Germans, and as war loomed, theCipher Bureau, on the PolishGeneral Staff's instructions, on 25 July 1939, atWarsaw, initiated French and British intelligence representatives into the secrets of Enigma decryption.
Soon after theinvasion of Polandby Germany on 1 September 1939, keyCipher Bureaupersonnel were evacuated southeastward; on 17 September, as theSoviet Union attacked Polandfrom the East, they crossed intoRomania. From there they reached Paris, France; atPC Bruno, near Paris, they continued working toward breaking Enigma, collaborating with BritishcryptologistsatBletchley Parkas the British got up to speed on their work breaking Enigma. In due course, the British cryptographers – whose ranks included many chess masters and mathematics dons such asGordon Welchman,Max Newman, andAlan Turing(the conceptual founder of moderncomputing) – made substantial breakthroughs in the scale and technology ofEnigma decryption.
German code breaking in World War IIalso had some success, most importantly bybreaking the Naval Cipher No. 3. This enabled them to track and sink Atlantic convoys. It was onlyUltraintelligence that finally persuaded the admiralty to change their codes in June 1943. This is surprising given the success of the BritishRoom 40code breakers in the previous world war.
At the end of the War, on 19 April 1945, Britain's highest level civilian and military officials were told that they could never reveal that the German Enigma cipher had been broken because it would give the defeated enemy the chance to say they "were not well and fairly beaten".[33]
The German military also deployed severalteleprinterstream ciphers. Bletchley Park called them theFish ciphers;Max Newmanand colleagues designed and deployed theHeath Robinson, and then the world's first programmable digital electronic computer, theColossus, to help with their cryptanalysis. The German Foreign Office began to use theone-time padin 1919; some of this traffic was read in World War II partly as the result of recovery of some key material in South America that was discarded without sufficient care by a German courier.
TheSchlüsselgerät 41was developed late in the war as a more secure replacement for Enigma, but only saw limited use.
A US Army group, theSIS, managed to break the highest security Japanese diplomatic cipher system (an electromechanicalstepping switchmachine calledPurpleby the Americans) in 1940, before the attack on Pearl Harbor. The locally developed Purple machine replaced the earlier "Red" machine used by the Japanese Foreign Ministry, and a related machine, the M-1, used by Naval attachés which was broken by the U.S. Navy'sAgnes Driscoll. All the Japanese machine ciphers were broken, to one degree or another, by the Allies.
The Japanese Navy and Army largely used code book systems, later with a separate numerical additive.US Navycryptographers (with cooperation from British and Dutch cryptographers after 1940) broke into severalJapanese Navycrypto systems. The break into one of them,JN-25, famously led to the US victory in theBattle of Midway; and to the publication of that fact in theChicago Tribuneshortly after the battle, though the Japanese seem not to have noticed for they kept using the JN-25 system.
The Americans referred to the intelligence resulting from cryptanalysis, perhaps especially that from the Purple machine, as 'Magic'. The British eventually settled on 'Ultra' for intelligence resulting from cryptanalysis, particularly that from message traffic protected by the various Enigmas. An earlier British term for Ultra had been 'Boniface' in an attempt to suggest, if betrayed, that it might have an individual agent as a source.
Alliedcipher machines used in World War II included the BritishTypeXand the AmericanSIGABA; both were electromechanical rotor designs similar in spirit to the Enigma, albeit with major improvements. Neither is known to have been broken by anyone during the War. The Poles used theLacidamachine, but its security was found to be less than intended (by Polish Army cryptographers in the UK), and its use was discontinued. US troops in the field used theM-209and the still less secureM-94family machines. BritishSOEagents initially used 'poem ciphers' (memorized poems were the encryption/decryption keys), but later in the War, they began toswitchtoone-time pads.
TheVIC cipher(used at least until 1957 in connection withRudolf Abel's NY spy ring) was a very complex hand cipher, and is claimed to be the most complicated known to have been used by the Soviets, according to David Kahn inKahn on Codes. For the decrypting of Soviet ciphers (particularly whenone-time padswere reused), seeVenona project.
The UK and US employed large numbers of women in their code-breaking operation, with close to 7,000 reporting to Bletchley Park[34]and 11,000 to the separate US Army and Navy operations, around Washington, DC.[35]By tradition in Japan andNazi doctrinein Germany, women were excluded from war work, at least until late in the war. Even after encryption systems were broken, large amounts of work were needed to respond to changes made, recover daily key settings for multiple networks, and intercept, process, translate, prioritize and analyze the huge volume of enemy messages generated in a global conflict. A few women, includingElizabeth FriedmanandAgnes Meyer Driscoll, had been major contributors to US code-breaking in the 1930s and the Navy and Army began actively recruiting top graduates of women's colleges shortly before the attack on Pearl Harbor. Liza Mundy argues that this disparity in utilizing the talents of women between the Allies and Axis made a strategic difference in the war.[35]: p.29
Encryption in modern times is achieved by using algorithms that have a key to encrypt and decrypt information. These keys convert the messages and data into "digital gibberish" through encryption and then return them to the original form through decryption. In general, the longer the key is, the more difficult it is to crack the code. This holds true because deciphering an encrypted message by brute force would require the attacker to try every possible key. To put this in context, each binary unit of information, or bit, has a value of 0 or 1. An 8-bit key would then have 256 or 2^8 possible keys. A 56-bit key would have 2^56, or 72 quadrillion, possible keys to try and decipher the message. With modern technology, cyphers using keys with these lengths are becoming easier to decipher. DES, an early US Government approved cypher, has an effective key length of 56 bits, and test messages using that cypher have been broken by brute force key search. However, as technology advances, so does the quality of encryption. Since World War II, one of the most notable advances in the study of cryptography is the introduction of the asymmetric key cyphers (sometimes termed public-key cyphers). These are algorithms which use two mathematically related keys for encryption of the same message. Some of these algorithms permit publication of one of the keys, due to it being extremely difficult to determine one key simply from knowledge of the other.[36]
Beginning around 1990, the use of theInternetfor commercial purposes and the introduction of commercial transactions over the Internet called for a widespread standard for encryption. Before the introduction of theAdvanced Encryption Standard(AES), information sent over the Internet, such as financial data, was encrypted if at all, most commonly using the Data Encryption Standard (DES). This had been approved by NBS (a US Government agency) for its security, after public call for, and a competition among, candidates for such a cypher algorithm. DES was approved for a short period, but saw extended use due to complex wrangles over the use by the public of high quality encryption. DES was finally replaced by the AES after another public competition organized by the NBS successor agency, NIST. Around the late 1990s to early 2000s, the use of public-key algorithms became a more common approach for encryption, and soon ahybrid of the two schemesbecame the most accepted way for e-commerce operations to proceed. Additionally, the creation of a new protocol known as the Secure Socket Layer, or SSL, led the way for online transactions to take place. Transactions ranging from purchasing goods to online bill pay and banking used SSL. Furthermore, as wireless Internet connections became more common among households, the need for encryption grew, as a level of security was needed in these everyday situations.[37]
Claude E. Shannonis considered by many[weasel words]to be the father of mathematical cryptography. Shannon worked for several years at Bell Labs, and during his time there, he produced an article entitled "A mathematical theory of cryptography". This article was written in 1945 and eventually was published in the Bell System Technical Journal in 1949.[38]It is commonly accepted that this paper was the starting point for development of modern cryptography. Shannon was inspired during the war to address "[t]he problems of cryptography [because] secrecy systems furnish an interesting application of communication theory". Shannon identified the two main goals of cryptography: secrecy and authenticity. His focus was on exploring secrecy and thirty-five years later, G.J. Simmons would address the issue of authenticity. Shannon wrote a further article entitled "A mathematical theory of communication" which highlights one of the most significant aspects of his work: cryptography's transition from art to science.[39]
In his works, Shannon described the two basic types of systems for secrecy. The first are those designed with the intent to protect against hackers and attackers who have infinite resources with which to decode a message (theoretical secrecy, now unconditional security), and the second are those designed to protect against hackers and attacks with finite resources with which to decode a message (practical secrecy, now computational security). Most of Shannon's work focused around theoretical secrecy; here, Shannon introduced a definition for the "unbreakability" of a cipher. If a cipher was determined "unbreakable", it was considered to have "perfect secrecy". In proving "perfect secrecy", Shannon determined that this could only be obtained with a secret key whose length given in binary digits was greater than or equal to the number of bits contained in the information being encrypted. Furthermore, Shannon developed the "unicity distance", defined as the "amount of plaintext that… determines the secret key."[39]
Shannon's work influenced further cryptography research in the 1970s, as the public-key cryptography developers, M. E. Hellman and W. Diffie cited Shannon's research as a major influence. His work also impacted modern designs of secret-key ciphers. At the end of Shannon's work with cryptography, progress slowed until Hellman and Diffie introduced their paper involving "public-key cryptography".[39]
The mid-1970s saw two major public (i.e., non-secret) advances. First was the publication of the draftData Encryption Standardin the U.S.Federal Registeron 17 March 1975. The proposed DES cipher was submitted by a research group atIBM, at the invitation of the National Bureau of Standards (nowNIST), in an effort to develop secure electronic communication facilities for businesses such as banks and other large financial organizations. After advice and modification by theNSA, acting behind the scenes, it was adopted and published as aFederal Information Processing StandardPublication in 1977 (currently atFIPS 46-3). DES was the first publicly accessible cipher to be 'blessed' by a national agency such as the NSA. The release of its specification by NBS stimulated an explosion of public and academic interest in cryptography.
The aging DES was officially replaced by theAdvanced Encryption Standard(AES) in 2001 when NIST announced FIPS 197. After an open competition, NIST selectedRijndael, submitted by two Belgian cryptographers, to be the AES. DES, and more secure variants of it (such asTriple DES), are still used today, having been incorporated into many national and organizational standards. However, its 56-bit key-size has been shown to be insufficient to guard againstbrute force attacks(one such attack, undertaken by the cyber civil-rights groupElectronic Frontier Foundationin 1997, succeeded in 56 hours.[40]) As a result, use of straight DES encryption is now without doubt insecure for use in new cryptosystem designs, and messages protected by older cryptosystems using DES, and indeed all messages sent since 1976 using DES, are also at risk. Regardless of DES' inherent quality, the DES key size (56-bits) was thought to be too small by some even in 1976, perhaps most publicly byWhitfield Diffie. There was suspicion that government organizations even then had sufficient computing power to break DES messages; clearly others have achieved this capability.
The second development, in 1976, was perhaps even more important, for it fundamentally changed the way cryptosystems might work. This was the publication of the paper "New Directions in Cryptography" byWhitfield DiffieandMartin Hellman.[41]It introduced a radically new method of distributing cryptographic keys, which went far toward solving one of the fundamental problems of cryptography, key distribution, and has become known asDiffie–Hellman key exchange. The article also stimulated the almost immediate public development of a new class of enciphering algorithms, theasymmetric key algorithms.
Prior to that time, all useful modern encryption algorithms had beensymmetric key algorithms, in which the samecryptographic keyis used with the underlying algorithm by both the sender and the recipient, who must both keep it secret. All of the electromechanical machines used in World War II were of this logical class, as were theCaesarandAtbashciphers and essentially all cipher systems throughout history. The 'key' for a code is, of course, the codebook, which must likewise be distributed and kept secret, and so shares most of the same problems in practice.
Of necessity, the key in every such system had to be exchanged between the communicating parties in some secure way prior to any use of the system (the term usually used is 'via asecure channel') such as a trustworthy courier with a briefcase handcuffed to a wrist, or face-to-face contact, or a loyal carrier pigeon. This requirement is never trivial and very rapidly becomes unmanageable as the number of participants increases, or when secure channels aren't available for key exchange, or when, as is sensible cryptographic practice, keys are frequently changed. In particular, if messages are meant to be secure from other users, a separate key is required for each possible pair of users. A system of this kind is known as a secret key, orsymmetric keycryptosystem. D-H key exchange (and succeeding improvements and variants) made operation of these systems much easier, and more secure, than had ever been possible before in all of history.
In contrast,asymmetric keyencryption uses a pair of mathematically related keys, each of which decrypts the encryption performed using the other. Some, but not all, of these algorithms have the additional property that one of the paired keys cannot be deduced from the other by any known method other than trial and error. An algorithm of this kind is known as a public key orasymmetric keysystem. Using such an algorithm, only one key pair is needed per user. By designating one key of the pair as private (always secret), and the other as public (often widely available), no secure channel is needed for key exchange. So long as the private key stays secret, the public key can be widely known for a very long time without compromising security, making it safe to reuse the same key pair indefinitely.
For two users of an asymmetric key algorithm to communicate securely over an insecure channel, each user will need to know their own public and private keys as well as the other user's public key. Take this basic scenario:Alice and Bobeach have a pair of keys they've been using for years with many other users. At the start of their message, they exchange public keys, unencrypted over an insecure line. Alice then encrypts a message using her private key, and then re-encrypts that result using Bob's public key. The double-encrypted message is then sent as digital data over a wire from Alice to Bob. Bob receives the bit stream and decrypts it using his own private key, and then decrypts that bit stream using Alice's public key. If the final result is recognizable as a message, Bob can be confident that the message actually came from someone who knows Alice's private key (presumably actually her if she's been careful with her private key), and that anyone eavesdropping on the channel will need Bob's private key in order to understand the message.
Asymmetric algorithms rely for their effectiveness on a class of problems in mathematics called one-way functions, which require relatively little computational power to execute, but vast amounts of power to reverse, if reversal is possible at all. A classic example of a one-way function is multiplication of very large prime numbers. It's fairly quick to multiply two large primes, but very difficult to find the factors of the product of two large primes. Because of the mathematics of one-way functions, most possible keys are bad choices as cryptographic keys; only a small fraction of the possible keys of a given length are suitable, and so asymmetric algorithms require very long keys to reach the samelevel of securityprovided by relatively shorter symmetric keys. The need to both generate the key pairs, and perform the encryption/decryption operations make asymmetric algorithms computationally expensive, compared to most symmetric algorithms. Since symmetric algorithms can often use any sequence of (random, or at least unpredictable) bits as a key, a disposablesession keycan be quickly generated for short-term use. Consequently, it is common practice to use a long asymmetric key to exchange a disposable, much shorter (but just as strong) symmetric key. The slower asymmetric algorithm securely sends a symmetric session key, and the faster symmetric algorithm takes over for the remainder of the message.
Asymmetric key cryptography, Diffie–Hellman key exchange, and the best known of the public key / private key algorithms (i.e., what is usually called the RSA algorithm), all seem to have been independently developed at a UK intelligence agency before the public announcement by Diffie and Hellman in 1976. GCHQ has released documents claiming they had developed public key cryptography before the publication of Diffie and Hellman's paper.[42]Various classified papers were written atGCHQduring the 1960s and 1970s which eventually led to schemes essentially identical to RSA encryption and to Diffie–Hellman key exchange in 1973 and 1974. Some of these have now been published, and the inventors (James H. Ellis,Clifford Cocks, andMalcolm Williamson) have made public (some of) their work.
Hashingis a common technique used in cryptography to encode information quickly using typical algorithms. Generally, analgorithmis applied to a string of text, and the resulting string becomes the "hash value". This creates a "digital fingerprint" of the message, as the specific hash value is used to identify a specific message. The output from the algorithm is also referred to as a "message digest" or a "check sum". Hashing is good for determining if information has been changed in transmission. If the hash value is different upon reception than upon sending, there is evidence the message has been altered. Once the algorithm has been applied to the data to be hashed, the hash function produces a fixed-length output. Essentially, anything passed through the hash function should resolve to the same length output as anything else passed through the same hash function. It is important to note that hashing is not the same as encrypting. Hashing is a one-way operation that is used to transform data into the compressed message digest. Additionally, the integrity of the message can be measured with hashing. Conversely, encryption is a two-way operation that is used to transform plaintext into cipher-text and then vice versa. In encryption, the confidentiality of a message is guaranteed.[43]
Hash functions can be used to verify digital signatures, so that when signing documents via the Internet, the signature is applied to one particular individual. Much like a hand-written signature, these signatures are verified by assigning their exact hash code to a person. Furthermore, hashing is applied to passwords for computer systems. Hashing for passwords began with theUNIXoperating system. A user on the system would first create a password. That password would be hashed, using an algorithm or key, and then stored in a password file. This is still prominent today, as web applications that require passwords will often hash user's passwords and store them in a database.[44]
The public developments of the 1970s broke the near monopoly on high quality cryptography held by government organizations (see S Levy'sCryptofor a journalistic account of some of the policy controversy of the time in the US). For the first time ever, those outside government organizations had access to cryptography not readily breakable by anyone (including governments). Considerable controversy, and conflict, both public and private, began more or less immediately, sometimes called thecrypto wars. They have not yet subsided. In many countries, for example,export of cryptographyis subject to restrictions. Until 1996 export from the U.S. of cryptography using keys longer than 40 bits (too small to be very secure against a knowledgeable attacker) was sharply limited. As recently as 2004, formerFBIDirectorLouis Freeh, testifying before the9/11 Commission, called for new laws against public use of encryption.
One of the most significant people favoring strong encryption for public use wasPhil Zimmermann. He wrote and then in 1991 releasedPGP(Pretty Good Privacy), a very high qualitycrypto system. He distributed a freeware version of PGP when he felt threatened by legislation then under consideration by the US Government that would require backdoors to be included in all cryptographic products developed within the US. His system was released worldwide shortly after he released it in the US, and that began a long criminal investigation of him by theUS Department of Justice(DOJ) for the alleged violation of export restrictions. The DOJ eventually dropped its case against Zimmermann, and the freeware distribution of PGP has continued around the world. PGP even eventually became an openInternetstandard (RFC 2440 orOpenPGP).
While modern ciphers likeAESand the higher quality asymmetric ciphers are widely considered unbreakable, poor designs and implementations are still sometimes adopted and there have been important cryptanalytic breaks of deployed crypto systems in recent years. Notable examples of broken crypto designs include the firstWi-Fiencryption schemeWEP, theContent Scrambling Systemused for encrypting and controlling DVD use, theA5/1andA5/2ciphers used inGSMcell phones, and theCRYPTO1cipher used in the widely deployedMIFAREClassicsmart cardsfromNXP Semiconductors, a spun off division ofPhilips Electronics. All of these are symmetric ciphers. Thus far, not one of the mathematical ideas underlying public key cryptography has been proven to be 'unbreakable', and so some future mathematical analysis advance might render systems relying on them insecure. While few informed observers foresee such a breakthrough, the key size recommended for security as best practice keeps increasing as increased computing power required for breaking codes becomes cheaper and more available.Quantum computers, if ever constructed with enough capacity, could break existing public key algorithms and efforts are underway to develop and standardizepost-quantum cryptography.
Even without breaking encryption in the traditional sense,side-channel attackscan be mounted that exploit information gained from the way a computer system is implemented, such as cache memory usage, timing information, power consumption, electromagnetic leaks or even sounds emitted. Newer cryptographic algorithms are being developed that make such attacks more difficult.
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ArpON(ARP handler inspection)[1]is acomputer softwareproject to improve network security.[2]It has attracted interest among network managers[3][4][5][6][7]and academic researchers[8][9][10][11][12][13]and is frequently cited as a means of protecting against ARP-based attacks.[14][15][16]
TheAddress Resolution Protocol(ARP) has many security issues. These include theMan In The Middle(MITM) attack through theARP spoofing,[17]ARP cache poisoning,[18][19][20]Denial of Service[21]andARP poison routingattacks.[22][23][24]
ArpON is a host-based solution that makes the ARP secure and avoids theman-in-the-middle attackthrough ARP spoofing, ARP cache poisoning or ARP poison routing. This is possible using three kinds of anti-ARP-spoofing techniques:
The goal of ArpON is therefore to provide a secure and efficient network daemon that provides the SARPI, DARPI and HARPI anti-ARP-spoofing technique, thus making the ARP standardized protocol secure from any foreign intrusion.[citation needed]
This security software article is astub. You can help Wikipedia byexpanding it.
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Cross-site request forgery, also known asone-click attackorsession ridingand abbreviated asCSRF(sometimes pronouncedsea-surf[1]) orXSRF, is a type of maliciousexploitof awebsiteor web application where unauthorized commands are submitted from auserthat the web application trusts.[2]There are many ways in which a malicious website can transmit such commands; specially-crafted image tags, hidden forms, andJavaScriptfetchor XMLHttpRequests, for example, can all work without the user's interaction or even knowledge. Unlikecross-site scripting(XSS), which exploits the trust a user has for a particular site, CSRF exploits the trust that a site has in a user's browser.[3]In a CSRF attack, an innocent end user is tricked by an attacker into submitting a web request that they did not intend. This may cause actions to be performed on the website that can include inadvertent client or server data leakage, change of session state, or manipulation of an end user's account.
The term "CSRF" is also used as an abbreviation in defences against CSRF attacks, such as techniques that use header data, form data, or cookies, to test for and prevent such attacks.
In a CSRF attack, the attacker's goal is to cause an innocent victim to unknowingly submit a maliciously crafted web request to a website that the victim has privileged access to. This web request can be crafted to include URL parameters, cookies and other data that appear normal to the web server processing the request. At risk areweb applicationsthat perform actions based on input from trusted andauthenticatedusers without requiring the user toauthorize(e.g. via a popup confirmation) the specific action. A user who is authenticated by acookiesaved in the user'sweb browsercould unknowingly send anHTTPrequest to a site that trusts the user and thereby cause an unwanted action.
A general property of web browsers is that they will automatically and invisibly include any cookies (including session cookies and others) used by a given domain in any web request sent to that domain. This property is exploited by CSRF attacks. In the event that a user is tricked into inadvertently submitting a request through their browser these automatically included cookies will cause the forged request to appear real to the web server and it will perform any appropriately requested actions including returning data, manipulating session state, or making changes to the victim's account.
In order for a CSRF attack to work, an attacker must identify a reproducible web request that executes a specific action such as changing an account password on the target page. Once such a request is identified, a link can be created that generates this malicious request and that link can be embedded on a page within the attacker's control.[1][4]This link may be placed in such a way that it is not even necessary for the victim to click the link. For example, it may be embedded within an html image tag on an email sent to the victim which will automatically be loaded when the victim opens their email. Once the victim has clicked the link, their browser will automatically include any cookies used by that website and submit the request to the web server. The web server will not be able to identify the forgery because the request was made by a user that was logged in, and submitted all the requisite cookies.
Cross-site request forgery is an example of aconfused deputy attackagainst a web browser because the web browser is tricked into submitting a forged request by a less privileged attacker.
CSRF commonly has the following characteristics:
CSRF Token vulnerabilities have been known and in some cases exploited since 2001.[5]Because it is carried out from the user'sIP address, some website logs might not have evidence of CSRF.[2]Exploits are under-reported, at least publicly, and as of 2007[6]there were few well-documented examples:
Attackers who can find a reproducible link that executes a specific action on the target page while the victim is logged in can embed such link on a page they control and trick the victim into opening it.[1]The attack carrier link may be placed in a location that the victim is likely to visit while logged into the target site (for example, a discussion forum), or sent in anHTML emailbody or attachment. A real CSRF vulnerability inuTorrent(CVE-2008-6586) exploited the fact that its web console accessible atlocalhost:8080 allowed critical actions to be executed using a simple GET request:
Attacks were launched by placing malicious, automatic-actionHTML image elementson forums andemail spam, so that browsers visiting these pages would open them automatically, without much user action. People running vulnerable uTorrent version at the same time as opening these pages were susceptible to the attack.
CSRF attacks using image tags are often made fromInternet forums, where users are allowed to post images but notJavaScript, for example usingBBCode:
When accessing the attack link to the local uTorrent application atlocalhost:8080, the browser would also always automatically send any existingcookiesfor that domain. This general property of web browsers enables CSRF attacks to exploit their targeted vulnerabilities and execute hostile actions as long as the user is logged into the target website (in this example, the local uTorrent web interface) at the time of the attack.
In the uTorrent example described above, the attack was facilitated by the fact that uTorrent's web interface usedGET requestfor critical state-changing operations (change credentials, download a file etc.), whichRFC2616explicitly discourages:
In particular, the convention has been established that the GET and HEAD methods SHOULD NOT have the significance of taking an action other than retrieval. These methods ought to be considered "safe". This allows user agents to represent other methods, such as POST, PUT and DELETE, in a special way, so that the user is made aware of the fact that a possibly unsafe action is being requested.
Because of this assumption, many existing CSRF prevention mechanisms inweb frameworkswillnotcoverGET requests, but rather apply the protection only to HTTP methods that are intended to be state-changing.[10]
An attacker may forge a request to log the victim into a target website using the attacker's credentials; this is known aslogin CSRF. Login CSRF makes various novel attacks possible; for instance, an attacker can later log into the site with their legitimate credentials and view private information like activity history that has been saved in the account. This attack has been demonstrated againstGoogle[11]andYahoo.[12]
Depending on the type, theHTTPrequest methodsvary in their susceptibility to the CSRF attacks (due to the differences in their handling by theweb browsers). Therefore, the protective measures against an attack depend on the method of the HTTP request.
Additionally, while typically described as a static type of attack, CSRF can also be dynamically constructed as part of a payload for across-site scriptingattack, as demonstrated by theSamyworm, or constructed on the fly from session information leaked via offsite content and sent to a target as a malicious URL. CSRF tokens could also be sent to a client by an attacker due tosession fixationor other vulnerabilities, or guessed via abrute-force attack, rendered on a malicious page that generates thousands of failed requests. The attack class of "Dynamic CSRF", or using a per-client payload for session-specific forgery, was described[15]in 2009 by Nathan Hamiel and Shawn Moyer at the BlackHat Briefings,[16]though the taxonomy has yet to gain wider adoption.
A new vector for composing dynamic CSRF attacks was presented by Oren Ofer at a local OWASP chapter meeting in January 2012 – "AJAX Hammer – Dynamic CSRF".[17][18]
Severity metrics have been issued for CSRF token vulnerabilities that result inremote code executionwithroot privileges[19]as well as a vulnerability that can compromise aroot certificate, which will completely undermine apublic key infrastructure.[20]
Several things have to happen for cross-site request forgery to succeed:
The attack is blind: the attacker cannot see what the target website sends back to the victim in response to the forged requests, unless they exploit across-site scriptingor other bug at the target website. Similarly, the attacker can only target any links or submit any forms that come up after the initial forged request if those subsequent links or forms are similarly predictable. (Multiple targets can be simulated by including multiple images on a page, or by using JavaScript to introduce a delay between clicks.)[22]
Most CSRF prevention techniques work by embedding additional authentication data into requests that allows the web application to detect requests from unauthorized locations.
Synchronizer token pattern(STP) is a technique where a token, a secret and unique value for each request, is embedded by the web application in all HTML forms and verified on the server side. The token may be generated by any method that ensures unpredictability and uniqueness (e.g. using ahash chainof random seed). This is called a anti-forgery token inASP.NET. The attacker is thus unable to place a correct token in their requests to authenticate them.[1][23][24]
Example of STP set byDjangoin a HTML form:
STP is the most compatible as it only relies on HTML, but introduces some complexity on the server side, due to the burden associated with checking validity of the token on each request. As the token is unique and unpredictable, it also enforces proper sequence of events (e.g. screen 1, then 2, then 3) which raises usability problem (e.g. user opens multiple tabs). It can be relaxed by using per session CSRF token instead of per request CSRF token.
Web applications that useJavaScriptfor the majority of their operations may use the following anti-CSRF technique:
Security of this technique is based on the assumption that onlyJavaScriptrunning on the client side of an HTTPS connection to the server that initially set the cookie will be able to read the cookie's value. JavaScript running from a rogue file or email should not be able to successfully read the cookie value to copy into the custom header. Even though thecsrf-tokencookiemay be automatically sent with the rogue request, subject to the cookies SameSite policy, the server will still expect a validX-Csrf-Tokenheader.
The CSRF token itself should be unique and unpredictable. It may be generated randomly, or it may be derived from thesession tokenusingHMAC:
The CSRF token cookie must not havehttpOnlyflag, as it is intended to be read byJavaScriptby design.
This technique is implemented by many modern frameworks, such asDjango[25]andAngularJS.[26]Because the token remains constant over the whole user session, it works well withAJAXapplications, but does not enforce sequence of events in the web application.
The protection provided by this technique can be thwarted if the target websitedisablesitssame-origin policyusing one of the following techniques:
Similarly to the cookie-to-header approach, but without involving JavaScript, a site can set a CSRF token as a cookie, and also insert it as a hidden field in each HTML form. When the form is submitted, the site can check that the cookie token matches the form token. The same-origin policy prevents an attacker from reading or setting cookies on the target domain, so they cannot put a valid token in their crafted form.[29]
The advantage of this technique over the Synchronizer pattern is that the token does not need to be stored on the server. However, if the site in question has cookie setting functionality, this protection can be bypassed.
An additional "SameSite" attribute can be included when the server sets a cookie, instructing the browser on whether to attach the cookie to cross-site requests. If this attribute is set to "strict", then the cookie will only be sent on same-site requests, making CSRF ineffective. However, this requires the browser to recognise and correctly implement the attribute.[30]
Browser extensions such as RequestPolicy (forMozilla Firefox) or uMatrix (for both Firefox andGoogle Chrome/Chromium) can prevent CSRF by providing a default-deny policy for cross-site requests. However, this can significantly interfere with the normal operation of many websites. The CsFire extension (also for Firefox) can mitigate the impact of CSRF with less impact on normal browsing, by removing authentication information from cross-site requests.
TheNoScriptextension for Firefox mitigates CSRF threats by distinguishing trusted from untrusted sites, and removing authentication & payloads from POST requests sent by untrusted sites to trusted ones. The Application Boundary Enforcer module in NoScript also blocks requests sent from internet pages to local sites (e.g. localhost), preventing CSRF attacks on local services (such as uTorrent) or routers.
The Self Destructing Cookies extension for Firefox does not directly protect from CSRF, but can reduce the attack window, by deleting cookies as soon as they are no longer associated with an open tab.
Various other techniques have been used or proposed for CSRF prevention historically:
Cross-site scripting(XSS) vulnerabilities (even in other applications running on the same domain) allow attackers to bypass essentially all CSRF preventions.[33]
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https://en.wikipedia.org/wiki/Cross-site_request_forgery
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HTTP cookie(also calledweb cookie,Internet cookie,browser cookie, or simplycookie) is a small block ofdatacreated by aweb serverwhile auserisbrowsingawebsiteand placed on the user's computer or other device by the user'sweb browser. Cookies are placed on the device used to access a website, and more than one cookie may be placed on a user's device during a session.
Cookies serve useful and sometimes essential functions on theweb. They enable web servers to storestatefulinformation (such as items added in the shopping cart in anonline store) on the user's device or to track the user's browsing activity (including clicking particular buttons,logging in, or recording whichpages were visited in the past).[1]They can also be used to save information that the user previously entered intoform fields, such as names, addresses,passwords, andpayment card numbersfor subsequent use.
Authentication cookiesare commonly used by web servers toauthenticatethat a user is logged in, and with whichaccountthey are logged in. Without the cookie, users would need to authenticate themselves by logging in on each page containing sensitive information that they wish to access. The security of an authentication cookie generally depends on the security of the issuing website and the user's web browser, and on whether the cookie data isencrypted.Security vulnerabilitiesmay allow a cookie's data to be read by anattacker, used to gain access touser data, or used to gain access (with the user's credentials) to the website to which the cookie belongs (seecross-site scriptingandcross-site request forgeryfor examples).[2]
Tracking cookies, and especiallythird-party tracking cookies, are commonly used as ways to compile long-term records of individuals'browsing histories— a potentialprivacy concernthat prompted European[3]and U.S. lawmakers to take action in 2011.[4][5]European law requires that all websites targetingEuropean Unionmember states gain "informed consent" from users before storing non-essential cookies on their device.
The termcookiewas coined by web-browser programmerLou Montulli. It was derived from the termmagic cookie, which is a packet of data a program receives and sends back unchanged, used byUnixprogrammers.[6][7]
Magic cookies were already used in computing when computer programmerLou Montullihad the idea of using them in web communications in June 1994.[8]At the time, he was an employee ofNetscape Communications, which was developing ane-commerceapplication forMCI.Vint CerfandJohn Klensinrepresented MCI in technical discussions with Netscape Communications. MCI did not want its servers to have to retain partial transaction states, which led them to ask Netscape to find a way to store that state in each user's computer instead. Cookies provided a solution to the problem of reliably implementing avirtual shopping cart.[9][10]
Together with John Giannandrea, Montulli wrote the initial Netscape cookie specification the same year. Version 0.9beta ofMosaic Netscape, released on October 13, 1994,[11][12]supported cookies.[10]The first use of cookies (out of the labs) was checking whether visitors to the Netscape website had already visited the site. Montulli applied for a patent for the cookie technology in 1995, which was granted in 1998.[13]Support for cookies was integrated withInternet Explorerin version 2, released in October 1995.[14]
The introduction of cookies was not widely known to the public at the time. In particular, cookies were accepted by default, and users were not notified of their presence.[15]The public learned about cookies after theFinancial Timespublished an article about them on February 12, 1996.[16]In the same year, cookies received a lot of media attention, especially because of potential privacy implications. Cookies were discussed in two U.S.Federal Trade Commissionhearings in 1996 and 1997.[2]
The development of the formal cookie specifications was already ongoing. In particular, the first discussions about a formal specification started in April 1995 on the www-talkmailing list. A special working group within theInternet Engineering Task Force(IETF) was formed. Two alternative proposals for introducing state in HTTP transactions had been proposed byBrian Behlendorfand David Kristol respectively. But the group, headed by Kristol himself and Lou Montulli, soon decided to use the Netscape specification as a starting point. In February 1996, the working group identified third-party cookies as a considerable privacy threat. The specification produced by the group was eventually published as RFC 2109 in February 1997. It specifies that third-party cookies were either not allowed at all, or at least not enabled by default.[17]At this time, advertising companies were already using third-party cookies. The recommendation about third-party cookies of RFC 2109 was not followed by Netscape and Internet Explorer. RFC 2109 was superseded by RFC 2965 in October 2000.
RFC 2965 added aSet-Cookie2header field, which informally came to be called "RFC 2965-style cookies" as opposed to the originalSet-Cookieheader field which was called "Netscape-style cookies".[18][19]Set-Cookie2was seldom used, however, and wasdeprecatedin RFC 6265 in April 2011 which was written as a definitive specification for cookies as used in the real world.[20]No modern browser recognizes theSet-Cookie2header field.[21]
Asession cookie(also known as anin-memory cookie,transient cookieornon-persistent cookie) exists only in temporary memory while the user navigates a website.[22]Session cookies expire or are deleted when the user closes the web browser.[23]Session cookies are identified by the browser by the absence of an expiration date assigned to them.
Apersistent cookieexpires at a specific date or after a specific length of time. For the persistent cookie's lifespan set by its creator, its information will be transmitted to the server every time the user visits the website that it belongs to, or every time the user views a resource belonging to that website from another website (such as an advertisement).
For this reason, persistent cookies are sometimes referred to astracking cookies[24][25]because they can be used by advertisers to record information about a user's web browsing habits over an extended period of time. Persistent cookies are also used for reasons such as keeping users logged into their accounts on websites, to avoid re-entering login credentials at every visit.(See§ Uses, below.)
Asecure cookiecan only be transmitted over an encrypted connection (i.e.HTTPS). They cannot be transmitted over unencrypted connections (i.e.HTTP). This makes the cookie less likely to be exposed to cookie theft viaeavesdropping. A cookie is made secure by adding theSecureflag to the cookie.
Anhttp-only cookiecannot be accessed by client-side APIs, such asJavaScript. This restriction eliminates the threat of cookie theft viacross-site scripting(XSS).[26]However, the cookie remains vulnerable tocross-site tracing(XST) andcross-site request forgery(CSRF) attacks. A cookie is given this characteristic by adding theHttpOnlyflag to the cookie.
In 2016Google Chromeversion 51 introduced[27]a new kind of cookie with attributeSameSitewith possible values ofStrict,LaxorNone.[28]With attributeSameSite=Strict, the browsers would only send cookies to a target domain that is the same as the origin domain. This would effectively mitigatecross-site request forgery(CSRF) attacks. WithSameSite=Lax, browsers would send cookies with requests to a target domain even it is different from the origin domain, but only forsaferequests such as GET (POST is unsafe) and not third-party cookies (inside iframe). AttributeSameSite=Nonewould allow third-party (cross-site) cookies, however, most browsers requiresecure attributeon SameSite=None cookies.[29]
The Same-site cookie is incorporated into a new RFC draft for "Cookies: HTTP State Management Mechanism"[30]to update RFC 6265 (if approved).
Chrome, Firefox, and Edge started to support Same-site cookies.[31]The key of rollout is the treatment of existing cookies without the SameSite attribute defined, Chrome has been treating those existing cookies as if SameSite=None, this would let all website/applications run as before. Google intended to change that default toSameSite=Laxin Chrome 80 planned to be released in February 2020,[32]but due to potential for breakage of those applications/websites that rely on third-party/cross-site cookies andCOVID-19circumstances, Google postponed this change to Chrome 84.[33][34]
Asupercookieis a cookie with an origin of atop-level domain(such as.com) or a public suffix (such as.co.uk). Ordinary cookies, by contrast, have an origin of a specific domain name, such asexample.com.
Supercookies can be a potential security concern and are therefore often blocked by web browsers. If unblocked by the browser, an attacker in control of a malicious website could set a supercookie and potentially disrupt or impersonate legitimate user requests to another website that shares the same top-level domain or public suffix as the malicious website. For example, a supercookie with an origin of.comcould maliciously affect a request made toexample.com, even if the cookie did not originate fromexample.com. This can be used to fake logins or change user information.
ThePublic Suffix List[35]helps to mitigate the risk that supercookies pose. The Public Suffix List is a cross-vendor initiative that aims to provide an accurate and up-to-date list of domain name suffixes. Older versions of browsers may not have an up-to-date list, and will therefore be vulnerable to supercookies from certain domains.
The termsupercookieis sometimes used for tracking technologies that do not rely on HTTP cookies. Two suchsupercookiemechanisms were found on Microsoft websites in August 2011:cookie syncingthat respawned MUID (machine unique identifier) cookies, andETagcookies.[36]Due to media attention, Microsoft later disabled this code.[37]In a 2021 blog post, Mozilla used the termsupercookieto refer tothe use of browser cacheas a means of tracking users across sites.[38]
Azombie cookieis data and code that has been placed by aweb serveron a visitor's computer or other device in a hidden location outside the visitor'sweb browser's dedicated cookie storage location, and that automatically recreates a HTTP cookie as a regular cookie after the original cookie had been deleted. The zombie cookie may be stored in multiple locations, such asFlash Local shared object,HTML5 Web storage, and other client-side and even server-side locations, and when absence is detected in one of the locations, the missing instance is recreated by the JavaScript code using the data stored in other locations.[39][40]
A cookie wall pops up on a website and informs the user of the website's cookie usage. It has no reject option, and the website is not accessible without tracking cookies.
A cookie consists of the following components:[41][42][43]
Cookies were originally introduced to provide a way for users to record items they want to purchase as they navigate throughout a website (a virtualshopping cartorshopping basket).[9][10]Today, however, the contents of a user's shopping cart are usually stored in a database on the server, rather than in a cookie on the client. To keep track of which user is assigned to which shopping cart, the server sends a cookie to the client that contains aunique session identifier(typically, a long string of random letters and numbers). Because cookies are sent to the server with every request the client makes, that session identifier will be sent back to the server every time the user visits a new page on the website, which lets the server know which shopping cart to display to the user.
Another popular use of cookies is for logging into websites. When the user visits a website's login page, the web server typically sends the client a cookie containing a unique session identifier. When the user successfully logs in, the server remembers that that particular session identifier has been authenticated and grants the user access to its services.
Because session cookies only contain a unique session identifier, this makes the amount of personal information that a website can save about each user virtually limitless—the website is not limited to restrictions concerning how large a cookie can be. Session cookies also help to improve page load times, since the amount of information in a session cookie is small and requires little bandwidth.
Cookies can be used to remember information about the user in order to show relevant content to that user over time. For example, a web server might send a cookie containing the username that was last used to log into a website, so that it may be filled in automatically the next time the user logs in.
Many websites use cookies for personalization based on the user's preferences. Users select their preferences by entering them in a web form and submitting the form to the server. The server encodes the preferences in a cookie and sends the cookie back to the browser. This way, every time the user accesses a page on the website, the server can personalize the page according to the user's preferences. For example, theGooglesearch engine once used cookies to allow users (even non-registered ones) to decide how many search results per page they wanted to see.
Also,DuckDuckGouses cookies to allow users to set the viewing preferences like colors of the web page.
Tracking cookies are used to track users' web browsing habits. This can also be done to some extent by using theIP addressof the computer requesting the page or therefererfield of theHTTPrequest header, but cookies allow for greater precision. This can be demonstrated as follows:
By analyzing this log file, it is then possible to find out which pages the user has visited, in what sequence, and for how long.
Corporations exploit users' web habits by tracking cookies to collect information about buying habits. TheWall Street Journalfound that America's top fifty websites installed an average of sixty-four pieces of tracking technology onto computers, resulting in a total of 3,180 tracking files.[44]The data can then be collected and sold to bidding corporations.
Cookies are arbitrary pieces of data, usually chosen and first sent by the web server, and stored on the client computer by the web browser. The browser then sends them back to the server with every request, introducingstates(memory of previous events) into otherwise statelessHTTPtransactions. Without cookies, each retrieval of aweb pageor component of a web page would be an isolated event, largely unrelated to all other page views made by the user on the website. Although cookies are usually set by the web server, they can also be set by the client using a scripting language such asJavaScript(unless the cookie'sHttpOnlyflag is set, in which case the cookie cannot be modified by scripting languages).
The cookie specifications[45][46]require that browsers meet the following requirements in order to support cookies:
Cookies are set using theSet-Cookieheader field, sent in an HTTP response from the web server. This header field instructs the web browser to store the cookie and send it back in future requests to the server (the browser will ignore this header field if it does not support cookies or has disabled cookies).
As an example, the browser sends its first HTTP request for the homepage of thewww.example.orgwebsite:
The server responds with twoSet-Cookieheader fields:
The server's HTTP response contains the contents of the website's homepage. But it also instructs the browser to set two cookies. The first,theme, is considered to be asession cookiesince it does not have anExpiresorMax-Ageattribute. Session cookies are intended to be deleted by the browser when the browser closes. The second,sessionToken, is considered to be apersistent cookiesince it contains anExpiresattribute, which instructs the browser to delete the cookie at a specific date and time.
Next, the browser sends another request to visit thespec.htmlpage on the website. This request contains aCookieheader field, which contains the two cookies that the server instructed the browser to set:
This way, the server knows that this HTTP request is related to the previous one. The server would answer by sending the requested page, possibly including moreSet-Cookieheader fields in the HTTP response in order to instruct the browser to add new cookies, modify existing cookies, or remove existing cookies. To remove a cookie, the server must include aSet-Cookieheader field with an expiration date in the past.
The value of a cookie may consist of any printableASCIIcharacter (!through~,Unicode\u0021through\u007E) excluding,and;andwhitespace characters. The name of a cookie excludes the same characters, as well as=, since that is the delimiter between the name and value. The cookie standard RFC 2965 is more restrictive but not implemented by browsers.
The termcookie crumbis sometimes used to refer to a cookie's name–value pair.[47]
Cookies can also be set by scripting languages such asJavaScriptthat run within the browser. In JavaScript, the objectdocument.cookieis used for this purpose. For example, the instructiondocument.cookie = "temperature=20"creates a cookie of nametemperatureand value20.[48]
In addition to a name and value, cookies can also have one or more attributes. Browsers do not include cookie attributes in requests to the server—they only send the cookie's name and value. Cookie attributes are used by browsers to determine when to delete a cookie, block a cookie or whether to send a cookie to the server.
TheDomainandPathattributes define the scope of the cookie. They essentially tell the browser what website the cookie belongs to. For security reasons, cookies can only be set on the current resource's top domain and its subdomains, and not for another domain and its subdomains. For example, the websiteexample.orgcannot set a cookie that has a domain offoo.combecause this would allow the websiteexample.orgto control the cookies of the domainfoo.com.
If a cookie'sDomainandPathattributes are not specified by the server, they default to the domain and path of the resource that was requested.[49]However, in most browsers there is a difference between a cookie set fromfoo.comwithout a domain, and a cookie set with thefoo.comdomain. In the former case, the cookie will only be sent for requests tofoo.com, also known as a host-only cookie. In the latter case, all subdomains are also included (for example,docs.foo.com).[50][51]A notable exception to this general rule is Edge prior to Windows 10 RS3 and Internet Explorer prior to IE 11 and Windows 10 RS4 (April 2018), which always sends cookies to subdomains regardless of whether the cookie was set with or without a domain.[52]
Below is an example of someSet-Cookieheader fields in the HTTP response of a website after a user logged in. The HTTP request was sent to a webpage within thedocs.foo.comsubdomain:
The first cookie,LSID, has noDomainattribute, and has aPathattribute set to/accounts. This tells the browser to use the cookie only when requesting pages contained indocs.foo.com/accounts(the domain is derived from the request domain). The other two cookies,HSIDandSSID, would be used when the browser requests any subdomain in.foo.comon any path (for examplewww.foo.com/bar). The prepending dot is optional in recent standards, but can be added for compatibility with RFC 2109 based implementations.[53]
TheExpiresattribute defines a specific date and time for when the browser should delete the cookie. The date and time are specified in the formWdy, DD Mon YYYY HH:MM:SS GMT, or in the formWdy, DD Mon YY HH:MM:SS GMTfor values of YY where YY is greater than or equal to 0 and less than or equal to 69.[54]
Alternatively, theMax-Ageattribute can be used to set the cookie's expiration as an interval of seconds in the future, relative to the time the browser received the cookie. Below is an example of threeSet-Cookieheader fields that were received from a website after a user logged in:
The first cookie,lu, is set to expire sometime on 15 January 2013. It will be used by the client browser until that time. The second cookie,made_write_conn, does not have an expiration date, making it a session cookie. It will be deleted after the user closes their browser. The third cookie,reg_fb_gate, has its value changed todeleted, with an expiration time in the past. The browser will delete this cookie right away because its expiration time is in the past. Note that cookie will only be deleted if the domain and path attributes in theSet-Cookiefield match the values used when the cookie was created.
As of 2016[update]Internet Explorer did not supportMax-Age.[55][56]
TheSecureandHttpOnlyattributes do not have associated values. Rather, the presence of just their attribute names indicates that their behaviors should be enabled.
TheSecureattribute is meant to keep cookie communication limited to encrypted transmission, directing browsers to use cookies only viasecure/encryptedconnections. However, if a web server sets a cookie with a secure attribute from a non-secure connection, the cookie can still be intercepted when it is sent to the user byman-in-the-middle attacks. Therefore, for maximum security, cookies with the Secure attribute should only be set over a secure connection.
TheHttpOnlyattribute directs browsers not to expose cookies through channels other than HTTP (and HTTPS) requests. This means that the cookie cannot be accessed via client-side scripting languages (notablyJavaScript), and therefore cannot be stolen easily viacross-site scripting(a pervasive attack technique).[57]
Most modern browsers support cookies and allow the user to disable them. The following are common options:[58]
Add-on tools for managing cookie permissions also exist.[59][60][61][62]
Cookies have some important implications for the privacy and anonymity of web users. While cookies are sent only to the server setting them or a server in the same Internet domain, a web page may contain images or other components stored on servers in other domains. Cookies that are set during retrieval of these components are calledthird-party cookies. A third-party cookie, belongs to a domain different from the one shown in the address bar. This sort of cookie typically appears when web pages feature content from external websites, such asbanner advertisements. This opens up the potential fortrackingthe user's browsing history and is used by advertisers toserve relevant advertisementsto each user.
As an example, suppose a user visitswww.example.org. This website contains an advertisement fromad.foxytracking.com, which, when downloaded, sets a cookie belonging to the advertisement's domain (ad.foxytracking.com). Then, the user visits another website,www.foo.com, which also contains an advertisement fromad.foxytracking.comand sets a cookie belonging to that domain (ad.foxytracking.com). Eventually, both of these cookies will be sent to the advertiser when loading their advertisements or visiting their website. The advertiser can then use these cookies to build up a browsing history of the user across all the websites that have ads from this advertiser, through the use of theHTTP refererheader field.
As of 2014[update], some websites were setting cookies readable for over 100 third-party domains.[63]On average, a single website was setting 10 cookies, with a maximum number of cookies (first- and third-party) reaching over 800.[64]
The older standards for cookies, RFC 2109[17]and RFC 2965, recommend that browsers should protect user privacy and not allow sharing of cookies between servers by default. However, the newer standard, RFC 6265, explicitly allows user agents to implement whichever third-party cookie policy they wish. Most modern web browsers containprivacy settingsthat canblockthird-party cookies. Since 2020,Apple Safari,[65]Firefox,[66]andBrave[67]block all third-party cookies by default. Safari allows embedded sites to use Storage Access API to request permission to set first-party cookies. In May 2020,Google Chrome83 introduced new features to block third-party cookies by default in its Incognito mode for private browsing, making blocking optional during normal browsing. The same update also added an option to block first-party cookies.[68]In April 2024, Chrome postponed third-party cookie blocking by default to 2025.[69]In July 2024, Google announced plan to avoid blocking third-party cookies by default and instead prompt users to allow third-party cookies.[70]
The possibility of building a profile of users is a privacy threat, especially when tracking is done across multiple domains using third-party cookies. For this reason, some countries have legislation about cookies.
Website operators who do not disclose third-party cookie use to consumers run the risk of harming consumer trust if cookie use is discovered. Having clear disclosure (such as in aprivacy policy) tends to eliminate any negative effects of such cookie discovery.[71][failed verification]
TheUnited Statesgovernment set strict rules on setting cookies in 2000 after it was disclosed that the White Housedrug policy officeused cookies to track computer users viewing its online anti-drug advertising. In 2002, privacy activist Daniel Brandt found that theCIAhad been leaving persistent cookies on computers that had visited its website. When notified it was violating policy, CIA stated that these cookies were not intentionally set and stopped setting them. On December 25, 2005, Brandt discovered that theNational Security Agency(NSA) had been leaving two persistent cookies on visitors' computers due to a software upgrade. After being informed, the NSA immediately disabled the cookies.[72]
In 2002, the European Union launched theDirective on Privacy and Electronic Communications(e-Privacy Directive), a policy requiring end users' consent for the placement of cookies, and similar technologies for storing and accessing information on users' equipment.[73][74]In particular, Article 5 Paragraph 3 mandates that storing technically unnecessary data on a user's computer can only be done if the user is provided information about how this data is used, and the user is given the possibility of denying this storage operation. The Directive does not require users to authorise or be provided notice of cookie usage that are functionally required for delivering a service they have requested, for example to retain settings, store log-in sessions, or remember what is in a user's shopping basket.[75]
In 2009, the law was amended by Directive 2009/136/EC, which included a change to Article 5, Paragraph 3. Instead of having an option for users to opt out of cookie storage, the revised Directive requires consent to be obtained for cookie storage.[74]The definition of consent is cross-referenced to the definition in European data protection law, firstly the Data Protection Directive 1995 and subsequently theGeneral Data Protection Regulation(GDPR). As the definition of consent was strengthened in the text of the GDPR, this had the effect of increasing the quality of consent required by those storing and accessing information such as cookies on users devices. In a case decided under the Data Protection Directive however, theCourt of Justice of the European Unionlater confirmed however that the previous law implied the same strong quality of consent as the current instrument.[76]In addition to the requirement of consent which stems from storing or accessing information on a user's terminal device, the information in many cookies will be considered personal data under the GDPR alone, and will require a legal basis to process. This has been the case since the 1995 Data Protection Directive, which used an identical definition of personal data, although the GDPR in interpretative Recital 30 clarifies that cookie identifiers are included. While not all data processing under the GDPR requires consent, the characteristics of behavioural advertising mean that it is difficult or impossible to justify under any other ground.[77][78]
Consent under the combination of the GDPR and e-Privacy Directive has to meet a number of conditions in relation to cookies.[79]It must be freely given and unambiguous: preticked boxes were banned under both the Data Protection Directive 1995[76]and the GDPR (Recital 32).[80]The GDPR is specific that consent must be as 'easy to withdraw as to give',[80]meaning that a reject-all button must be as easy to access in terms of clicks and visibility as an 'accept all' button.[79]It must be specific and informed, meaning that consent relates to particular purposes for the use of this data, and all organisations seeking to use this consent must be specifically named.[81][82]TheCourt of Justice of the European Unionhas also ruled that consent must be 'efficient and timely', meaning that it must be gained before cookies are laid and data processing begins instead of afterwards.[83]
The industry's response has been largely negative. Robert Bond of the law firm Speechly Bircham describes the effects as "far-reaching and incredibly onerous" for "all UK companies". Simon Davis ofPrivacy Internationalargues that proper enforcement would "destroy the entire industry".[84]However, scholars note that the onerous nature of cookie pop-ups stems from an attempt to continue to operate a business model through convoluted requests that may be incompatible with the GDPR.[77]
Academic studies and regulators both describe widespread non-compliance with the law. A study scraping 10,000 UK websites found that only 11.8% of sites adhered to minimal legal requirements, with only 33.4% of websites studied providing a mechanism to reject cookies that was as easy to use as accepting them.[79]A study of 17,000 websites found that 84% of sites breached this criterion, finding additionally that many laid third party cookies with no notice at all.[85]The UK regulator, theInformation Commissioner's Office, stated in 2019 that the industry's 'Transparency and Consent Framework' from the advertising technology group theInteractive Advertising Bureauwas 'insufficient to ensure transparency and fair processing of the personal data in question and therefore also insufficient to provide for free and informed consent, with attendant implications for PECR [e-Privacy] compliance.'[81]Many companies that sell compliance solutions (Consent Management Platforms) permit them to be configured in manifestly illegal ways, which scholars have noted creates questions around the appropriate allocation of liability.[86]
AW3Cspecification calledP3Pwas proposed for servers to communicate their privacy policy to browsers, allowing automatic, user-configurable handling. However, few websites implement the specification, and the W3C has discontinued work on the specification.[87]
Third-party cookies can be blocked by most browsers to increase privacy and reduce tracking by advertising and tracking companies without negatively affecting the user's web experience on all sites. Some sites operate 'cookie walls', which make access to a site conditional on allowing cookies either technically in a browser, through pressing 'accept', or both.[88]In 2020, theEuropean Data Protection Board, composed of all EU data protection regulators, stated that cookie walls were illegal.
In order for consent to be freely given, access to services and functionalities must not be made conditional on the consent of a user to the storing of information, or gaining of access to information already stored, in the terminal equipment of a user (so called cookie walls).[89]
Many advertising operators have an opt-out option to behavioural advertising, with a generic cookie in the browser stopping behavioural advertising.[90][91]However, this is often ineffective against many forms of tracking, such as first-party tracking that is growing in popularity to avoid the impact of browsers blocking third party cookies.[92][93]Furthermore, if such a setting is more difficult to place than the acceptance of tracking, it remains in breach of the conditions of the e-Privacy Directive.[79]
Most websites use cookies as the only identifiers for user sessions, because other methods of identifying web users have limitations and vulnerabilities. If a website uses cookies as session identifiers, attackers can impersonate users' requests by stealing a full set of victims' cookies. From the web server's point of view, a request from an attacker then has the same authentication as the victim's requests; thus the request is performed on behalf of the victim's session.
Listed here are various scenarios of cookie theft and user session hijacking (even without stealing user cookies) that work with websites relying solely on HTTP cookies for user identification.
Traffic on a network can be intercepted and read by computers on the network other than the sender and receiver (particularly overunencryptedopenWi-Fi). This traffic includes cookies sent on ordinary unencryptedHTTP sessions. Where network traffic is not encrypted, attackers can therefore read the communications of other users on the network, including HTTP cookies as well as the entire contents of the conversations, for the purpose of aman-in-the-middle attack.
An attacker could use intercepted cookies to impersonate a user and perform a malicious task, such as transferring money out of the victim's bank account.
This issue can be resolved by securing the communication between the user's computer and the server by employingTransport Layer Security(HTTPSprotocol) to encrypt the connection. A server can specify theSecureflag while setting a cookie, which will cause the browser to send the cookie only over an encrypted channel, such as a TLS connection.[45]
If an attacker is able to cause aDNS serverto cache a fabricated DNS entry (calledDNS cache poisoning), then this could allow the attacker to gain access to a user's cookies. For example, an attacker could use DNS cache poisoning to create a fabricated DNS entry off12345.www.example.comthat points to theIP addressof the attacker's server. The attacker can then post an image URL from his own server (for example,http://f12345.www.example.com/img_4_cookie.jpg). Victims reading the attacker's message would download this image fromf12345.www.example.com. Sincef12345.www.example.comis a sub-domain ofwww.example.com, victims' browsers would submit allexample.com-related cookies to the attacker's server.
If an attacker is able to accomplish this, it is usually the fault of theInternet Service Providersfor not properly securing their DNS servers. However, the severity of this attack can be lessened if the target website uses secure cookies. In this case, the attacker would have the extra challenge[94]of obtaining the target website's TLS certificate from acertificate authority, since secure cookies can only be transmitted over an encrypted connection. Without a matching TLS certificate, victims' browsers would display a warning message about the attacker's invalid certificate, which would help deter users from visiting the attacker's fraudulent website and sending the attacker their cookies.
Cookies can also be stolen using a technique called cross-site scripting. This occurs when an attacker takes advantage of a website that allows its users to post unfilteredHTMLandJavaScriptcontent. By posting malicious HTML and JavaScript code, the attacker can cause the victim's web browser to send the victim's cookies to a website the attacker controls.
As an example, an attacker may post a message onwww.example.comwith the following link:
When another user clicks on this link, the browser executes the piece of code within theonclickattribute, thus replacing the stringdocument.cookiewith the list of cookies that are accessible from the current page. As a result, this list of cookies is sent to theattacker.comserver. If the attacker's malicious posting is on an HTTPS websitehttps://www.example.com, secure cookies will also be sent to attacker.com in plain text.
It is the responsibility of the website developers to filter out such malicious code.
Such attacks can be mitigated by using HttpOnly cookies. These cookies will not be accessible by client-side scripting languages like JavaScript, and therefore, the attacker will not be able to gather these cookies.
In older versions of many browsers, there were security holes in the implementation of theXMLHttpRequestAPI. This API allows pages to specify a proxy server that would get the reply, and this proxy server is not subject to thesame-origin policy. For example, a victim is reading an attacker's posting onwww.example.com, and the attacker's script is executed in the victim's browser. The script generates a request towww.example.comwith the proxy serverattacker.com. Since the request is forwww.example.com, allexample.comcookies will be sent along with the request, but routed through the attacker's proxy server. Hence, the attacker would be able to harvest the victim's cookies.
This attack would not work with secure cookies, since they can only be transmitted overHTTPSconnections, and the HTTPS protocol dictatesend-to-end encryption(i.e. the information is encrypted on the user's browser and decrypted on the destination server). In this case, the proxy server would only see the raw, encrypted bytes of the HTTP request.
For example, Bob might be browsing a chat forum where another user, Mallory, has posted a message. Suppose that Mallory has crafted an HTML image element that references an action on Bob's bank's website (rather than an image file), e.g.,
If Bob's bank keeps his authentication information in a cookie, and if the cookie hasn't expired, then the attempt by Bob's browser to load the image will submit the withdrawal form with his cookie, thus authorizing a transaction without Bob's approval.
Cookiejackingis an attack againstInternet Explorerwhich allows the attacker to stealsession cookiesof a user by tricking a user into dragging an object across the screen.[95]Microsoft deemed the flaw low-risk because of "the level of required user interaction",[95]and the necessity of having a user already logged into the website whose cookie is stolen.[96]Despite this, a researcher tried the attack on 150 of their Facebook friends and obtained cookies of 80 of them viasocial engineering.[95]
Besides privacy concerns, cookies also have some technical drawbacks. In particular, they do not always accurately identify users, they can be used for security attacks, and they are often at odds with the Representational State Transfer (REST) software architectural style.[97][98]
If more than one browser is used on a computer, each usually has a separate storage area for cookies. Hence, cookies do not identify a person, but a combination of a user account, a computer, and a web browser. Thus, anyone who uses multiple accounts, computers, or browsers has multiple sets of cookies.[99]
Likewise, cookies do not differentiate between multiple users who share the sameuser account, computer, and browser.
Some of the operations that can be done using cookies can also be done using other mechanisms.
AJSON Web Token(JWT) is a self-contained packet of information that can be used to store user identity and authenticity information. This allows them to be used in place of session cookies. Unlike cookies, which are automatically attached to each HTTP request by the browser, JWTs must be explicitly attached to each HTTP request by the web application.
The HTTP protocol includes thebasic access authenticationand thedigest access authenticationprotocols, which allow access to a web page only when the user has provided the correct username and password. If the server requires such credentials for granting access to a web page, the browser requests them from the user and, once obtained, the browser stores and sends them in every subsequent page request. This information can be used to track the user.
Thequery stringpart of theURLis the part that is typically used for this purpose, but other parts can be used as well. TheJava ServletandPHPsession mechanisms both use this method if cookies are not enabled.
This method consists of the web server appending query strings containing a unique session identifier to all the links inside of a web page. When the user follows a link, the browser sends the query string to the server, allowing the server to identify the user and maintain state.
These kinds of query strings are very similar to cookies in that both contain arbitrary pieces of information chosen by the server and both are sent back to the server on every request. However, there are some differences. Since a query string is part of a URL, if that URL is later reused, the same attached piece of information will be sent to the server, which could lead to confusion. For example, if the preferences of a user are encoded in the query string of a URL and the user sends this URL to another user bye-mail, those preferences will be used for that other user as well.
Moreover, if the same user accesses the same page multiple times from different sources, there is no guarantee that the same query string will be used each time. For example, if a user visits a page by coming from a pageinternal to the sitethe first time, and then visits the same page by coming from anexternalsearch enginethe second time, the query strings would likely be different. If cookies were used in this situation, the cookies would be the same.
Other drawbacks of query strings are related to security. Storing data that identifies a session in a query string enablessession fixationattacks,refererlogging attacks and othersecurity exploits. Transferring session identifiers as HTTP cookies is more secure.
Another form of session tracking is to useweb formswith hidden fields. This technique is very similar to using URL query strings to hold the information and has many of the same advantages and drawbacks. In fact, if the form is handled with theHTTPGET method, then this technique is similar to using URL query strings, since the GET method adds the form fields to the URL as a query string. But most forms are handled with HTTP POST, which causes the form information, including the hidden fields, to be sent in the HTTP request body, which is neither part of the URL, nor of a cookie.
This approach presents two advantages from the point of view of the tracker. First, having the tracking information placed in the HTTP request body rather than in the URL means it will not be noticed by the average user. Second, the session information is not copied when the user copies the URL (to bookmark the page or send it via email, for example).
All current web browsers can store a fairly large amount of data (2–32 MB) via JavaScript using theDOMpropertywindow.name. This data can be used instead of session cookies. The technique can be coupled withJSON/JavaScript objects to store complex sets of session variables on the client side.
The downside is that every separate window ortabwill initially have an emptywindow.nameproperty when opened.
In some respects, this can be more secure than cookies due to the fact that its contents are not automatically sent to the server on every request like cookies are, so it is not vulnerable to network cookie sniffing attacks.
Some users may be tracked based on theIP addressof the computer requesting the page. The server knows the IP address of the computer running the browser (or theproxy, if any is used) and could theoretically link a user's session to this IP address.
However, IP addresses are generally not a reliable way to track a session or identify a user. Many computers designed to be used by a single user, such as office PCs or home PCs, are behind a network address translator (NAT). This means that several PCs will share a public IP address. Furthermore, some systems, such asTor, are designed to retainInternet anonymity, rendering tracking by IP address impractical, impossible, or a security risk.
Because ETags are cached by the browser, and returned with subsequent requests for the same resource, a tracking server can simply repeat any ETag received from the browser to ensure an assigned ETag persists indefinitely (in a similar way to persistent cookies). Additional caching header fields can also enhance the preservation of ETag data.
ETags can be flushed in some browsers by clearing thebrowser cache.
The browser cache can also be used to store information that can be used to track individual users. This technique takes advantage of the fact that the web browser will use resources stored within the cache instead of downloading them from the website when it determines that the cache already has the most up-to-date version of the resource.
For example, a website could serve a JavaScript file with code that sets a unique identifier for the user (for example,var userId = 3243242;). After the user's initial visit, every time the user accesses the page, this file will be loaded from the cache instead of downloaded from the server. Thus, its content will never change.
Abrowser fingerprintis information collected about a browser's configuration, such as version number, screen resolution, and operating system, for the purpose of identification. Fingerprints can be used to fully or partially identify individual users or devices even when cookies are turned off.
Basicweb browserconfiguration information has long been collected byweb analyticsservices in an effort to accurately measure real humanweb trafficand discount various forms ofclick fraud. With the assistance ofclient-side scriptinglanguages, collection of much more esoteric parameters is possible.[100][101]Assimilation of such information into a single string constitutes a device fingerprint. In 2010,EFFmeasured at least 18.1 bits ofentropypossible from browser fingerprinting.[102]Canvas fingerprinting, a more recent technique, claims to add another 5.7 bits.
Some web browsers support persistence mechanisms which allow the page to store the information locally for later use.
TheHTML5standard (which most modern web browsers support to some extent) includes a JavaScript API calledWeb storagethat allows two types of storage: local storage and session storage. Local storage behaves similarly topersistent cookieswhile session storage behaves similarly tosession cookies, except that session storage is tied to an individual tab/window's lifetime (AKA a page session), not to a whole browser session like session cookies.[103]
Internet Explorer supports persistent information[104]in the browser's history, in the browser's favorites, in an XML store ("user data"), or directly within a web page saved to disk.
Some web browser plugins include persistence mechanisms as well. For example,Adobe FlashhasLocal shared objectandMicrosoft Silverlighthas Isolated storage.[105]
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The economics ofinformation securityaddresses the economic aspects ofprivacyandcomputer security. Economics of information security includes models of the strictly rational “homo economicus” as well asbehavioral economics.Economics of securitiesaddresses individual and organizational decisions and behaviors with respect to security and privacy as market decisions.
Economics of security addresses a core question: why do agents choose technical risks when there exists technical solutions to mitigate security and privacy risks? Economics addresses not only this question, but also inform design decisions insecurity engineering.
National securityis the canonicalpublic good. The economic status of information security came to the intellectual fore around 2000. As is the case with innovations it arose simultaneously in multiple venues.
In 2000,Ross Andersonwrote,Why Information Security is Hard. Anderson explained that a significant difficulty in optimal development of security technology is that incentives must be aligned with the technology to enable rational adoption. Thus, economic insights should be integrated into technical design. A security technology should enable the party at risk to invest to limit that risk. Otherwise, the designers are simply counting onaltruismfor adoption and diffusion. Many consider this publication the birth of economics of security.
Also in 2000 at Harvard, Camp at the School of Government and Wolfram in the Department of Economics argued that security is not apublic goodbut rather each extant vulnerabilities has an associated negativeexternalityvalue. Vulnerabilities were defined in this work as tradable goods. Six years later,iDEFENSE,ZDIandMozillahave extant markets for vulnerabilities.
In 2000, the scientists at the Computer Emergency Response Team atCarnegie Mellon Universityproposed an early mechanism forrisk assessment. The Hierarchical Holographic Model provided the first multi-faceted evaluation tool to guide security investments using the science of risk. Since that time, CERT has developed a suite of systematic mechanism for organizations to use in risk evaluations, depending on the size and expertise of the organization:OCTAVE. The study of computer security as an investment in risk avoidance has become standard practice.
In 2001, in an unrelated development,Lawrence A. GordonandMartin P. LoebpublishedUsing Information Security as a Response to Competitor Analysis System. A working paper of the published article was written in 2000. These professors, from Maryland's Smith School of Business, present agame-theoreticframework that demonstrates how information security can prevent rival firms from gaining sensitive information. In this context, the article considers the economic (i.e., cost-benefit) aspects of information security.
The authors came together to develop and expand a series of flagship events under the name Workshop on the Economics of Information Security.
Proof of work is a security technology designed to stop spam by altering the economics. An early paper in economics of information security argued thatproof of workcannot work. In fact, the finding was thatproof of workcannot work withoutprice discriminationas illustrated by a later paper,Proof of Work can Work.
Another finding, one that is critical to an understanding of current American data practices, is that the opposite ofprivacyis not, in economic termsanonymity, but ratherprice discrimination.Privacy and price discriminationwas authored byAndrew Odlyzkoand illustrates that what may appear as information pathology in collection of data is in fact rational organizational behavior.
Hal Varianpresented three models of security using the metaphor of the height of walls around a town to show security as a normal good, public good, or good with externalities.Free ridingis the end result, in any case.
Lawrence A. GordonandMartin P. Loebwrote the "Economics of Information Security Investment".[1]TheGordon–Loeb modelis considered by many as the first economic model that determines the optimal amount to invest to protect a given set of information. The model takes into account the vulnerability of the information to a security breach and the potential loss should such a breach occur.
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Information assurance(IA) is the practice of assuring information and managing risks related to the use, processing, storage, andtransmissionof information. Information assurance includes protection of theintegrity, availability, authenticity,non-repudiationandconfidentialityof user data.[1]IA encompasses both digital protections and physical techniques. These methods apply todata in transit, both physical and electronic forms, as well asdata at rest. IA is best thought of as a superset ofinformation security(i.e. umbrella term), and as the business outcome ofinformation risk management.
Information assurance (IA) is the process of processing, storing, and transmitting the right information to the right people at the right time.[1]IA relates to the business level andstrategicrisk management of information and related systems, rather than the creation and application of security controls. IA is used to benefit business through the use of informationrisk management,trust management, resilience, appropriate architecture, system safety, and security, which increases the utility of information to only their authorized users.
Besides defending against malicioushackersand code (e.g.,viruses), IA practitioners considercorporate governanceissues such asprivacy, regulatory and standardscompliance,auditing,business continuity, anddisaster recoveryas they relate to information systems. Further, IA is an interdisciplinary field requiring expertise inbusiness,accounting, user experience,fraudexamination,forensic science,management science,systems engineering,security engineering, andcriminology, in addition to computer science.
With the growth of telecommunication networks also comes the dependency on networks, which makes communities increasing vulnerable to cyber attacks that could interrupt, degrade or destroy vital services.[2]Starting from the 1950s the role and use of information assurance has grown and evolved. These feedback loop practices were employed while developingWWMCCSmilitary decision support systems.
In the beginning information assurance involved just the backing up of data.[3]However once the volume of information increased, the act of information assurance began to become automated, reducing the use of operator intervention, allowing for the creation of instant backups.[3]The last main development of information assurance is implementingdistributed systemsfor the processing and storage of data through techniques likeSANsandNASplus usingcloud computing.[4][5][3]
These three main developments of information assurance parallel the three generations of information technologies, the first used to prevent intrusions, the 2nd to detect intrusion and the 3rd for survivability.[6][7]Information assurance is a collaborative effort of all sectors of life to allow a free and equal exchange of ideas.[citation needed]
Information assurance is built between five pillars:availability,integrity,authentication,confidentialityandnonrepudiation.[8]These pillars are taken into account to protect systems while still allowing them to efficiently provide services; However, these pillars do not act independently from one another, rather they interfere with the goal of the other pillars.[8]These pillars of information assurance have slowly changed to become referred to as thepillars of Cyber Security.As an administrator it is important to emphasize the pillars that you want in order to achieve your desired result for their information system, balancing the aspects of service, andprivacy.
Authentication refers to the verification of the validity of a transmission, originator, or process within an information system.[9]Authentication provides the recipient confidence in the data senders validity as well as the validity of their message.[8]There exists many ways to bolster authentication, mainly breaking down into three main ways,personally identifiable informationsuch as a person's name, address telephone number, access to akey token, or known information, like passwords.[10]
Integrity refers to the protection of information from unauthorized alteration.[3]The goal of information integrity is to ensure data is accurate throughout its entire lifespan.[11][12]User authentication is a critical enabler for information integrity.[8]Information integrity is a function of the number ofdegrees-of-trustexisting between the ends of an information exchange .[12]One way information integrity risk is mitigated is through the use of redundant chip and software designs.[13]A failure of authentication could pose a risk to information integrity as it would allow an unauthorized party to alter content. For example, if a hospital has inadequate password policies, an unauthorized user could gain access to an information systems governing the delivery of medication to patients and risk altering the treatment course to the detriment of a particular patient.[12]
The pillar of availability refers to the preservation of data to be retrieved or modified from authorized individuals. Higher availability is preserved through an increase in storage system or channel reliability.[8]Breaches in information availability can result from power outages, hardware failures,DDOS, etc. The goal of high availability is to preserve access to information. Availability of information can be bolstered by the use ofbackup power, sparedata channels, off site capabilities andcontinuous signal.[12]
Confidentiality is in essence the opposite of Integrity. Confidentiality is a security measure which protects against who is able to access the data, which is done by shielding who has access to the information.[8]This is different from Integrity as integrity is shielding who can change the information. Confidentiality is often ensured with the use of cryptography and steganography of data.[3]Confidentiality can be seen within the classification and information superiority with international operations such as NATO[14]Information assurance confidentiality in the United States need to follow HIPAA and healthcare provider security policyinformation labelingandneed-to-knowregulations to ensurenondisclosureof information.[12]
Nonrepudiationis the integrity of the data to be true to its origin, which prevents possible denial that an action occurred.[3][1]Increasing non-repudiation makes it more difficult to deny that the information comes from a certain source. In other words, it making it so that you can not dispute the source/ authenticity of data. Non-repudiation involves the reduction todata integritywhile that data is in transit, usually through the use of aman-in-the-middle attackorphishing.[15]
As stated earlier the pillars do not interact independently of one another, with some pillars impeding on the functioning of other pillars or in the opposite case where they boost other pillars.[8]For example, the increasing the availability of information works directly against the goals of three other pillars: integrity, authentication and confidentiality.[8]
The information assurance process typically begins with the enumeration and classification of the informationassetsto be protected. Next, the IA practitioner will perform arisk assessmentfor those assets.[16]Vulnerabilities in the information assets are determined in order to enumerate the threats capable of exploiting the assets. The assessment then considers both the probability and impact of a threat exploiting a vulnerability in an asset, with impact usually measured in terms of cost to the asset's stakeholders.[17]The sum of the products of the threats' impact and the probability of their occurring is the total risk to the information asset.
With the risk assessment complete, the IA practitioner then develops arisk management plan. This plan proposes countermeasures that involve mitigating, eliminating, accepting, or transferring the risks, and considers prevention, detection, and response to threats.
A framework published by a standards organization, such as NIST RMF,Risk IT,CobiT,PCI DSSorISO/IEC 27002, may guide development.Countermeasuresmay include technical tools such asfirewallsandanti-virus software, policies and procedures requiring such controls as regular backups and configuration hardening, employee training in security awareness, or organizing personnel into dedicatedcomputer emergency response team(CERT) or computer security incident response team (CSIRT). The cost and benefit of each countermeasure is carefully considered. Thus, the IA practitioner does not seek to eliminate all risks; but, to manage them in the mostcost-effectiveway.[18]
After the risk management plan is implemented, it is tested and evaluated, often by means of formal audits.[16]The IA process is an iterative one, in that the risk assessment and risk management plan are meant to be periodically revised and improved based on data gathered about their completeness and effectiveness.[2]
There are two meta-techniques with information assurance:auditand risk assessment.[16]
Business Risk Managementbreaks down into three main processes Risk Assessment, Risk Mitigation and Evaluation and assessment.[citation needed]Information Assurance is one of the methodologies which organizations use to implement business risk management. Through the use of information assurance policies like the "BRICK" frame work.[1]Additionally, Business Risk Management also occurs to comply with federal and international laws regarding the release and security of information such asHIPAA.[19]
Information assurance can be aligned with corporates strategies through training and awareness, senior management involvement and support, and intra-organizational communication allowing for greater internal control and business risk management.[20]
Many security executives in are firms are moving to a reliance on information assurance to protect intellectual property, protect against potential data leakage, and protect users against themselves.[17]While the use of information assurance is good ensuring certain pillars like, confidentiality, non-repudiation, etc. because of their conflicting nature an increase in security often comes at the expense of speed.[8][17]Using information assurance in the business model improves reliable management decision-making, customer trust, business continuity and good governance in both public and private sectors.[21]
There are a number of international and national bodies that issue standards on information assurance practices, policies, and procedures. In the UK, these include the Information Assurance Advisory Council and theInformation Assurance Collaboration Group.[4]
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TheNational Cipher Challengeis an annualcryptographiccompetition organised by theUniversity of SouthamptonSchool of Mathematics. Competitors attempt to breakcryptogramspublished on the competition website.[1]In the 2017, more than 7,500 students took part in the competition.[2]Participants must be in full-time school level education in order to qualify for prizes.[3]
The competition is organised into eight to ten challenges, which are further subdivided into parts A and B. The part A challenge consists of a comparatively simpler cryptogram, and usually provides some useful information to assist in the solving of part B. Part B is usually more complex. In later challenges the cryptograms become harder to break.[3]In the past, part A cryptograms have been encrypted with theCaesar cipher, theAffine cipher, theKeyword cipher, theTransposition cipher, theVigenère cipherand the 2x2Hill cipher.
The part B challenges are intended to be harder. These begin with relatively simplesubstitution ciphers, including theBacon cipherandPolybius square, before moving on totransposition ciphers,Playfair ciphersand polyalphabetic ciphers such as theVigenère cipher, theAutokey cipherand theAlberti cipher. In the later stages of the competition, theADFGVX cipher, theSolitaire cipher, the Double Playfair cipher, theHill cipher, theBook cipherand versions of theEnigmaandFialkacipher machines have all been used. The 2009 challenge ended with a Jefferson Disk cipher, the 2012 challenge ended with the ADFGVX Cipher, the 2014 with thePlayfair Cipher, and the most recent challenge ended with a sectioned Cadenus transposition.
£25 cash prizes are awarded to eight random entrants who submit a correct solution for each part A of the challenge. Leaderboards for the part B challenges are also compiled, based on how accurate solutions are and how quickly the entrant broke the cipher. Prizes are awarded to the top three entrants at the end of the challenge. In the 2009/10 challenge, the sponsors provided several prizes: IBM provided iPod Touches to each member of the team winning the Team Prize, Trinity College provided a cash prize of £700, and GCHQ provided a cash prize of £1000. In previous years prizes such as an IBM Thinkpad laptop have been awarded.
After the challenge the winners of the top prizes and other randomly selected entrants are invited to a day held atBletchley Parkconsisting of lectures (with subjects such as the Semantic Web,World War IIcryptography andcomputer programming) and the prize-giving ceremony.
Current sponsors of the competition includeGCHQ,IBM,British Computer Society,Trinity College, Cambridge,Cambridge University Press,Winton Capital ManagementandEPSRC.
The websites for the challenges earlier than this are no longer available.
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https://en.wikipedia.org/wiki/National_Cipher_Challenge
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Vulnerabilitiesare flaws or weaknesses in a system's design, implementation, or management that can be exploited by a malicious actor to compromise its security.
Despite intentions to achieve complete correctness, virtually all hardware and software contain bugs where the system does not behave as expected. If the bug could enable an attacker to compromise the confidentiality, integrity, or availability of system resources, it is called a vulnerability. Insecuresoftware developmentpractices as well as design factors such as complexity can increase the burden of vulnerabilities. There are different types most common in different components such as hardware, operating systems, and applications.
Vulnerability managementis a process that includes identifying systems and prioritizing which are most important, scanning for vulnerabilities, and taking action to secure the system. Vulnerability management typically is a combination of remediation (fixing the vulnerability), mitigation (increasing the difficulty or reducing the danger of exploits), and accepting risks that are not economical or practical to eliminate. Vulnerabilities can be scored for risk according to theCommon Vulnerability Scoring Systemor other systems, and added to vulnerability databases. As of November 2024[update], there are more than 240,000 vulnerabilities[1]catalogued in theCommon Vulnerabilities and Exposures(CVE) database.
A vulnerability is initiated when it is introduced into hardware or software. It becomes active and exploitable when the software or hardware containing the vulnerability is running. The vulnerability may be discovered by the vendor or a third party. Disclosing the vulnerability (as apatchor otherwise) is associated with an increased risk of compromise because attackers often move faster than patches are rolled out. Regardless of whether a patch is ever released to remediate the vulnerability, its lifecycle will eventually end when the system, or older versions of it, fall out of use.
Despite developers' goal of delivering a product that works entirely as intended, virtually allsoftwareandhardwarecontain bugs.[2]If a bug creates a security risk, it is called a vulnerability.[3][4][5]Software patchesare often released to fix identified vulnerabilities, but those that remain unknown (zero days) as well as those that have not been patched are still liable for exploitation.[6]Vulnerabilities vary in their ability to beexploitedby malicious actors,[3]and the actual risk is dependent on the nature of the vulnerability as well as the value of the surrounding system.[7]Although some vulnerabilities can only be used fordenial of serviceattacks, more dangerous ones allow the attacker toinjectand run their own code (calledmalware), without the user being aware of it.[3]Only a minority of vulnerabilities allow forprivilege escalation, which is necessary for more severe attacks.[8]Without a vulnerability, the exploit cannot gain access.[9]It is also possible formalwareto be installed directly, without an exploit, if the attacker usessocial engineeringor implants the malware in legitimate software that is downloaded deliberately.[10]
Fundamental design factors that can increase the burden of vulnerabilities include:
Somesoftware developmentpractices can affect the risk of vulnerabilities being introduced to a code base. Lack of knowledge about secure software development or excessive pressure to deliver features quickly can lead to avoidable vulnerabilities to enter production code, especially if security is not prioritized by thecompany culture. This can lead to unintended vulnerabilities. The more complex the system is, the easier it is for vulnerabilities to go undetected. Some vulnerabilities are deliberately planted, which could be for any reason from a disgruntled employee selling access to cyber criminals, to sophisticated state-sponsored schemes to introduce vulnerabilities to software.[15]Inadequatecode reviewscan lead to missed bugs, but there are alsostatic code analysistools that can be used as part of code reviews and may find some vulnerabilities.[16]
DevOps, a development workflow that emphasizes automated testing and deployment to speed up the deployment of new features, often requires that many developers be granted access to change configurations, which can lead to deliberate or inadvertent inclusion of vulnerabilities.[17]Compartmentalizing dependencies, which is often part of DevOps workflows, can reduce theattack surfaceby paring down dependencies to only what is necessary.[18]Ifsoftware as a serviceis used, rather than the organization's own hardware and software, the organization is dependent on the cloud services provider to prevent vulnerabilities.[19]
TheNational Vulnerability Databaseclassifies vulnerabilities into eight root causes that may be overlapping, including:[20]
Deliberate security bugs can be introduced during or after manufacturing and cause theintegrated circuitnot to behave as expected under certain specific circumstances. Testing for security bugs in hardware is quite difficult due to limited time and the complexity of twenty-first century chips,[23]while the globalization of design and manufacturing has increased the opportunity for these bugs to be introduced by malicious actors.[24]
Althoughoperating system vulnerabilitiesvary depending on theoperating systemin use, a common problem isprivilege escalationbugs that enable the attacker to gain more access than they should be allowed.Open-sourceoperating systems such asLinuxandAndroidhave a freely accessiblesource codeand allow anyone to contribute, which could enable the introduction of vulnerabilities. However, the same vulnerabilities also occur in proprietary operating systems such asMicrosoft WindowsandApple operating systems.[25]All reputable vendors of operating systems provide patches regularly.[26]
Client–server applicationsare downloaded onto the end user's computers and are typically updated less frequently than web applications. Unlike web applications, they interact directly with a user'soperating system. Common vulnerabilities in these applications include:[27]
Web applicationsrun on many websites. Because they are inherently less secure than other applications, they are a leading source ofdata breachesand other security incidents.[28][29]They can include:
Attacks used against vulnerabilities in web applications include:
There is little evidence about the effectiveness and cost-effectiveness of different cyberattack prevention measures.[32]Although estimating the risk of an attack is not straightforward, the mean time to breach and expected cost can be considered to determine the priority for remediating or mitigating an identified vulnerability and whether it is cost effective to do so.[33]Although attention to security can reduce the risk of attack, achieving perfect security for a complex system is impossible, and many security measures have unacceptable cost or usability downsides.[34]For example, reducing the complexity and functionality of the system is effective at reducing theattack surface.[35]
Successful vulnerability management usually involves a combination of remediation (closing a vulnerability), mitigation (increasing the difficulty, and reducing the consequences, of exploits), and accepting some residual risk. Often adefense in depthstrategy is used for multiple barriers to attack.[36]Some organizations scan for only the highest-risk vulnerabilities as this enables prioritization in the context of lacking the resources to fix every vulnerability.[37]Increasing expenses is likely to havediminishing returns.[33]
Remediation fixes vulnerabilities, for example by downloading asoftware patch.[38]Software vulnerability scannersare typically unable to detect zero-day vulnerabilities, but are more effective at finding known vulnerabilities based on a database. These systems can find some known vulnerabilities and advise fixes, such as a patch.[39][40]However, they have limitations includingfalse positives.[38]
Vulnerabilities can only be exploited when they are active-the software in which they are embedded is actively running on the system.[41]Before the code containing the vulnerability is configured to run on the system, it is considered a carrier.[42]Dormant vulnerabilities can run, but are not currently running. Software containing dormant and carrier vulnerabilities can sometimes be uninstalled or disabled, removing the risk.[43]Active vulnerabilities, if distinguished from the other types, can be prioritized for patching.[41]
Vulnerability mitigation is measures that do not close the vulnerability, but make it more difficult to exploit or reduce the consequences of an attack.[44]Reducing theattack surface, particularly for parts of the system withroot(administrator) access, and closing off opportunities for exploits to engage inprivilege exploitationis a common strategy for reducing the harm that a cyberattack can cause.[38]If a patch for third-party software is unavailable, it may be possible to temporarily disable the software.[45]
Apenetration testattempts to enter the system via an exploit to see if the system is insecure.[46]If a penetration test fails, it does not necessarily mean that the system is secure.[47]Some penetration tests can be conducted with automated software that tests against existing exploits for known vulnerabilities.[48]Other penetration tests are conducted by trained hackers. Many companies prefer to contract out this work as it simulates an outsider attack.[47]
The vulnerability lifecycle begins when vulnerabilities are introduced into hardware or software.[49]Detection of vulnerabilities can be by the software vendor, or by a third party. In the latter case, it is considered most ethical to immediately disclose the vulnerability to the vendor so it can be fixed.[50]Government or intelligence agencies buy vulnerabilities that have not been publicly disclosed and may use them in an attack, stockpile them, or notify the vendor.[51]As of 2013, theFive Eyes(United States, United Kingdom, Canada, Australia, and New Zealand) captured the plurality of the market and other significant purchasers included Russia, India, Brazil, Malaysia, Singapore, North Korea, and Iran.[52]Organized criminal groups also buy vulnerabilities, although they typically preferexploit kits.[53]
Even vulnerabilities that are publicly known or patched are often exploitable for an extended period.[54][55]Security patches can take months to develop,[56]or may never be developed.[55]A patch can have negative effects on the functionality of software[55]and users may need totestthe patch to confirm functionality and compatibility.[57]Larger organizations may fail to identify and patch all dependencies, while smaller enterprises and personal users may not install patches.[55]Research suggests that risk of cyberattack increases if the vulnerability is made publicly known or a patch is released.[58]Cybercriminals canreverse engineerthe patch to find the underlying vulnerability and develop exploits,[59]often faster than users install the patch.[58]
Vulnerabilities become deprecated when the software or vulnerable versions fall out of use.[50]This can take an extended period of time; in particular, industrial software may not be feasible to replace even if the manufacturer stops supporting it.[60]
A commonly used scale for assessing the severity of vulnerabilities is the open-source specificationCommon Vulnerability Scoring System(CVSS). CVSS evaluates the possibility to exploit the vulnerability and compromise data confidentiality, availability, and integrity. It also considers how the vulnerability could be used and how complex an exploit would need to be. The amount of access needed for exploitation and whether it could take place without user interaction are also factored in to the overall score.[61][62]
Someone who discovers a vulnerability may disclose it immediately (full disclosure) or wait until a patch has been developed (responsible disclosure, or coordinated disclosure). The former approach is praised for its transparency, but the drawback is that the risk of attack is likely to be increased after disclosure with no patch available.[63]Some vendors paybug bountiesto those who report vulnerabilities to them.[64][65]Not all companies respond positively to disclosures, as they can cause legal liability and operational overhead.[66]There is no law requiring disclosure of vulnerabilities.[67]If a vulnerability is discovered by a third party that does not disclose to the vendor or the public, it is called azero-day vulnerability, often considered the most dangerous type because fewer defenses exist.[68]
The most commonly used vulnerability dataset isCommon Vulnerabilities and Exposures(CVE), maintained byMitre Corporation.[69]As of November 2024[update], it has over 240,000 entries[1]This information is shared into other databases, including the United States'National Vulnerability Database,[69]where each vulnerability is given a risk score usingCommon Vulnerability Scoring System(CVSS),Common Platform Enumeration(CPE) scheme, andCommon Weakness Enumeration.[citation needed]CVE and other databases typically do not track vulnerabilities insoftware as a serviceproducts.[39]Submitting a CVE is voluntary for companies that discovered a vulnerability.[67]
The software vendor is usually not legally liable for the cost if a vulnerability is used in an attack, which creates an incentive to make cheaper but less secure software.[70]Some companies are covered by laws, such asPCI,HIPAA, andSarbanes-Oxley, that place legal requirements on vulnerability management.[71]
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https://en.wikipedia.org/wiki/Security_vulnerability
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TheZendian problemwas an exercise in communication intelligence operations (mainlytraffic analysisandcryptanalysis) devised byLambros D. Callimahosas part of an advanced course, CA-400, that Callimahos taught toNational Security Agencycryptanalystsstarting in the 1950s.
The scenario involves 375 radio messages said to have been intercepted on December 23 by theUS Armycontingent of aUnited Nationsforce landed on the fictional island of Zendia in thePacific Ocean.
A typical intercept looks like this:
For each message, the first line is provided by theinterceptoperator, givingcall signs,frequency, time, and reference number. The rest of the message is a transcript of theMorse codetransmission.
At the beginning of the intercepted message there is a header which consists of 8 four-digit groups. Initially, the meaning of the numeric header is not known; the meanings of various components of this header (such as aserial numberassigned by the transmitting organization's message center) can be worked out through traffic analysis.
The rest of the message consists of "indicators" andciphertext; the first group is evidently a "discriminant" indicating the cryptosystem used, and (depending on the cryptosystem) some or all of the second group may contain a message-specifickeyingelement such as initialrotorsettings. The first two groups are repeated at the end of the message, which allows correction of garbled indicators. The remaining characters are encrypted text. Since the transmissions always use complete groups, "nulls" may have been used topad outthe text.
Cryptosystems employed includetransposition, dinome, androtor-basedciphersand aone-part code. While these can be successfully tackled without use of a computer, solution is not easy. The practical exercise reinforces many basic principles, including ways to exploit having acollectionof message traffic.
A certain amount ofPlaintextinter-operator "chatter" is also provided, and may help with the analysis.
Headers and discriminants are also given for intercepts from the next three days; these may be used for traffic analysis and in determining daily operating procedures.
The Zendian problem has beendeclassifiedand is available either as part ofMilitary Cryptanalytics[1]or as a book in itself.[2]Both were published as reprints byAegean Park Press, Walnut Creek, CA, USA. There are a few small differences in the ciphertext between these sources, with neither being entirely error-free; indeed, some errors were intentional, and dealing with them is part of the learning process.
Karl Heinz-Everts' Web site provides a link to a downloadable archive[3]of the "intercepts." (The site also provides analysis and partial solutions, which shouldnotbe viewed by anyone wishing to learn from working on this exercise.)
Cryptanalists who successfully finished CA-400 became members of theDundee Society. This society was founded by Lambros D. Callimahos[4]and was so named after the empty Dundee marmalade jar on his desk, as he couldn't disclose the society's real purpose.
A large print showing the fictional nation of Zendia hangs on the wall of the library at theNational Cryptologic Museum, which is operated by the NSA. Copies of this map, shrunken to such a degree that most feature labels are illegible, appear in the previously cited books.
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https://en.wikipedia.org/wiki/Zendian_Problem
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Incomputer security,shoulder surfingis a type ofsocial engineeringtechnique used to obtaininformationsuch aspersonal identification numbers (PINs),passwordsand other confidential data by looking over the victim's shoulder. Unauthorized users watch the keystrokes inputted on a device or listen to sensitive information being spoken, which is also known aseavesdropping.[1]
This attack can be performed either at close range (by directly looking over the victim's shoulder) or from a longer range with, for example, a pair ofbinocularsor similar hardware.[2]Attackers do not need any technical skills in order to perform this method, and keen observation of victims' surroundings and the typing pattern is sufficient. In the early 1980s, shoulder surfing was practiced near public pay phones to steal calling card digits and make long-distance calls or sell them in the market for cheaper prices than the original purchaser paid. However, the advent of modern-day technologies likehidden camerasand secret microphones makes shoulder surfing easier and gives the attacker more scope to perform long-range shoulder surfing. A hidden camera allows the attacker to capture the whole login process and other confidential data of the victim, which ultimately could lead to financial loss oridentity theft.[3]Shoulder surfing is more likely to occur in crowded places because it is easier to observe the information without getting the victim's attention.[4]There are two types of shoulder-surfing attack: direct observation attacks, in which authentication information is obtained by a person who is directly monitoring the authentication sequence, and recording attacks, in which the authentication information is obtained by recording the authentication sequence for later analysis to open the device. Apart from threats to password or PIN entry, shoulder surfing also occurs in daily situations to uncover private content on handheld mobile devices; shoulder surfing visual content was found to leak sensitive information of the user and even private information about third parties.[5]
The basic procedure for gaze-based password entry is similar to normal password entry, except that in place of typing a key or touching the screen, the user looks at each desired character or trigger region in sequence (same as eye typing). The approach can, therefore, be used both withcharacter-basedpasswords by using an on-screen keyboard and with graphical password schemes as surveyed in.[6]A variety of considerations is important for ensuring usability and security. Eye-tracking technology has progressed significantly since its origins in the early 1900s.[7]State of the art eye trackers offers non-encumbering, remotevideo-basedeye tracking with an accuracy of 1˚ of visual angle. Eye trackers are a specialized application of computer vision. A camera is used to monitor the user's eyes. One or more infrared light sources illuminate the user's face and produce a glint – a reflection of the light source on the cornea. As the user looks in different directions the pupil moves but the location of the glint on the cornea remains fixed. The relative motion and position of the center of the pupil and the glint are used to estimate the gaze vector, which is then mapped to coordinates on the screen plane.
Researchers proposed ways to counter shoulder surfing on mobile devices by leveraging the front-facing camera for gaze-based password entry. For example, GazeTouchPIN[8]and GazeTouchPass[9]combine gaze input in the form of eye movements to the left/right, and touch input by tapping on-screen buttons. These methods are more secure than traditional touch-based input (e.g., PIN and Lock Patterns) because they require shoulder surfers to (1) observe the user's eyes, (2) observe the user's touch input, and (3) combine the observations.
Painting album mechanismis an anti-shoulder surfing mechanism, which has characteristics of both recall and recognitiongraphical techniques. Rather than using a regular PIN or password involvingalphanumericcharacters, users select a sequence of colors or pictures to unlock the system. The order of the colors and pictures selected during the sign-in process has to match with the order at registration.[10]This anti-shoulder surfing security method was developed based on survey results of users' affinity of choices,[11]and through observation on the way children paint pictures. The resulting mechanism was developed from the survey of user choices, and the outcome created three input schemes named Swipe Scheme, Colour Scheme, and Scot Scheme. Swipe Scheme is implemented inMicrosoftWindows 8, and in later versions, it is known as Picture Password; however it has drawn criticism for requiring the user to use a secure enough gesture.[12]
For access to sensitive information with a low risk of shoulder surfing, the secret tap method is a technique that does not expose the authentication information during entry, even if other individuals try to view the input process. Additionally, the risk of camera recordings also poses athreat. Therefore, it is necessary to make the authentication process more complex in order to prevent authentication information from being stolen. For example,smartphonesuse biometrics such asfingerprint scanningor facial recognition which cannot be replicated by a shoulder surfer.
The secret tap authentication method can use icons or some other form of system. The goals of a secret tap system are:
The primary benefit ofgraphical passwordscompared toalphanumeric passwordsis improved memorability. However, the potential detriment of this advantage is the increased risk of shoulder-surfing. Graphical passwords that use graphics or pictures[13]such as PassFaces, Jiminy,[14]VIP, Passpoints[13]or a combination of graphics and audio such as AVAP are likely all subject to this increased risk unless somehow mitigated in implementation. The results indicate the fact that both alphanumeric and graphical password-based authentication mechanisms may have a significant vulnerability to shoulder-surfing unless certain precautions are taken. Despite the common belief that nondictionary passwords are the most secure type of password-based authentication, the results demonstrate that it is, in fact, the most vulnerable configuration to shoulder-surfing.
Personal identification number(or PIN for short) is used to authenticate oneself in various situations, while withdrawing or depositing money from anautomatic teller machine, unlocking a phone, a door, alaptopor aPDA. Though this method of authentication is atwo step verification processin some situations, it is vulnerable to shoulder surfing attacks. An attacker can obtain the PIN either by directly looking over the victim's shoulder or by recording the wholeloginprocess. On items such as mobile phones with glass, glossy screens, the user could leave smudges on the screen, revealing a PIN.[15]Some highly advanced attacks use thermal cameras to see the thermal signature of the PIN entered.[16]Thermal attacks take advantage of heat fingerprints remaining on keys after the authenticating person is done entering the secret.[17]So, various shoulder surfing resistant PIN entry methodologies are used to make theauthenticationprocess secure.[18]Examples include PIN pads with built-in privacy shields. Another example used in ATMs and some entry systems is that of the use of metal PIN pads, making thermal camera attacks nearly impossible due to their material,[19]shielding, reflectivity or internal heating.[17]The transfer of heat through wiping with warm objects or hands is found effective to counter thermal attacks in experiments.[17]
The cognitive trapdoor game has three groups involved in it: a machine verifier, a human prover, and a human observer. The goal of each group is that a human prover has to input the PIN by answering the questions posed by the machine verifier while an observer attempts to shoulder surf the PIN. As the countermeasures are by design harder to easily usurp, it is not easy for the observer to remember the whole login process unless the observer has a recording device.[20]
A user could wear avirtual reality headsetto mitigate the issues of shoulder surfing; however, gesture controls, buttons pressed, and voice commands could still be attacked.[21]
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https://en.wikipedia.org/wiki/Shoulder_surfing_(computer_security)
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Opportunistic encryption(OE) refers to anysystemthat, when connecting to another system, attempts toencryptcommunications channels, otherwise falling back to unencrypted communications. This method requires no pre-arrangement between the two systems.
Opportunistic encryption can be used to combatpassive wiretapping.[1](anactivewiretapper, on the other hand, can disrupt encryption negotiation to either force an unencrypted channel or perform aman-in-the-middle attackon the encrypted link.) It does not provide a strong level ofsecurityas authentication may be difficult to establish and secure communications are not mandatory. However, it does make the encryption of mostInternet trafficeasy to implement, which removes a significant impediment to the mass adoption of Internet traffic security.
Opportunistic encryption on the Internet is described inRFC4322"Opportunistic Encryption using the Internet Key Exchange (IKE)",RFC7435"Opportunistic Security: Some Protection Most of the Time", and inRFC8164"Opportunistic Security for HTTP/2".
TheFreeS/WANproject was one of the early proponents of OE.[2]The effort is continued by the former freeswan developers now working onLibreswan. Libreswan aims to support different authentication hooks for opportunistic encryption withIPsec. Version 3.16, which was released in December 2015, had support for Opportunistic IPsec using AUTH-NULL[3]which is based onRFC 7619. The Libreswan Project is currently working on (forward)Domain Name System Security Extensions(DNSSEC) andKerberossupport for Opportunistic IPsec.[citation needed]
Openswanhas also been ported to theOpenWrtproject.[4]Openswan used reverseDNSrecords to facilitate the key exchange between the systems.
It is possible to useOpenVPNand networking protocols to set up dynamic VPN links which act similar to OE for specific domains.[5]
The FreeS/WAN and forks such as Openswan andstrongSwanoffer VPNs that can also operate in OE mode using IPsec-based technology.Obfuscated TCPis another method of implementing OE.
Microsoft Windowsplatforms have an implementation of OE installed by default. This method uses IPsec to secure the traffic and is a simple procedure to turn on. It is accessed via theMMCand "IP Security Policies on Local Computer" and then editing the properties to assign the "(Request Security)" policy.[6]This will turn on optional IPsec in a Kerberos environment.
Many systems also have problems when either side is behind aNAT. This problem is addressed byNAT traversal(NAT-T) and is accomplished by editing aregistryitem.[7]Using the filtering options provided in MMC, it is possible to tailor the networking to require, request or permit traffic to various domains and protocols to use encryption.
Opportunistic encryption can also be used for specific traffic likee-mailusing theSMTPSTARTTLSextension for relaying messages across the Internet, or theInternet Message Access Protocol(IMAP) STARTTLS extension for reading e-mail. With this implementation, it is not necessary to obtain a certificate from acertificate authority, as aself-signed certificatecan be used.
Many systems employ a variant with third-party add-ons to traditional email packages by first attempting to obtain an encryption key and if unsuccessful, then sending the email in the clear.PGP,p≡p,Hushmail, and Ciphire, among others can all be set up to work in this mode.
In practice, STARTTLS in SMTP is often deployed with self-signed certificates,[8]which represents a minimal one-time task for a system administrator, and results in most email traffic being opportunistically encrypted.[9]
SomeVoice over IP(VoIP) solutions provide for painless encryption of voice traffic when possible. Some versions of theSipura TechnologyandLinksyslines ofanalog telephony adapters(ATA) include a hardware implementation ofSRTPwith the installation of a certificate from Voxilla, a VoIP information site. When the call is placed an attempt is made to use SRTP, if successful a series of tones are played into the handset, if not the call proceeds without using encryption.SkypeandAmicimause only secure connections andGizmo5attempts a secure connection between its clients.Phil Zimmermann, Alan Johnston, andJon Callashave proposed a new VoIP encryption protocol calledZRTP.[10]They have an implementation of it calledZfonewhose source and compiled binaries are available.
For encryptingWWW/HTTPconnections,HTTPSis typically used, which requires strict encryption and has significant administrative costs, both in terms of initial setup and continued maintenance costs for thewebsiteoperator. Most browsers verify thewebserver's identity to make sure that anSSL certificateis signed by a trustedcertificate authorityand has not expired, usually requiring the website operator to manually change the certificate every one or two years. The easiest way to enable some sort of opportunistic website encryption is by using self-signed certificates, but this causesbrowsersto display a warning each time the website is visited unless the user manually marks the website's certificate as trusted. Because unencrypted websites do not currently display any such warnings, the use of self-signed certificates is not well received.
In 2015,Mozillastarted to roll out opportunistic encryption inFirefoxversion 37.[11]This was quickly rolled back (in update 37.0.1) due to a seriousvulnerabilitythat could bypassSSL certificateverification.[12]
Browser extensions likeHTTPS Everywhereand HTTPSfinder[13]find and automatically switch the connection to HTTPS when possible.
Several proposals were available for true, seamless opportunistic encryption ofHTTP/2protocol.[14]These proposals were later rejected.Poul-Henning Kamp, lead developer ofVarnishand a seniorFreeBSDkernel developer, has criticized theIETFfor following a particularpolitical agendawith HTTP/2 for not implementing opportunistic encryption in the standard.[15][16]
STARTTLSimplementations often used withSMTPare vulnerable toSTRIPTLSattacks when subject toactive wiretapping.
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https://en.wikipedia.org/wiki/Opportunistic_encryption
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Form grabbingis a form ofmalwarethat works by retrieving authorization and log-in credentials from a web data form before it is passed over the Internet to a secure server. This allows themalwareto avoid HTTPSencryption. This method is more effective thankeylogger softwarebecause it will acquire the user’s credentials even if they are input using virtual keyboard, auto-fill, or copy and paste.[1]It can then sort the information based on its variable names, such asemail, account name, andpassword. Additionally, the form grabber will log theURLand title of the website the data was gathered from.[2]
The method was invented in 2003 by the developer of a variant of atrojan horsecalled Downloader.Barbew, which attempts to download Backdoor.Barbew from the Internet and bring it over to the local system for execution. However, it was not popularized as a well known type ofmalwareattack until the emergence of the infamous banking trojanZeusin 2007.[3]Zeus was used to steal banking information by man-in-the-browserkeystroke loggingand form grabbing. Like Zeus, the Barbew trojan was initially spammed to large numbers of individuals through e-mails masquerading as big-name banking companies.[4]Form grabbing as a method first advanced through iterations of Zeus that allowed the module to not only detect the grabbed form data but to also determine how useful the information taken was. In later versions, the form grabber was also privy to the website where the actual data was submitted, leaving sensitive information more vulnerable than before.[5]
A trojan known as Tinba (Tiny Banker Trojan) has been built with form grabbing and is able to steal online banking credentials and was first discovered in 2012. Another program calledWeyland-Yutani BOTwas the first software designed to attack themacOSplatform and can work onFirefox. The web injects templates in Weyland-Yutani BOT were different from existing ones such asZeusandSpyEye.[6]
Another known version is British Airways breach in September 2018. In the British Airways’ case, the organizations’ servers appeared to have been compromised directly, with the attackers modifying one of the JavaScript files (Modernizr JavaScript library, version 2.6.2) to include a PII/credit card logging script that would grab the payment information and send the information to the server controlled by the attacker hosted on “baways[.]com” domain with an SSL certificate issued by “Comodo” Certificate Authority.
The British Airways mobile application also loads a webpage built with the same CSS and JavaScript components as the main website, including the malicious script installed by Magecart. Thus, the payments made using the British Airways mobile app were also affected.
Due to the recent increase in keylogging and form grabbing,antiviruscompanies are adding additional protection to counter the efforts of key-loggers and prevent collecting passwords. These efforts have taken different forms varying from antivirus companies, such as safepay, password manager, and others.[1]To further counter form grabbing, users' privileges can become limited which would prevent them from installingBrowser Helper Objects(BHOs) and other form grabbing software. Administrators should create a list of maliciousserversto theirfirewalls.[2]
New countermeasures, such as usingOut-of-bandcommunication, to circumvent form grabbers andMan-in-the-browserare also emerging; examples include FormL3SS.;[7]those that circumvent the threat use a different communication channel to send the sensitive data to the trusted server. Thus, no information is entered on the compromised device. Alternative Initiatives such asFideliususe added hardware to protect the input/output to the compromised or believed compromised device.
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https://en.wikipedia.org/wiki/Form_grabbing
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Incomputer security, athreatis a potential negative action or event enabled by avulnerabilitythat results in an unwanted impact to a computer system or application.
A threat can be either a negative "intentional" event (i.e. hacking: an individual cracker or a criminal organization) or an "accidental" negative event (e.g. the possibility of a computer malfunctioning, or the possibility of anatural disasterevent such as anearthquake, afire, or atornado) or otherwise a circumstance, capability, action, or event (incidentis often used as a blanket term).[1]Athreat actorwho is an individual or group that can perform the threat action, such as exploiting a vulnerability to actualise a negative impact. Anexploitis a vulnerability that a threat actor used to cause an incident.
A more comprehensive definition, tied to anInformation assurancepoint of view, can be found in "Federal Information Processing Standards (FIPS) 200, Minimum Security Requirements for Federal Information and Information Systems" byNISTofUnited States of America[2]
National Information Assurance Glossarydefinesthreatas:
ENISAgives a similar definition:[3]
The Open Groupdefinesthreatas:[4]
Factor analysis of information riskdefinesthreatas:[5]
National Information Assurance Training and Education Centergives a more articulated definition ofthreat:[6][7]
The term "threat" relates to some other basic security terms as shown in the following diagram:[1]A resource (both physical or logical) can have one or more vulnerabilities that can be exploited by a threat agent in a threat action. The result can potentially compromise theconfidentiality,integrityoravailabilityproperties of resources (potentially different than the vulnerable one) of the organization and others involved parties (customers, suppliers).The so-calledCIA triadis the basis ofinformation security.
Theattackcan beactivewhen it attempts to alter system resources or affect their operation: so it compromises Integrity or Availability. A "passive attack" attempts to learn or make use of information from the system but does not affect system resources: so it compromises Confidentiality.[1]
OWASP(see figure) depicts the same phenomenon in slightly different terms: a threat agent through an attack vector exploits a weakness (vulnerability) of the system and the relatedsecurity controlscausing a technical impact on an IT resource (asset) connected to a business impact.
A set of policies concerned with information security management, theInformation security management systems(ISMS), has been developed to manage, according torisk managementprinciples, thecountermeasuresin order to accomplish to a security strategy set up following rules and regulations applicable in a country. Countermeasures are also called security controls; when applied to the transmission of information are namedsecurity services.[8]
The overall picture represents therisk factorsof the risk scenario.[9]
The widespread of computer dependencies and the consequent raising of the consequence of a successful attack, led to a new termcyberwarfare.
Nowadays the many real attacks exploitPsychologyat least as much as technology.PhishingandPretextingand other methods are calledsocial engineeringtechniques.[10]TheWeb 2.0applications, specificallySocial network services, can be a mean to get in touch with people in charge of system administration or even system security, inducing them to reveal sensitive information.[11]One famous case isRobin Sage.[12]
The most widespread documentation oncomputer insecurityis about technical threats such as acomputer virus,trojanand othermalware, but a serious study to apply cost effective countermeasures can only be conducted following a rigorousIT risk analysisin the framework of an ISMS: a pure technical approach will let out the psychological attacks that are increasing threats.
Threats can be classified according to their type and origin:[13]
Note that a threat type can have multiple origins.
Recent trends in computer threats show an increase in ransomware attacks, supply chain attacks, and fileless malware. Ransomware attacks involve the encryption of a victim's files and a demand for payment to restore access. Supply chain attacks target the weakest links in a supply chain to gain access to high-value targets. Fileless malware attacks use techniques that allow malware to run in memory, making it difficult to detect.[14]
Below are the few common emerging threats:
Microsoftpublished a mnemonic,STRIDE,[15]from the initials of threat groups:
Microsoft previously rated the risk of security threats using five categories in a classification calledDREAD: Risk assessment model. The model is considered obsolete by Microsoft.
The categories were:
The DREAD name comes from the initials of the five categories listed.
The spread over a network of threats can lead to dangerous situations. In military and civil fields, threat level has been defined: for exampleINFOCONis a threat level used by the US. Leadingantivirus softwarevendors publish global threat level on their websites.[16][17]
The termThreat Agentis used to indicate an individual or group that can manifest a threat. It is fundamental to identify who would want to exploit the assets of a company, and how they might use them against the company.[18]
Individuals within a threat population; Practically anyone and anything can, under the right circumstances, be a threat agent – the well-intentioned, but inept, computer operator who trashes a daily batch job by typing the wrong command, the regulator performing an audit, or the squirrel that chews through a data cable.[5]
Threat agents can take one or more of the following actions against an asset:[5]
Each of these actions affects different assets differently, which drives the degree and nature of loss. For example, the potential for productivity loss resulting from a destroyed or stolen asset depends upon how critical that asset is to the organization's productivity. If a critical asset is simply illicitly accessed, there is no direct productivity loss. Similarly, the destruction of a highly sensitive asset that does not play a critical role in productivity would not directly result in a significant productivity loss. Yet that same asset, if disclosed, can result in significant loss of competitive advantage or reputation, and generate legal costs. The point is that it is the combination of the asset and type of action against the asset that determines the fundamental nature and degree of loss. Which action(s) a threat agent takes will be driven primarily by that agent's motive (e.g., financial gain, revenge, recreation, etc.) and the nature of the asset. For example, a threat agent bent on financial gain is less likely to destroy a critical server than they are to steal an easilypawnedasset like a laptop.[5]
It is important to separate the concept of the event that a threat agent get in contact with the asset (even virtually, i.e. through the network) and the event that a threat agent act against the asset.[5]
OWASP collects a list of potential threat agents to prevent system designers, and programmers insert vulnerabilities in the software.[18]
Threat Agent = Capabilities + Intentions + Past Activities
These individuals and groups can be classified as follows:[18]
Threat sources are those who wish a compromise to occur. It is a term used to distinguish them from threat agents/actors who are those who carry out the attack and who may be commissioned or persuaded by the threat source to knowingly or unknowingly carry out the attack.[19]
Threat actionis an assault on system security.A completesecurity architecturedeals with both intentional acts (i.e. attacks) and accidental events.[20]
Various kinds of threat actions are defined as subentries under "threat consequence".
Threat analysisis the analysis of the probability of occurrences and consequences of damaging actions to a system.[1]It is the basis ofrisk analysis.
Threat modelingis a process that helps organizations identify and prioritize potential threats to their systems. It involves analyzing the system's architecture, identifying potential threats, and prioritizing them based on their impact and likelihood. By using threat modeling, organizations can develop a proactive approach to security and prioritize their resources to address the most significant risks.[21]
Threat intelligenceis the practice of collecting and analyzing information about potential and current threats to an organization. This information can include indicators of compromise, attack techniques, and threat actor profiles. By using threat intelligence, organizations can develop a better understanding of the threat landscape and improve their ability to detect and respond to threats.[22]
Threat consequenceis a security violation that results from a threat action.[1]Includes disclosure, deception, disruption, and usurpation.
The following subentries describe four kinds of threat consequences, and also list and describe the kinds of threat actions that cause each consequence.[1]Threat actions that are accidental events are marked by "*".
A collection of threats in a particular domain or context, with information on identified vulnerable assets, threats, risks, threat actors and observed trends.[23][24]
Threats should be managed by operating an ISMS, performing all theIT risk managementactivities foreseen by laws, standards and methodologies.
Very large organizations tend to adoptbusiness continuity managementplans in order to protect, maintain and recover business-critical processes and systems. Some of these plans are implemented bycomputer security incident response team(CSIRT).
Threat management must identify, evaluate, and categorize threats. There are two primary methods ofthreat assessment:
Many organizations perform only a subset of these methods, adopting countermeasures based on a non-systematic approach, resulting incomputer insecurity.
Informationsecurity awarenessis a significant market. There has been a lot of software developed to deal with IT threats, including bothopen-source softwareandproprietary software.[25]
Threat management involves a wide variety of threats including physical threats like flood and fire. While ISMS risk assessment process does incorporate threat management for cyber threats such as remote buffer overflows the risk assessment process doesn't include processes such as threat intelligence management or response procedures.
Cyber threat management (CTM) is emerging as the best practice for managing cyber threats beyond the basic risk assessment found in ISMS. It enables early identification of threats, data-driven situational awareness, accurate decision-making, and timely threat mitigating actions.[26]
CTM includes:
Cyber threat huntingis "the process of proactively and iteratively searching through networks to detect and isolate advanced threats that evade existing security solutions."[27]This is in contrast to traditional threat management measures, such asfirewalls,intrusion detection systems, andSIEMs, which typically involve an investigationafterthere has been a warning of a potential threat, or an incident has occurred.
Threat hunting can be a manual process, in which a security analyst sifts through various data information using their knowledge and familiarity with the network to create hypotheses about potential threats. To be even more effective and efficient, however, threat hunting can be partially automated, or machine-assisted, as well. In this case, the analyst utilizes software that harnessesmachine learninganduser and entity behaviour analytics(UEBA) to inform the analyst of potential risks. The analyst then investigates these potential risks, tracking suspicious behaviour in the network. Thus hunting is an iterative process, meaning that it must be continuously carried out in a loop, beginning with a hypothesis. There are three types of hypotheses:
The analyst researches their hypothesis by going through vast amounts of data about the network. The results are then stored so that they can be used to improve the automated portion of the detection system and to serve as a foundation for future hypotheses.
TheSANS Institutehas conducted research and surveys on the effectiveness of threat hunting to track and disrupt cyber adversaries as early in their process as possible. According to a survey performed in 2019, "61% [of the respondents] report at least an 11% measurable improvement in their overall security posture" and 23.6% of the respondents have experienced a 'significant improvement' in reducing thedwell time.[29]
To protect yourself from computer threats, it's essential to keep your software up-to-date, use strong and unique passwords, and be cautious when clicking on links or downloading attachments. Additionally, using antivirus software and regularly backing up your data can help mitigate the impact of a threat.
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Thistimeline of computer viruses and wormspresents a chronological timeline of noteworthycomputer viruses,computer worms,Trojan horses, similarmalware, related research and events.
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DNS hijacking,DNS poisoning, orDNS redirectionis the practice of subverting the resolution ofDomain Name System(DNS) queries.[1]This can be achieved by malware that overrides a computer'sTCP/IPconfiguration to point at a rogue DNS server under the control of an attacker, or through modifying the behaviour of a trusted DNS server so that it does not comply withinternet standards.
These modifications may be made for malicious purposes such asphishing, for self-serving purposes byInternet service providers(ISPs), by theGreat Firewall of Chinaand public/router-based onlineDNS server providersto direct users' web traffic to the ISP's ownweb serverswhere advertisements can be served, statistics collected, or other purposes of the ISP; and by DNS service providers to block access to selected domains as a form ofcensorship.
One of the functions of a DNS server is to translate adomain nameinto anIP addressthatapplicationsneed to connect to an Internet resource such as awebsite. This functionality is defined in various formalinternet standardsthat define theprotocolin considerable detail. DNS servers are implicitly trusted by internet-facing computers and users to correctly resolve names to the actual addresses that are registered by the owners of an internet domain.
A rogue DNS server translates domain names of desirable websites (search engines, banks, brokers, etc.) into IP addresses of sites with unintended content, even malicious websites. Most users depend on DNS servers automatically assigned by theirISPs. A router's assigned DNS servers can also be altered through the remote exploitation of a vulnerability within the router's firmware.[2]When users try to visit websites, they are instead sent to a bogus website. This attack is termedpharming. If the site they are redirected to is a malicious website, masquerading as a legitimate website, in order to fraudulently obtain sensitive information, it is calledphishing.[3]
A number of consumer ISPs such asAT&T,[4]Cablevision'sOptimum Online,[5]CenturyLink,[6]Cox Communications,RCN,[7]Rogers,[8]Charter Communications (Spectrum),Plusnet,[9]Verizon,[10]Sprint,[11]T-Mobile US,[12]Virgin Media,[13][14]Frontier Communications,Bell Sympatico,[15]Deutsche Telekom AG,[16]Optus,[17]Mediacom,[18]ONO,[19]TalkTalk,[20]Bigpond(Telstra),[21][22][23][24]TTNET, Türksat, and all Indonesian customer ISPs use or used DNS hijacking for their own purposes, such as displaying advertisements[25]or collecting statistics. Dutch ISPs XS4ALL and Ziggo use DNS hijacking by court order: they were ordered to block access toThe Pirate Bayand display a warning page[26]while all customer ISP in Indonesia do DNS hijacking to comply with the National DNS law[27]which requires every customer Indonesian ISP to hijackport 53and redirect it to their own server to block website that are listed inTrustpositifbyKominfounder Internet Sehat campaign. These practices violate theRFCstandard for DNS (NXDOMAIN) responses,[28]and can potentially open users tocross-site scriptingattacks.[25]
The concern with DNS hijacking involves this hijacking of the NXDOMAIN response. Internet andintranetapplications rely on the NXDOMAIN response to describe the condition where the DNS has no entry for the specified host. If one were to query the invalid domain name (for example www.example.invalid), one should get an NXDOMAIN response – informing the application that the name is invalid and taking the appropriate action (for example, displaying an error or not attempting to connect to the server). However, if the domain name is queried on one of these non-compliant ISPs, one would always receive a fake IP address belonging to the ISP. In aweb browser, this behavior can be annoying or offensive as connections to this IP address display theISP redirect pageof the provider, sometimes with advertising, instead of a proper error message. However, other applications that rely on the NXDOMAIN error will instead attempt to initiate connections to this spoofed IP address, potentially exposing sensitive information.
Examples of functionality that breaks when an ISP hijacks DNS:
In some, but not most cases, the ISPs provide subscriber-configurable settings to disable hijacking of NXDOMAIN responses. Correctly implemented, such a setting reverts DNS to standard behavior. Other ISPs, however, instead use a web browsercookieto store the preference. In this case, the underlying behavior is not resolved: DNS queries continue to be redirected, while the ISP redirect page is replaced with a counterfeit DNS error page. Applications other than web browsers cannot be opted out of the scheme using cookies as the opt-out targets only theHTTPprotocol, when the scheme is actually implemented in the protocol-neutral DNS.
In the UK, the Information Commissioner's Office has acknowledged that the practice of involuntary DNS hijacking contravenesPECR, and EC Directive 95/46 on Data Protection which require explicit consent for processing of communication traffic.[13]In Germany, in 2019 it was revealed that the Deutsche Telekom AG not only manipulated their DNS servers, but also transmitted network traffic (such as non-secure cookies when users did not useHTTPS) to a third-party company because the web portal T-Online, at which users were redirected due to the DNS manipulation, was not (any more) owned by the Deutsche Telekom. After a user filed a criminal complaint, the Deutsche Telekom stopped further DNS manipulations.[32]
ICANN, the international body responsible for administering top-level domain names, has published a memorandum highlighting its concerns, and affirming:[31]
ICANN strongly discourages the use of DNS redirection, wildcards, synthesized responses and any other form of NXDOMAIN substitution in existing gTLDs, ccTLDs and any other level in the DNS tree for registry-class domain names.
End users, dissatisfied with poor "opt-out" options like cookies, have responded to the controversy by finding ways to avoid spoofed NXDOMAIN responses. DNS software such asBINDandDnsmasqoffer options to filter results, and can be run from a gateway or router to protect an entire network. Google, among others, run open DNS servers that currently do not return spoofed results. So a user could useGoogle Public DNSinstead of their ISP's DNS servers if they are willing to accept that they use the service underGoogle's privacy policyand potentially be exposed to another method by which Google can track the user. One limitation of this approach is that some providers block or rewrite outside DNS requests.OpenDNS, owned by Cisco, is a similar popular service which does not alter NXDOMAIN responses.
Google in April 2016 launched DNS-over-HTTPS service.[33]This scheme can overcome the limitations of the legacy DNS protocol. It performs remote DNSSEC check and transfers the results in a secure HTTPS tunnel.
There are also application-level work-arounds, such as the NoRedirect[34]Firefox extension, that mitigate some of the behavior. An approach like that only fixes one application (in this example, Firefox) and will not address any other issues caused. Website owners may be able to fool some hijackers by using certain DNS settings. For example, setting a TXT record of "unused" on their wildcard address (e.g. *.example.com). Alternatively, they can try setting the CNAME of the wildcard to "example.invalid", making use of the fact that ".invalid" is guaranteed not to exist per the RFC. The limitation of that approach is that it only prevents hijacking on those particular domains, but it may address some VPN security issues caused by DNS hijacking.
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IEEE P1363is anInstitute of Electrical and Electronics Engineers(IEEE) standardization project forpublic-key cryptography. It includes specifications for:
The chair of the working group as of October 2008 isWilliam Whyteof NTRU Cryptosystems, Inc., who has served since August 2001. Former chairs wereAri Singer, also of NTRU (1999–2001), andBurt KaliskiofRSA Security(1994–1999).
The IEEE Standard Association withdrew all of the 1363 standards except 1363.3-2013 on 7 November 2019.[1]
This specification includes key agreement, signature, and encryption
schemes using several mathematical approaches:integer factorization,discrete logarithm, andelliptic curve discrete logarithm.
This document includes a number ofpassword-authenticated key agreementschemes, and a password-authenticated key retrieval scheme.
This standard was published on 15 November 2013. It includes techniques for identity-based encryption, signatures, signcryption, key agreement, and proxy re-encryption, all based on bilinear pairings.
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Incryptography,Simultaneous Authentication of Equals(SAE) is apassword-based authentication andpassword-authenticated key agreementmethod.[1]
SAE is a variant of theDragonfly Key Exchangedefined inRFC7664,[2]based onDiffie–Hellman key exchangeusingfinite cyclic groupswhich can be aprimary cyclic groupor anelliptic curve.[1]The problem of using Diffie–Hellman key exchange is that it does not have an authentication mechanism. So the resulting key is influenced by apre-shared keyand theMAC addressesof both peers to solve theauthentication problem.
SAE was originally implemented for use betweenpeersinIEEE 802.11s.[1]When peers discover each other (and security is enabled) they take part in an SAE exchange. If SAE completes successfully, each peer knows the other party possesses the mesh password and, as a by-product of the SAE exchange, the two peers establish a cryptographically strong key. This key is used with the "Authenticated Mesh Peering Exchange" (AMPE) to establish a secure peering and derive a session key to protect mesh traffic, including routing traffic.
In January 2018, theWi-Fi AllianceannouncedWPA3as a replacement toWPA2.[3][4]The new standard uses 128-bit encryption in WPA3-Personal mode (192-bit in WPA3-Enterprise)[5]andforward secrecy.[6]The WPA3 standard also replaces thepre-shared key(PSK) exchange with Simultaneous Authentication of Equals as defined inIEEE 802.11-2016resulting in a more secure initial key exchange in personal mode.[7][8]The Wi-Fi Alliance also claims that WPA3 will mitigate security issues posed by weak passwords and simplify the process of setting up devices with no display interface.[9]
In 2019 Eyal Ronen and Mathy Vanhoef (co-author of theKRACKattack) released an analysis of WPA3's Dragonfly handshake and found that "an attacker within range of a victim can still recover the password" and the bugs found "allow an adversary to impersonate any user, and thereby access the Wi-Fi network, without knowing the user's password."[10][11]
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Intelecommunications, acryptochannelis a completesystemofcrypto-communicationsbetween two or more holders or parties. It includes: (a) the cryptographic aids prescribed; (b) the holders thereof; (c) the indicators or other means of identification; (d) the area or areas in which effective; (e) the special purpose, if any, for which provided; and (f) pertinent notes as to distribution, usage,etc.A cryptochannel is analogous to aradiocircuit.[1][2][3][4]
This article related totelecommunicationsis astub. You can help Wikipedia byexpanding it.
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Incryptography, ahybrid cryptosystemis one which combines the convenience of apublic-key cryptosystemwith the efficiency of asymmetric-key cryptosystem.[1]Public-key cryptosystems are convenient in that they do not require the sender and receiver to share acommon secretin order to communicate securely.[2]However, they often rely on complicated mathematical computations and are thus generally much more inefficient than comparable symmetric-key cryptosystems. In many applications, the high cost of encrypting long messages in a public-key cryptosystem can be prohibitive. This is addressed by hybrid systems by using a combination of both.[3]
A hybrid cryptosystem can be constructed using any two separate cryptosystems:
The hybrid cryptosystem is itself a public-key system, whose public and private keys are the same as in the key encapsulation scheme.[4]
Note that for very long messages the bulk of the work in encryption/decryption is done by the more efficient symmetric-key scheme, while the inefficient public-key scheme is used only to encrypt/decrypt a short key value.[3]
All practical implementations of public key cryptography today employ the use of a hybrid system. Examples include theTLSprotocol[5]and theSSHprotocol,[6]that use a public-key mechanism for key exchange (such asDiffie-Hellman) and a symmetric-key mechanism for data encapsulation (such asAES). TheOpenPGP[7]file format and thePKCS#7[8]file format are other examples.
Hybrid Public Key Encryption (HPKE, published asRFC 9180) is a modern standard for generic hybrid encryption. HPKE is used within multiple IETF protocols, includingMLSand TLS Encrypted Hello.
Envelope encryption is an example of a usage of hybrid cryptosystems incloud computing. In a cloud context, hybrid cryptosystems also enable centralizedkey management.[9][10]
To encrypt a message addressed to Alice in a hybrid cryptosystem, Bob does the following:
To decrypt this hybrid ciphertext, Alice does the following:
If both the key encapsulation and data encapsulation schemes in a hybrid cryptosystem are secure againstadaptive chosen ciphertext attacks, then the hybrid scheme inherits that property as well.[4]However, it is possible to construct a hybrid scheme secure against adaptive chosen ciphertext attacks even if the key encapsulation has a slightly weakened security definition (though the security of the data encapsulation must be slightly stronger).[12]
Envelope encryption is term used for encrypting with a hybrid cryptosystem used by all majorcloud service providers,[9]often as part of a centralizedkey managementsystem in cloud computing.[13]
Envelope encryption gives names to the keys used in hybrid encryption: Data Encryption Keys (abbreviated DEK, and used to encrypt data) and Key Encryption Keys (abbreviated KEK, and used to encrypt the DEKs). In a cloud environment, encryption with envelope encryption involves generating a DEK locally, encrypting one's data using the DEK, and then issuing a request to wrap (encrypt) the DEK with a KEK stored in a potentially more secureservice. Then, this wrapped DEK and encrypted message constitute aciphertextfor the scheme. To decrypt a ciphertext, the wrapped DEK is unwrapped (decrypted) via a call to a service, and then the unwrapped DEK is used to decrypt the encrypted message.[10]In addition to the normal advantages of a hybrid cryptosystem, using asymmetric encryption for the KEK in a cloud context provides easier key management and separation of roles, but can be slower.[13]
In cloud systems, such asGoogle Cloud PlatformandAmazon Web Services, a key management system (KMS) can be available as a service.[13][10][14]In some cases, the key management system will store keys inhardware security modules, which are hardware systems that protect keys with hardware features like intrusion resistance.[15]This means that KEKs can also be more secure because they are stored on secure specialized hardware.[13]Envelope encryption makes centralized key management easier because a centralized key management system only needs to store KEKs, which occupy less space, and requests to the KMS only involve sending wrapped and unwrapped DEKs, which use less bandwidth than transmitting entire messages. Since one KEK can be used to encrypt many DEKs, this also allows for less storage space to be used in the KMS. This also allows for centralized auditing and access control at one point of access.[10]
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Secure communicationis when two entities are communicating and do not want a third party to listen in. For this to be the case, the entities need to communicate in a way that is unsusceptible toeavesdroppingorinterception.[1][2]Secure communication includes means by which people can share information with varying degrees of certainty that third parties cannot intercept what is said. Other than spoken face-to-face communication with no possible eavesdropper, it is probable that no communication is guaranteed to be secure in this sense, although practical obstacles such as legislation, resources, technical issues (interception and encryption), and the sheer volume of communication serve to limitsurveillance.
With many communications taking place over long distance and mediated by technology, and increasing awareness of the importance of interception issues, technology and its compromise are at the heart of this debate. For this reason, this article focuses on communications mediated or intercepted by technology.
Also seeTrusted Computing, an approach under present development that achieves security in general at the potential cost of compelling obligatory trust in corporate and government bodies.
In 1898,Nikola Teslademonstrated aradio controlledboat inMadison Square Gardenthat allowed secure communication betweentransmitterandreceiver.[3]
One of the most famous systems of secure communication was theGreen Hornet. During WWII,Winston Churchillhad to discuss vital matters withFranklin D. Roosevelt. In the beginning, the calls were made using a voice scrambler, as this was thought to be secure. When this was found to be untrue, engineers started to work on a whole new system, which resulted in the Green Hornet orSIGSALY. With the Green Hornet, any unauthorized party listening in would just hearwhite noise, but the conversation would remain clear to authorized parties. As secrecy was paramount, the location of the Green Hornet was only known by the people who built it and Winston Churchill. To maintain secrecy, the Green Hornet was kept in a closet labeled 'Broom Cupboard.'' The Green Hornet used aone-time pad. SIGSALY was also never broken.[4]
Security can be broadly categorized under the following headings, with examples:
Each of the three types of security is important, and depending on the circumstances, any of these may be critical. For example, if a communication is not readily identifiable, then it is unlikely to attract attention for identification of parties, and the mere fact a communication has taken place (regardless of content) is often enough by itself to establish an evidential link in legal prosecutions. It is also important with computers, to be sure where the security is applied, and what is covered.
A further category, which touches upon secure communication, is software intended to take advantage of security openings at the end-points. This software category includestrojan horses,keyloggersand otherspyware.
These types of activity are usually addressed with everyday mainstream security methods, such asantivirussoftware,firewalls, programs that identify or neutralizeadwareandspyware, and web filtering programs such asProxomitronandPrivoxywhich check all web pages being read and identify and remove common nuisances contained. As a rule they fall undercomputer securityrather than secure communications.
Encryptionis a method in which data is rendered hard to read by an unauthorized party. Since encryption methods are created to be extremely hard to break, many communication methods either use deliberately weaker encryption than possible, or havebackdoorsinserted to permit rapid decryption. In some cases government authorities have required backdoors be installed in secret. Many methods of encryption are also subject to "man in the middle" attack whereby a third party who can 'see' the establishment of the secure communication is made privy to the encryption method, this would apply for example to the interception of computer use at an ISP. Provided it is correctly programmed, sufficiently powerful, and the keys not intercepted, encryption would usually be considered secure. The article onkey sizeexamines the key requirements for certain degrees of encryption security.
Encryption can be implemented in a way that requires the use of encryption, i.e. if encrypted communication is impossible then no traffic is sent, or opportunistically.Opportunistic encryptionis a lower security method to generally increase the percentage of generic traffic which is encrypted. This is analogous to beginning every conversation with "Do you speakNavajo?" If the response is affirmative, then the conversation proceeds in Navajo, otherwise it uses the common language of the two speakers. This method does not generally provideauthenticationoranonymitybut it does protect the content of the conversation fromeavesdropping.
AnInformation-theoretic securitytechnique known asphysical layer encryptionensures that a wireless communication link is provably secure with communications and coding techniques.
Steganography("hidden writing") is the means by which data can be hidden within other more innocuous data. Thus a watermark proving ownership embedded in the data of a picture, in such a way it is hard to find or remove unless you know how to find it. Or, for communication, the hiding of important data (such as a telephone number) in apparently innocuous data (an MP3 music file). An advantage of steganography isplausible deniability, that is, unless one can prove the data is there (which is usually not easy), it is deniable that the file contains any.
Unwanted or malicious activities are possible on the web since the internet is effectively anonymous. True identity-based networks replace the ability to remain anonymous and are inherently more trustworthy since the identity of the sender and recipient are known. (The telephone system is an example of an identity-based network.)
Recently, anonymous networking has been used to secure communications. In principle, a large number of users running the same system, can have communications routed between them in such a way that it is very difficult to detect what the complete message is, which user sent it, and where it is ultimately coming from or going to. Examples areCrowds,Tor,I2P,Mixminion, variousanonymous P2Pnetworks, and others.
Typically, an unknown device would not be noticed, since so many other devices are in use. This is not assured in reality, due to the presence of systems such asCarnivoreandunzak, which can monitor communications over entire networks, and the fact that the far end may be monitored as before. Examples includepayphones,Internet cafe, etc.
The placing covertly of monitoring and/or transmission devices either within the communication device, or in the premises concerned.
Any security obtained from a computer is limited by the many ways it can be compromised – by hacking,keystroke logging,backdoors, or even in extreme cases by monitoring the tiny electrical signals given off by keyboard or monitors to reconstruct what is typed or seen (TEMPEST, which is complex).
Sounds, including speech, inside rooms can be sensed by bouncing alaserbeam off a window of the room where a conversation is held, and detecting and decoding the vibrations in the glass caused by thesound waves.[5]
Cellphones can easily be obtained, but are also easily traced and "tapped". There is no (or only limited) encryption, the phones are traceable – often even when switched off[citation needed]– since the phone and SIM card broadcast their International Mobile Subscriber Identity (IMSI). It is possible for a cellphone company to turn on some cellphones when the user is unaware and use the microphone to listen in on you, and according to James Atkinson, acounter-surveillancespecialist cited in the same source, "Security-conscious corporate executives routinely remove the batteries from their cell phones" since many phones' software can be used "as-is", or modified, to enable transmission without user awareness and the user can be located within a small distance using signaltriangulationand now using built in GPS features for newer models. Transceivers may also be defeated byjammingorFaraday cage.
Some cellphones (Apple'siPhone,Google'sAndroid) track and store users' position information, so that movements for months or years can be determined by examining the phone.[6]
The U.S. Government also has access to cellphone surveillance technologies, mostly applied for law enforcement.[7]
Analogue landlines are not encrypted, it lends itself to being easily tapped. Such tapping requires physical access to the line which can be easily obtained from a number of places, e.g. the phone location, distribution points, cabinets and the exchange itself. Tapping a landline in this way can enable an attacker to make calls which appear to originate from the tapped line.
Using athird partysystem of any kind (payphone, Internet cafe) is often secure, however if that system is used to access known locations (a known email account or 3rd party) then it may be tapped at the far end, or noted, and this will remove any security benefit obtained. Some countries also impose mandatory registration of Internet cafe users.
Anonymous proxiesare another common type of protection, which allow one to access the net via a third party (often in a different country) and make tracing difficult. Note that there is seldom any guarantee that theplaintextis not tappable, nor that the proxy does not keep its own records of users or entire dialogs. As a result, anonymous proxies are a generally useful tool but may not be as secure as other systems whose security can be better assured. Their most common use is to prevent a record of the originatingIP, or address, being left on the target site's own records. Typical anonymous proxies are found at both regular websites such as Anonymizer.com and spynot.com, and on proxy sites which maintain up to date lists of large numbers of temporary proxies in operation.
A recent development on this theme arises when wireless Internet connections ("Wi-Fi") are left in their unsecured state. The effect of this is that any person in range of the base unit canpiggybackthe connection – that is, use it without the owner being aware. Since many connections are left open in this manner, situations where piggybacking might arise (willful or unaware) have successfully led to a defense in some cases, since it makes it difficult to prove the owner of the connection was the downloader, or had knowledge of the use to which unknown others might be putting their connection. An example of this was the Tammie Marson case, where neighbours and anyone else might have been the culprit in the sharing of copyright files.[8]Conversely, in other cases, people deliberately seek out businesses and households with unsecured connections, for illicit and anonymous Internet usage, or simply to obtain freebandwidth.[9]
Several secure communications networks, which were predominantly used by criminals, have been shut down by law enforcement agencies, including:EncroChat,Sky Global / Sky ECC, andPhantom Secure.
In September 2024 Eurojust, Europol, and law enforcement agencies from a number of countries took down a secure communication service used for organized crime. The encryption network was operated by equipment and personnel in Sweden, Ireland, the Netherlands, France, Spain, Italy, Australia, and Canada.[10]
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In theInternet, adomain nameis astringthat identifies a realm of administrative autonomy, authority or control. Domain names are often used to identify services provided through the Internet, such aswebsites,emailservices and more. Domain names are used in various networking contexts and for application-specific naming and addressing purposes. In general, a domain name identifies anetwork domainor anInternet Protocol(IP) resource, such as a personal computer used to access the Internet, or a server computer.
Domain names are formed by the rules and procedures of theDomain Name System(DNS). Any name registered in the DNS is a domain name. Domain names are organized in subordinate levels (subdomains) of theDNS rootdomain, which is nameless. The first-level set of domain names are thetop-level domains(TLDs), including thegeneric top-level domains(gTLDs), such as the prominent domainscom,info,net,edu, andorg, and thecountry code top-level domains(ccTLDs). Below these top-level domains in the DNS hierarchy are the second-level and third-level domain names that are typically open for reservation by end-users who wish to connect local area networks to the Internet, create other publicly accessible Internet resources or run websites, such as "wikipedia.org".
The registration of a second- or third-level domain name is usually administered by adomain name registrarwho sell its services to the public.
Afully qualified domain name(FQDN) is a domain name that is completely specified with all labels in the hierarchy of the DNS, having no parts omitted. Traditionally a FQDN ends in a dot (.) to denote the top of the DNS tree.[1]Labels in the Domain Name System arecase-insensitive, and may therefore be written in any desired capitalization method, but most commonly domain names are written in lowercase in technical contexts.[2]Ahostnameis a domain name that has at least one associatedIP address.
Domain names serve to identify Internet resources, such as computers, networks, and services, with a text-based label that is easier to memorize than the numerical addresses used in the Internet protocols. A domain name may represent entire collections of such resources or individual instances. Individual Internet host computers use domain names as host identifiers, also calledhostnames. The termhostnameis also used for the leaf labels in the domain name system, usually without further subordinate domain name space. Hostnames appear as a component inUniform Resource Locators(URLs) for Internet resources such aswebsites(e.g., en.wikipedia.org).
Domain names are also used as simple identification labels to indicate ownership or control of a resource. Such examples are the realm identifiers used in theSession Initiation Protocol(SIP), theDomain Keysused to verify DNS domains ine-mailsystems, and in many otherUniform Resource Identifiers(URIs).
An important function of domain names is to provide easily recognizable and memorizable names to numericallyaddressedInternet resources. This abstraction allows any resource to be moved to a different physical location in the address topology of the network, globally or locally in anintranet. Such a move usually requires changing the IP address of a resource and the corresponding translation of this IP address to and from its domain name.
Domain names are used to establish a unique identity. Organizations can choose a domain name that corresponds to their name, helping Internet users to reach them easily.
A generic domain is a name that defines a general category, rather than a specific or personal instance, for example, the name of an industry, rather than a company name. Some examples of generic names arebooks.com,music.com, andtravel.info. Companies have created brands based on generic names, and such generic domain names may be valuable.[3]
Domain names are often simply referred to asdomainsand domain name registrants are frequently referred to asdomain owners, although domain name registration with a registrar does not confer any legal ownership of the domain name, only an exclusive right of use for a particular duration of time. The use of domain names in commerce may subject them totrademark law.
The practice of using a simple memorable abstraction of a host's numerical address on a computer network dates back to theARPANETera, before the advent of today's commercial Internet. In the early network, each computer on the network retrieved the hosts file (host.txt) from a computer at SRI (nowSRI International),[4][5]which mapped computer hostnames to numerical addresses. The rapid growth of the network made it impossible to maintain a centrally organized hostname registry and in 1983 the Domain Name System was introduced on the ARPANET and published by theInternet Engineering Task Forceas RFC 882 and RFC 883.
The following table shows the first five.comdomains with the dates of their registration:[6]
and the first five.edudomains:[7]
Today, theInternet Corporation for Assigned Names and Numbers(ICANN) manages the top-level development and architecture of the Internet domain name space. It authorizesdomain name registrars, through which domain names may be registered and reassigned.
The domain name space consists of atreeof domain names. Each node in the tree holds information associated with the domain name. The tree sub-divides intozonesbeginning at theDNS root zone.
A domain name consists of one or more parts, technically calledlabels, that are conventionally concatenated, and delimited by dots, such asexample.com.
When the Domain Name System was devised in the 1980s, the domain name space was divided into two main groups of domains.[9]Thecountry code top-level domains(ccTLD) were primarily based on the two-character territory codes ofISO-3166country abbreviations. In addition, a group of sevengeneric top-level domains(gTLD) was implemented which represented a set of categories of names and multi-organizations.[10]These were the domainsgov,edu,com,mil,org,net, andint. These two types oftop-level domains(TLDs) are the highest level of domain names of the Internet. Top-level domains form theDNS root zoneof the hierarchicalDomain Name System. Every domain name ends with a top-level domain label.
During the growth of the Internet, it became desirable to create additional generic top-level domains. As of October 2009, 21 generic top-level domains and 250 two-letter country-code top-level domains existed.[11]In addition, theARPAdomain serves technical purposes in the infrastructure of the Domain Name System.
During the 32nd International Public ICANN Meeting in Paris in 2008,[12]ICANN started a new process of TLD naming policy to take a "significant step forward on the introduction of new generic top-level domains." This program envisions the availability of many new or already proposed domains, as well as a new application and implementation process.[13]Observers believed that the new rules could result in hundreds of new top-level domains to be registered.[14]In 2012, the program commenced, and received 1930 applications.[15]By 2016, the milestone of 1000 live gTLD was reached.
TheInternet Assigned Numbers Authority(IANA) maintains an annotated list of top-level domains in theDNS root zonedatabase.[16]
For special purposes, such as network testing, documentation, and other applications, IANA also reserves a set of special-use domain names.[17]This list contains domain names such asexample,local,localhost, andtest. Other top-level domain names containing trade marks are registered for corporate use. Cases include brands such asBMW,Google, andCanon.[18]
Below the top-level domains in the domain name hierarchy are thesecond-level domain(SLD) names. These are the names directly to the left of .com, .net, and the other top-level domains. As an example, in the domainexample.co.uk,cois the second-level domain.
Next are third-level domains, which are written immediately to the left of a second-level domain. There can be fourth- and fifth-level domains, and so on, with virtually no limitation. Each label is separated by afull stop(dot). An example of an operational domain name with four levels of domain labels issos.state.oh.us. 'sos' is said to be a sub-domain of 'state.oh.us', and 'state' a sub-domain of 'oh.us', etc. In general,subdomainsare domains subordinate to their parent domain. An example of very deep levels of subdomain ordering are theIPv6reverse resolutionDNS zones, e.g., 1.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.ip6.arpa, which is the reverse DNS resolution domain name for the IP address of aloopbackinterface, or thelocalhostname.
Second-level (or lower-level, depending on the established parent hierarchy) domain names are often created based on the name of a company (e.g.,bbc.co.uk), product or service (e.g.hotmail.com). Below these levels, the next domain name component has been used to designate a particular host server. Therefore,ftp.example.commight be an FTP server,www.example.comwould be aWorld Wide Webserver, andmail.example.comcould be an email server, each intended to perform only the implied function. Modern technology allows multiple physical servers with either different (cf.load balancing) or even identical addresses (cf.anycast) to serve a single hostname or domain name, or multiple domain names to be served by a single computer. The latter is very popular inWeb hosting servicecenters, where service providers host the websites of many organizations on just a few servers.
The hierarchicalDNS labelsor components of domain names are separated in a fully qualified name by thefull stop(dot,.).
The character set allowed in the Domain Name System is based onASCIIand does not allow the representation of names and words of many languages in their native scripts or alphabets.ICANNapproved theInternationalized domain name(IDNA) system, which mapsUnicodestrings used in application user interfaces into the valid DNS character set by an encoding calledPunycode. For example, københavn.eu is mapped to xn--kbenhavn-54a.eu. Manyregistrieshave adopted IDNA.
The first commercial Internet domain name, in the TLDcom, was registered on 15 March 1985 in the namesymbolics.comby Symbolics Inc., a computer systems firm in Massachusetts.[19][20]
By 1992, fewer than 15,000comdomains had been registered.
In the first quarter of 2015, 294 million domain names had been registered.[21]A large fraction of them are in thecomTLD, which as of December 21, 2014, had 115.6 million domain names,[22]including 11.9 million online business and e-commerce sites, 4.3 million entertainment sites, 3.1 million finance related sites, and 1.8 million sports sites.[23]As of July 15, 2012, thecomTLD had more registrations than all of the ccTLDs combined.[24]
As of December 31, 2023,[update]359.8 million domain names had been registered.[25]
The right to use a domain name is delegated bydomain name registrars, which are accredited by theInternet Corporation for Assigned Names and Numbers(ICANN), the organization charged with overseeing the name and number systems of the Internet. In addition to ICANN, each top-level domain (TLD) is maintained and serviced technically by an administrative organization operating a registry. A registry is responsible for maintaining the database of names registered within the TLD it administers. The registry receives registration information from each domain name registrar authorized to assign names in the corresponding TLD and publishes the information using a special service, theWHOISprotocol.
Registries and registrars usually charge an annual fee for the service of delegating a domain name to a user and providing a default set of name servers. Often, this transaction is termed a sale or lease of the domain name, and the registrant may sometimes be called an "owner", but no such legal relationship is actually associated with the transaction, only the exclusive right to use the domain name. More correctly, authorized users are known as "registrants" or as "domain holders".
ICANN publishes the complete list of TLD registries and domain name registrars. Registrant information associated with domain names is maintained in an online database accessible with the WHOIS protocol. For most of the 250country code top-level domains(ccTLDs), the domain registries maintain the WHOIS (Registrant, name servers, expiration dates, etc.) information.
Some domain name registries, often callednetwork information centers(NIC), also function as registrars to end-users. The major generic top-level domain registries, such as for thecom,net,org,infodomains and others, use a registry-registrar model consisting of hundreds of domain name registrars (see lists at ICANN[26]or VeriSign).[27]In this method of management, the registry only manages the domain name database and the relationship with the registrars. Theregistrants(users of a domain name) are customers of the registrar, in some cases through additional layers of resellers.
There are also a few otheralternative DNS rootproviders that try to compete or complement ICANN's role of domain name administration, however, most of them failed to receive wide recognition, and thus domain names offered by those alternative roots cannot be used universally on most other internet-connecting machines without additional dedicated configurations.
In the process of registering a domain name and maintaining authority over the new name space created, registrars use several key pieces of information connected with a domain:
A domain name consists of one or more labels, each of which is formed from the set of ASCII letters, digits, and hyphens (a–z, A–Z, 0–9, -), but not starting or ending with a hyphen. The labels are case-insensitive; for example, 'label' is equivalent to 'Label' or 'LABEL'. In the textual representation of a domain name, the labels are separated by afull stop(period).
Domain names are often seen in analogy toreal estatein that domain names are foundations on which a website can be built, and the highestqualitydomain names, like sought-after real estate, tend to carry significant value, usually due to their online brand-building potential, use in advertising,search engine optimization, and many other criteria.
A few companies have offered low-cost, below-cost, or even free domain registration with a variety of models adopted to recoup the costs to the provider. These usually require that domains be hosted on their website within a framework or portal that includes advertising wrapped around the domain holder's content, revenue from which allows the provider to recoup the costs. Domain registrations were free of charge when the DNS was new. A domain holder may provide an infinite number ofsubdomainsin their domain. For example, the owner of example.org could provide subdomains such as foo.example.org and foo.bar.example.org to interested parties.
Many desirable domain names are already assigned and users must search for other acceptable names, using Web-based search features, orWHOISanddigoperating system tools. Many registrars have implementeddomain name suggestiontools which search domain name databases and suggest available alternative domain names related to keywords provided by the user.
The business of resale of registered domain names is known as thedomain aftermarket. Various factors influence the perceived value or market value of a domain name. Most of the high-prize domain sales are carried out privately.[28]Also, it is called confidential domain acquiring or anonymous domain acquiring.[29]
Intercappingis often used to emphasize the meaning of a domain name, because DNS names are not case-sensitive. Some names may be misinterpreted in certain uses of capitalization. For example:Who Represents, a database of artists and agents, chosewhorepresents.com,[30]which can be misread. In such situations, the proper meaning may be clarified by placement of hyphens when registering a domain name. For instance,Experts Exchange, a programmers' discussion site, usedexpertsexchange.com, but changed its domain name toexperts-exchange.com.[31]
The domain name is a component of auniform resource locator(URL) used to accesswebsites, for example:
A domain name may point to multipleIP addressesto provide server redundancy for the services offered, a feature that is used to manage the traffic of large, popular websites.
Web hosting services, on the other hand, run servers that are typically assigned only one or a few addresses while serving websites for many domains, a technique referred to asvirtual web hosting. Such IP address overloading requires that each request identifies the domain name being referenced, for instance by using theHTTP request header fieldHost:, orServer Name Indication.
Critics often claim abuse of administrative power over domain names. Particularly noteworthy was the VeriSignSite Findersystem which redirected all unregistered .com and .net domains to a VeriSign webpage. For example, at a public meeting withVeriSignto air technical concerns aboutSite Finder,[32]numerous people, active in theIETFand other technical bodies, explained how they were surprised by VeriSign's changing the fundamental behavior of a major component of Internet infrastructure, not having obtained the customary consensus. Site Finder, at first, assumed every Internet query was for a website, and it monetized queries for incorrect domain names, taking the user to VeriSign's search site. Other applications, such as many implementations of email, treat a lack of response to a domain name query as an indication that the domain does not exist, and that the message can be treated as undeliverable. The original VeriSign implementation broke this assumption for mail, because it would always resolve an erroneous domain name to that of Site Finder. While VeriSign later changed Site Finder's behaviour with regard to email, there was still widespread protest about VeriSign's action being more in its financial interest than in the interest of the Internet infrastructure component for which VeriSign was the steward.
Despite widespread criticism, VeriSign only reluctantly removed it after theInternet Corporation for Assigned Names and Numbers(ICANN) threatened to revoke its contract to administer the root name servers. ICANN published the extensive set of letters exchanged, committee reports, and ICANN decisions.[33]
There is also significant disquiet regarding the United States Government's political influence over ICANN. This was a significant issue in the attempt to create a.xxxtop-level domainand sparked greater interest inalternative DNS rootsthat would be beyond the control of any single country.[34]
Additionally, there are numerous accusations ofdomain name front running, whereby registrars, when given whois queries, automatically register the domain name for themselves. Network Solutions has been accused of this.[35]
In the United States, theTruth in Domain Names Actof 2003, in combination with thePROTECT Act of 2003, forbids the use of a misleading domain name with the intention of attracting Internet users into visitingInternet pornographysites.
The Truth in Domain Names Act follows the more generalAnticybersquatting Consumer Protection Actpassed in 1999 aimed at preventingtyposquattingand deceptive use of names and trademarks in domain names.
In the early 21st century, the US Department of Justice (DOJ) pursued theseizureof domain names, based on the legal theory that domain names constitute property used to engage in criminal activity, and thus are subject toforfeiture. For example, in the seizure of the domain name of a gambling website, the DOJ referenced18 U.S.C.§ 981and18 U.S.C.§ 1955(d).[36][1]In 2013 the US government seizedLiberty Reserve, citing18 U.S.C.§ 982(a)(1).[37]
The U.S. Congress passed theCombating Online Infringement and Counterfeits Actin 2010. Consumer Electronics Association vice president Michael Petricone was worried that seizure was ablunt instrumentthat could harm legitimate businesses.[38][39]After a joint operation on February 15, 2011, the DOJ and the Department of Homeland Security claimed to have seized ten domains of websites involved in advertising and distributing child pornography, but also mistakenly seized the domain name of a large DNS provider, temporarily replacing 84,000 websites with seizure notices.[40]
In theUnited Kingdom, thePolice Intellectual Property Crime Unit(PIPCU) has been attempting to seize domain names from registrars without court orders.[41]
PIPCU and other UK law enforcement organisations make domain suspension requests toNominetwhich they process on the basis of breach of terms and conditions. Around 16,000 domains are suspended annually, and about 80% of the requests originate from PIPCU.[42]
Because of the economic value it represents, theEuropean Court of Human Rightshas ruled that the exclusive right to a domain name is protected as property under article 1 of Protocol 1 to theEuropean Convention on Human Rights.[43]
ICANNBusiness Constituency (BC) has spent decades trying to make IDN variants work at the second level, and in the last several years at the top level. Domain name variants are domain names recognized in different character encodings, like a single domain presented intraditional Chineseandsimplified Chinese. It is anInternationalization and localizationproblem. Under Domain Name Variants, the different encodings of the domain name (in simplified and traditional Chinese) would resolve to the same host.[44][45]
According toJohn Levine, an expert on Internet related topics, "Unfortunately, variants don't work. The problem isn't putting them in the DNS, it's that once they're in the DNS, they don't work anywhere else."[44]
Afictitious domain nameis a domain name used in a work of fiction or popular culture to refer to a domain that does not actually exist, often with invalid or unofficialtop-level domainssuch as ".web", a usage exactly analogous to the dummy555 telephone number prefixused in film and other media. The canonical fictitious domain name is "example.com", specifically set aside by IANA in RFC 2606 for such use, along with the.exampleTLD.
Domain names used in works of fiction have often been registered in the DNS, either by their creators or bycybersquattersattempting to profit from it. This phenomenon promptedNBCto purchase the domain nameHornymanatee.comafter talk-show hostConan O'Brienspoke the name while ad-libbing onhis show. O'Brien subsequently created a website based on the concept and used it as arunning gagon the show.[46]Companies whose works have used fictitious domain names have also employed firms such asMarkMonitorto park fictional domain names in order to prevent misuse by third parties.[47]
Misspelled domain names, also known astyposquattingorURL hijacking, are domain names that are intentionally or unintentionally misspelled versions of popular or well-known domain names. The goal of misspelled domain names is to capitalize on internet users who accidentally type in a misspelled domain name, and are then redirected to a different website.
Misspelled domain names are often used for malicious purposes, such asphishingscams or distributingmalware. In some cases, the owners of misspelled domain names may also attempt to sell the domain names to the owners of the legitimate domain names, or to individuals or organizations who are interested in capitalizing on the traffic generated by internet users who accidentally type in the misspelled domain names.
To avoid being caught by a misspelled domain name, internet users should be careful to type in domain names correctly, and should avoid clicking on links that appear suspicious or unfamiliar. Additionally, individuals and organizations who own popular or well-known domain names should consider registering common misspellings of their domain names in order to prevent others from using them for malicious purposes.
The termDomain name spoofing(or simply though less accurately,Domain spoofing) is used generically to describe one or more of a class ofphishingattacks that depend on falsifying or misrepresenting an internet domain name.[48][49]These are designed to persuade unsuspecting users into visiting a web site other than that intended, or opening an email that is not in reality from the address shown (or apparently shown).[50]Although website and email spoofing attacks are more widely known, any service that relies ondomain name resolutionmay be compromised.
There are a number of better-known types of domain spoofing:
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https://en.wikipedia.org/wiki/Domain_name#Domain_name_spoofing
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Email spoofingis the creation ofemailmessages with aforgedsender address.[1]The term applies to email purporting to be from an address which is not actually the sender's; mail sent in reply to that address may bounce or be delivered to an unrelated party whose identity has been faked.Disposable email addressor "masked" email is a different topic, providing a masked email address that is not the user's normal address, which is not disclosed (for example, so that it cannot beharvested), but forwards mail sent to it to the user's real address.[2]
The originaltransmission protocolsused for email do not have built-in authentication methods: this deficiency allowsspamandphishingemails to use spoofing in order to mislead the recipient. More recentcountermeasureshave made such spoofing from internet sources more difficult but they have not eliminated it completely; few internal networks have defences against a spoof email from a colleague'scompromised computeron that network. Individuals and businesses deceived by spoof emails may suffer significant financial losses; in particular, spoofed emails are often used to infect computers withransomware.
When aSimple Mail Transfer Protocol (SMTP)email is sent, the initial connection provides two pieces of address information:
Together, these are sometimes referred to as the "envelope" addressing – an analogy to a traditionalpaper envelope.[3]Unless the receiving mail server signals that it has problems with either of these items, the sending system sends the "DATA" command, and typically sends several header items, including:
The result is that the email recipient sees the email as having come from the address in theFrom:header. They may sometimes be able to find theMAIL FROMaddress, and if they reply to the email, it will go to either the address presented in theFrom:orReply-to:header, but none of these addresses are typically reliable,[4]so automatedbounce messagesmay generatebackscatter.
Although email spoofing is effective in forging the email address, theIP addressof the computer sending the mail can generally be identified from the "Received:" lines in the email header.[5]In malicious cases, however, this is likely to be the computer of an innocent third party infected bymalwarethat is sending the email without the owner's knowledge.
Phishingand business email compromisescamsgenerally involve an element of email spoofing.
Email spoofing has been responsible for public incidents with serious business and financial consequences. This was the case in an October 2013 email to a news agency which was spoofed to look as if it was from the Swedish companyFingerprint Cards. The email stated thatSamsungoffered to purchase the company. The news spread and the company's stock price surged by 50%.[6]
Malware such asKlezandSoberamong many more modern examples often search for email addresses within the computer they have infected, and they use those addresses both as targets for email, and also to create credible forgedFromfields in the emails that they send.[citation needed]This is to ensure that the emails are more likely to be opened. For example:
In this case, even if Bob's system detects the incoming mail as containing malware, he sees the source as being Charlie, even though it really came from Alice's computer. Meanwhile, Alice may remain unaware that her computer has been infected, and Charlie does not know anything about it at all, unless he receives an error message from Bob.
Traditionally, mail servers could accept a mail item, then later send aNon-Delivery Report or "bounce" messageif it could not be delivered or had been quarantined for any reason. These would be sent to the "MAIL FROM:"a.k.a."Return Path" address. With the massive rise in forged addresses, best practice is now tonotgenerate NDRs for detected spam, viruses etc.[7]but to reject the email during the SMTP transaction. When mail administrators fail to take this approach, their systems are guilty of sending "backscatter" emails to innocent parties – in itself a form of spam – or being used to perform "Joe job" attacks.
TheSSL/TLSsystem used to encrypt server-to-server email traffic can also be used to enforce authentication, but in practice it is seldom used,[8]and a range ofother potential solutionshave also failed to gain traction.
A number of defensive systems have come into wide use, including:
To effectively stop forged email being delivered, the sending domains, their mail servers, and the receiving system all need to be configured correctly for these higher standards of authentication. Although their use is increasing, estimates vary widely as to what percentage of emails have no form of domain authentication: from 8.6%[10]to "almost half".[11][12][13]For this reason, receiving mail systems typically have a range of settings to configure how they treat poorly-configured domains or email.[14][15]
While there has been research into improving email security, little emphasis has been placed on informing users whose email addresses have been used for spoofing. Currently, only the email recipient can identify a fake email, and users whose addresses are spoofed remain unaware unless the recipient manually scrutinizes the message.[citation needed]
Business email compromise attacksare a class ofcyber crimewhich useemail fraudto attack organizations. Examples include invoice scams andspear-phishingattacks which are designed to gather data for other criminal activities. A business deceived by an email spoof can suffer additional financial,business continuityand reputational damage. Fake emails can also be used to spreadmalware.
Typically, an attack targets specific employee roles within an organization by sending spoof emails which fraudulently represent a senior colleague, trusted customer, or supplier.[16](This type of attack is known asspear phishing). The email will issue instructions, such as approving payments or releasing client data. The emails often usesocial engineeringto trick the victim into making money transfers to the bank account of the fraudster.[17]
The United States'Federal Bureau of Investigationrecorded $26billion of US and international losses associated with BEC attacks between June 2016 and July 2019.[18]More recent figures estimate losses of over $50billion from 2013 to 2022.[19]
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https://en.wikipedia.org/wiki/Email_spoofing
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Website spoofingis the act of creating awebsitewith the intention of misleading readers that the website has been created by a different person or organization.
Normally, the spoof website will adopt the design of the target website, and it sometimes has a similarURL.[1]A more sophisticated attack results in an attacker creating a "shadow copy" of theWorld Wide Webby having all of the victim's traffic go through the attacker's machine, causing the attacker to obtain the victim's sensitive information.[2]
Another technique is to use a 'cloaked' URL.[3]By usingdomain forwarding, or insertingcontrol characters, the URL can appear to be genuine while concealing the actual address of the malicious website.Punycodecan also be used for this purpose. Punycode-based attacks exploit the similar characters in different writing systems in common fonts. For example, on one large font, the greek letter tau (τ) is similar in appearance to the Latin lowercase letter t. However, the greek letter tau is represented in punycode as 5xa, while the Latin lowercase letter is simply represented as t, since it is present on the ASCII system. In 2017, a security researcher managed to register the domain xn--80ak6aa92e.com and have it show on several mainstream browsers as apple.com. While the characters used didn't belong to the latin script, due to the default font on those browsers, the end result was non-latin characters that were indistinguishable from those on the latin script.[4][5]
The objective may be fraudulent, often associated withphishing,e-mail spoofingor to lure potential victims to scams such as aget-rich-quick scheme, like in the case offakenews articles with sensational titles purporting of incidents involving popular celebrities with a forged interview discussing about and leading victims to acryptocurrency scam.[6][7]Because the purpose is often malicious, "spoof" (an expression whose base meaning is innocent parody) is a poor term for this activity so that more accountable organisations such as government departments and banks tend to avoid it, preferring more explicit descriptors such as "fraud", "counterfeit" or "phishing".[8][9]
A relatively more benign use of website spoofing is to criticize or make fun of the person or body whose website the spoofed site purports to represent. As an example of the use of this technique toparodyan organisation, in November 2006 two spoof websites, www.msfirefox.com and www.msfirefox.net, were produced claiming thatMicrosofthad boughtFirefoxand released "Microsoft Firefox 2007."[10]A similar incident occurred in 2023 when the culture jamming collectiveBarbie Liberation Organizationcreated a satirical parody page closely resembling theMattelcorporate website using the URL mattel-corporate.com[11]where they announced a fictitious line ofBarbiedolls called "MyCelia EcoWarrior" alongside a series of hoax videos with actressDaryl Hannahposing as a spokesperson for Mattel to lend further legitimacy to the nonexistent dolls, leveraging the publicity surrounding the2023 live-action film.[12]The website's heavy resemblance to the legitimate Mattel corporate site led to a number of news outletsmistakenly reporting it as real, to which they eventually issued a correction and removed the articles in question.[13][12]
Spoofed websites predominate in efforts developinganti-phishing softwarethough there are concerns about their effectiveness. A majority of efforts are focused on the PC market leaving mobile devices lacking.[14]
DNS is the layer at whichbotnetscontrol drones. In 2006,OpenDNSbegan offering a free service to prevent users from entering website spoofing sites. Essentially, OpenDNS has gathered a large database from various anti-phishing and anti-botnet organizations as well as its own data to compile a list of known website spoofing offenders. When a user attempts to access one of these bad websites, they are blocked at theDNSlevel.APWGstatistics show that most phishing attacks use URLs, not domain names, so there would be a large amount of website spoofing that OpenDNS would be unable to track. At the time of release, OpenDNS is unable to prevent unnamed phishing exploits that sit on Yahoo, Google etc.[15]
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https://en.wikipedia.org/wiki/Website_spoofing
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ALAND(local area network denial) is adenial-of-service attackthat consists of sending a special poisonspoofedpacketto a computer, causing it to lock up. The security flaw was first discovered in 1997 by someone using the alias and has resurfaced many years later inoperating systemssuch asWindows Server 2003andWindows XPSP2.
The attack involves sending a spoofedTCPSYNpacket (connection initiation) with the target host'sIP addressto an open port as both source and destination. This causes the machine to reply to itself continuously. It is, however, distinct from theTCP SYN Flood vulnerability.
Other LAND attacks have since been found in services likeSNMPand Windows 88/tcp (kerberos/global services). Such systems had design flaws that would allow the device to accept request on the wire appearing to be from themselves, causing repeated replies.
Below is a list of vulnerable operating systems:[1]
Mostfirewallsshould intercept and discard the poison packet thus protecting the host from this attack. Some operating systems released updates fixing this security hole.
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https://en.wikipedia.org/wiki/LAND
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Modulusis thediminutivefrom the Latin wordmodusmeaning measure or manner. It, or its pluralmoduli, may refer to the following:
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https://en.wikipedia.org/wiki/Modulus_(disambiguation)
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Module,modularandmodularitymay refer to the concept ofmodularity. They may also refer to:
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https://en.wikipedia.org/wiki/Module_(disambiguation)
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