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Incomputing, awireless intrusion prevention system(WIPS) is anetworkdevice that monitors theradio spectrumfor the presence of unauthorizedaccess points(intrusion detection), and can automatically take countermeasures(intrusion prevention).
The primary purpose of a WIPS is to prevent unauthorized network access tolocal area networksand other information assets by wireless devices. These systems are typically implemented as an overlay to an existingWireless LANinfrastructure, although they may be deployed standalone to enforce no-wireless policies within an organization. Some advanced wireless infrastructure has integrated WIPS capabilities.
Large organizations with manyemployeesare particularly vulnerable to security breaches[1]caused byrogue access points. If an employee (trusted entity) in a location brings in an easily availablewireless router, the entire network can be exposed to anyone within range of the signals.
In July 2009, thePCI Security Standards Councilpublished wireless guidelines[2]forPCI DSSrecommending the use of WIPS to automate wireless scanning for large organizations.
Awirelessintrusion detectionsystem(WIDS) monitors the radio spectrum for the presence of unauthorized, rogue access points and the use of wireless attack tools. The system monitors the radio spectrum used by wireless LANs, and immediately alerts asystems administratorwhenever a rogue access point is detected. Conventionally it is achieved by comparing theMAC addressof the participating wireless devices.
Rogue devices canspoofMAC address of an authorized network device as their own. New research usesfingerprintingapproach to weed out devices with spoofed MAC addresses. The idea is to compare the unique signatures exhibited by the signals emitted by each wireless device against the known signatures of pre-authorized, known wireless devices.[3]
In addition to intrusion detection, a WIPS also includes features that prevent against the threatautomatically. For automatic
prevention, it is required that the WIPS is able to accurately detect and automatically classify a threat.
The following types of threats can be prevented by a good WIPS:
WIPS configurations consist of three components:
A simple intrusion detection system can be a single computer, connected to a wireless signal processing device, andantennasplaced throughout the facility. For huge organizations, a Multi Network Controller provides central control of multiple WIPS servers, while forSOHOor SMB customers, all the functionality of WIPS is available in single box.
In a WIPS implementation, users first define the operating wireless policies in the WIPS. The WIPS sensors then analyze the traffic in the air and send this information to WIPS server. The WIPS server correlates the information, validates it against the defined policies, and classifies if it is a threat. The administrator of the WIPS is then notified of the threat, or, if a policy has been set accordingly, the WIPS takes automatic protection measures.
WIPS is configured as either a network implementation or a hosted implementation.
In a network WIPS implementation, server, sensors and the console are all placed inside a private network and are not accessible from the Internet.
Sensors communicate with the server over a private network using a private port. Since the server resides on the private network, users can access the console only from within the private network.
A network implementation is suitable for organizations where all locations are within the private network.
In a hosted WIPS implementation, sensors are installed inside a private network. However, the server is hosted in secure data center and is accessible on the Internet. Users can access the WIPS console from anywhere on the Internet. A hosted WIPS implementation is as secure as a network implementation because the data flow is encrypted between sensors and server, as well as between server and console. A hosted WIPS implementation requires very little configuration because the sensors are programmed to automatically look for the server on the Internet over a secureTLSconnection.
For a large organization with locations that are not a part of a private network, a hosted WIPS implementation simplifies deployment significantly because sensors connect to the Server over the Internet without requiring any special configuration. Additionally, the Console can be accessed securely from anywhere on the Internet.
Hosted WIPS implementations are available in an on-demand, subscription-basedsoftware as a servicemodel.[4]Hosted implementations may be appropriate for organizations looking to fulfill the minimum scanning requirements of PCI DSS.
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https://en.wikipedia.org/wiki/Wireless_intrusion_prevention_system
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This is a technical featurecomparison of differentdisk encryption software.
TrueCrypt License Version 3.0 (legacy code only)
Different modes of operation supported by the software. Note that an encrypted volume can only use one mode of operation.
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https://en.wikipedia.org/wiki/Comparison_of_disk_encryption_software
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Digital forensics(sometimes known asdigital forensic science) is a branch offorensic scienceencompassing the recovery, investigation, examination, and analysis of material found in digital devices, often in relation to mobile devices andcomputer crime.[1][2]The term "digital forensics" was originally used as a synonym forcomputer forensicsbut has been expanded to cover investigation of all devices capable ofstoring digital data.[1]With roots in thepersonal computing revolutionof the late 1970s and early 1980s, the discipline evolved in a haphazard manner during the 1990s, and it was not until the early 21st century that national policies emerged.
Digital forensics investigations have a variety of applications. The most common is to support or refute a hypothesis beforecriminalorcivilcourts. Criminal cases involve the alleged breaking of laws that are defined by legislation and enforced by the police and prosecuted by the state, such as murder, theft, and assault against the person. Civil cases, on the other hand, deal with protecting the rights and property of individuals (often associated with family disputes), but may also be concerned with contractual disputes between commercial entities where a form of digital forensics referred to aselectronic discovery(ediscovery) may be involved.
Forensics may also feature in the private sector, such as during internal corporate investigations or intrusion investigations (a special probe into the nature and extent of an unauthorizednetwork intrusion).
The technical aspect of an investigation is divided into several sub-branches related to the type of digital devices involved: computer forensics,network forensics,forensic data analysis, andmobile device forensics.[3]The typical forensic process encompasses the seizure, forensic imaging (acquisition), and analysis of digital media, followed with the production of a report of the collected evidence.
As well as identifying direct evidence of a crime, digital forensics can be used to attribute evidence to specific suspects, confirmalibisor statements, determineintent, identify sources (for example, in copyright cases), or authenticate documents.[4]Investigations are much broader in scope than other areas of forensic analysis (where the usual aim is to provide answers to a series of simpler questions), often involving complex time-lines or hypotheses.[5]
Prior to the 1970s, crimes involving computers were dealt with using existing laws. The firstcomputer crimeswere recognized in the 1978 Florida Computer Crimes Act,[6]which included legislation against the unauthorized modification or deletion of data on a computer system.[7]Over the next few years, the range of computer crimes being committed increased, and laws were passed to deal with issues ofcopyright, privacy/harassment (e.g.,cyber bullying,happy slapping,cyber stalking, andonline predators), andchild pornography.[8][9]It was not until the 1980s that federal laws began to incorporate computer offences. Canada was the first country to pass legislation in 1983.[7]This was followed by the US FederalComputer Fraud and Abuse Actin 1986, Australian amendments to their crimes acts in 1989, and the BritishComputer Misuse Actin 1990.[7][9]Digital forensics methods are increasingly being applied to preserve and authenticate born-digital cultural materials in heritage institutions.[10]
The growth in computer crime during the 1980s and 1990s causedlaw enforcement agenciesto begin establishing specialized groups, usually at the national level, to handle the technical aspects of investigations. For example, in 1984, theFBIlaunched aComputer Analysis and Response Teamand the following year a computer crime department was set up within the BritishMetropolitan Policefraud squad. As well as being law enforcement professionals, many of the early members of these groups were also computer hobbyists and became responsible for the field's initial research and direction.[11][12]
One of the first practical (or at least publicized) examples of digital forensics wasCliff Stoll'spursuit of hackerMarkus Hessin 1986. Stoll, whose investigation made use of computer and network forensic techniques, was not a specialized examiner.[13]Many of the earliest forensic examinations followed the same profile.[14]
Throughout the 1990s, there was high demand for these new, and basic, investigative resources. The strain on central units lead to the creation of regional, and even local, level groups to help handle the load. For example, the BritishNational Hi-Tech Crime Unitwas set up in 2001 to provide a national infrastructure for computer crime, with personnel located both centrally in London and with the variousregional police forces(the unit was folded into theSerious Organised Crime Agency (SOCA)in 2006).[12]
During this period, the science of digital forensics grew from the ad-hoc tools and techniques developed by these hobbyist practitioners. This is in contrast to other forensics disciplines, which developed from work by the scientific community.[1][15]It was not until 1992 that the term "computer forensics" was used inacademic literature(although prior to this, it had been in informal use); a paper by Collier and Spaul attempted to justify this new discipline to the forensic science world.[16][17]This swift development resulted in a lack of standardization and training. In his 1995 book,High-Technology Crime: Investigating Cases Involving Computers, K. Rosenblatt wrote the following:
Seizing, preserving, and analyzing evidence stored on a computer is the greatest forensic challenge facing law enforcement in the 1990s. Although most forensic tests, such as fingerprinting and DNA testing, are performed by specially trained experts the task of collecting and analyzing computer evidence is often assigned to patrol officers and detectives.[18]
Since 2000, in response to the need for standardization, various bodies and agencies have published guidelines for digital forensics. TheScientific Working Group on Digital Evidence(SWGDE) produced a 2002 paper,Best practices for Computer Forensics, this was followed, in 2005, by the publication of anISOstandard (ISO 17025,General requirements for the competence of testing and calibration laboratories).[7][19][20]A European-led international treaty, theConvention on Cybercrime, came into force in 2004 with the aim of reconciling national computer crime laws, investigative techniques, and international co-operation. The treaty has been signed by 43 nations (including the US, Canada, Japan, South Africa, UK, and other European nations) and ratified by 16.
The issue of training also received attention. Commercial companies (often forensic software developers) began to offer certification programs, and digital forensic analysis was included as a topic at the UK specialist investigator training facility,Centrex.[7][12]
In the late 1990s, mobile devices became more widely available, advancing beyond simple communication devices, and were found to be rich forms of information, even for crime not traditionally associated with digital forensics.[21]Despite this, digital analysis of phones has lagged behind traditional computer media, largely due to problems over the proprietary nature of devices.[22]
Focus has also shifted onto internet crime, particularly the risk ofcyber warfareandcyberterrorism. A February 2010 report by theUnited States Joint Forces Commandconcluded the following:
Through cyberspace, enemies will target industry, academia, government, as well as the military in the air, land, maritime, and space domains. In much the same way that airpower transformed the battlefield of World War II, cyberspace has fractured the physical barriers that shield a nation from attacks on its commerce and communication.[23]
The field of digital forensics still faces unresolved issues. A 2009 paper, "Digital Forensic Research: The Good, the Bad and the Unaddressed" by Peterson and Shenoi, identified a bias towards Windows operating systems in digital forensics research.[24]In 2010,Simson Garfinkelidentified issues facing digital investigations in the future, including the increasing size of digital media, the wide availability of encryption to consumers, a growing variety of operating systems and file formats, an increasing number of individuals owning multiple devices, and legal limitations on investigators. The paper also identified continued training issues, as well as the prohibitively high cost of entering the field.[13]
During the 1980s, very few specialized digital forensic tools existed. Consequently, investigators often performedlive analysison media, examining computers from within the operating system using existingsysadmintools to extract evidence. This practice carried the risk of modifying data on the disk, either inadvertently or otherwise, which led to claims of evidence tampering. A number of tools were created during the early 1990s to address the problem.
The need for such software was first recognized in 1989 at theFederal Law Enforcement Training Center, resulting in the creation of IMDUMP[25](by Michael White) and in 1990, SafeBack[26](developed by Sydex). Similar software was developed in other countries; DIBS (a hardware and software solution) was released commercially in the UK in 1991, and Rob McKemmish releasedFixed Disk Imagefree to Australian law enforcement.[11]These tools allowed examiners to create an exact copy of a piece of digital media to work on, leaving the original disk intact for verification. By the end of the 1990s, as demand for digital evidence grew, more advanced commercial tools such asEnCaseandFTKwere developed, allowing analysts to examine copies of media without using any live forensics.[7]More recently, a trend towards "live memory forensics" has grown, resulting in the availability of tools such asWindowsSCOPE.
More recently, the same progression of tool development has occurred formobile devices; initially investigators accessed data directly on the device, but soon specialist tools such asXRYor Radio Tactics Aceso appeared.[7]
Police forces have begun implementing risk-based triage systems to manage the overwhelming demand for digital forensic services.[27]
A digital forensic investigation commonly consists of 3 stages:
Acquisition does not normally involve capturing an image of the computer's volatile memory (RAM) unless this is done as part of an incident response investigation.[30]Typically the task involves creating an exactsectorlevel duplicate (or "forensic duplicate") of the media, often using awrite blockingdevice to prevent modification of the original. However, the growth in size of storage media and developments such as cloud computing[31]have led to more use of 'live' acquisitions whereby a 'logical' copy of the data is acquired rather than a complete image of the physical storage device.[28]Both acquired image (or logical copy) and original media/data arehashed(using an algorithm such asSHA-1orMD5) and the values compared to verify the copy is accurate.[32]
An alternative (and patented) approach (that has been dubbed 'hybrid forensics'[33]or 'distributed forensics'[34]) combines digital forensics and ediscovery processes. This approach has been embodied in a commercial tool called ISEEK that was presented together with test results at a conference in 2017.[33]
During the analysis phase an investigator recovers evidence material using a number of different methodologies and tools. In 2002, an article in theInternational Journal of Digital Evidencereferred to this step as "an in-depth systematic search of evidence related to the suspected crime."[1]In 2006, forensics researcherBrian Carrierdescribed an "intuitive procedure" in which obvious evidence is first identified and then "exhaustive searches are conducted to start filling in the holes."[5]
The actual process of analysis can vary between investigations, but common methodologies include conducting keyword searches across the digital media (within files as well as unallocated andslack space), recovering deleted files and extraction of registry information (for example to list user accounts, or attached USB devices).
The evidence recovered is analyzed to reconstruct events or actions and to reach conclusions, work that can often be performed by less specialized staff.[1]When an investigation is complete the data is presented, usually in the form of a written report, inlay persons' terms.[1]
Digital forensics is commonly used in both criminal law and private investigation. Traditionally it has been associated with criminal law, where evidence is collected to support or oppose a hypothesis before the courts. As with other areas of forensics this is often a part of a wider investigation spanning a number of disciplines. In some cases, the collected evidence is used as a form of intelligence gathering, used for other purposes than court proceedings (for example to locate, identify or halt other crimes). As a result, intelligence gathering is sometimes held to a less strict forensic standard.
In civil litigation or corporate matters, digital forensics forms part of theelectronic discovery(or eDiscovery) process. Forensic procedures are similar to those used in criminal investigations, often with different legal requirements and limitations. Outside of the courts digital forensics can form a part of internal corporate investigations.
A common example might be following unauthorizednetwork intrusion. A specialist forensic examination, into the nature and extent of the attack, is performed as a damage limitation exercise, both to establish the extent of any intrusion and in an attempt to identify the attacker.[4][5]Such attacks were commonly conducted over phone lines during the 1980s, but in the modern era are usually propagated over the Internet.[35]
The main focus of digital forensics investigations is to recover objective evidence of a criminal activity (termedactus reusin legal parlance). However, the diverse range of data held in digital devices can help with other areas of inquiry.[4]
One major limitation to a forensic investigation is the use of encryption; this disrupts initial examination where pertinent evidence might be located using keywords. Laws to compel individuals todisclose encryption keysare still relatively new and controversial.[13]But always more frequently there are solutions tobrute forcepasswords or bypass encryption, such as in smartphones or PCs where by means ofbootloadertechniques the content of the device can be first acquired and later forced in order to find the password or encryption key. It is estimated that about 60% of cases that involve encrypted devices, often go unprocessed because there is no way to access the potential evidence.[36]
The examination of digital media is covered by national and international legislation. For civil investigations, in particular, laws may restrict the abilities of analysts to undertake examinations. Restrictions againstnetwork monitoringor reading of personal communications often exist.[37]During criminal investigation, national laws restrict how much information can be seized.[37]For example, in the United Kingdom seizure of evidence by law enforcement is governed by thePACE act.[7]During its existence early in the field, the "International Organization on Computer Evidence" (IOCE) was one agency that worked to establish compatible international standards for the seizure of evidence.[38]
In the UK, the same laws covering computer crime can also affect forensic investigators. The 1990Computer Misuse Actlegislates against unauthorized access to computer material. This is a particular concern for civil investigators who have more limitations than law enforcement.
An individual's right to privacy is one area of digital forensics which is still largely undecided by courts. The USElectronic Communications Privacy Actplaces limitations on the ability of law enforcement or civil investigators to intercept and access evidence. The act makes a distinction between stored communication (e.g. email archives) and transmitted communication (such asVOIP). The latter, being considered more of a privacy invasion, is harder to obtain a warrant for.[7][18]The ECPA also affects the ability of companies to investigate the computers and communications of their employees, an aspect that is still under debate as to the extent to which a company can perform such monitoring.[7]
Article 5 of the European Convention on Human Rightsasserts similar privacy limitations to the ECPA and limits the processing and sharing of personal data both within the EU and with external countries. The ability of UK law enforcement to conduct digital forensics investigations is legislated by theRegulation of Investigatory Powers Act.[7]
When used in acourt of law, digital evidence falls under the same legal guidelines as other forms of evidence, as courts do not usually require more stringent guidelines.[7][39]In the United States, theFederal Rules of Evidenceare used to evaluate theadmissibilityof digital evidence. The United Kingdom PACE andCivil Evidence actshave similar guidelines and many other countries have their own laws. US federal laws restrict seizures to items with only obvious evidential value. This is acknowledged as not always being possible to establish with digital media prior to an examination.[37]
Laws dealing with digital evidence are concerned with two issues:
The ease with which digital media can be modified means that documenting thechain of custodyfrom the crime scene, through analysis and, ultimately, to the court, (a form ofaudit trail) is important to establish the authenticity of evidence.[7]
Attorneys have argued that because digital evidence can theoretically be altered it undermines the reliability of the evidence. US judges are beginning to reject this theory, in the caseUS v. Bonallothe court ruled that "the fact that it is possible to alter data contained in a computer is plainly insufficient to establish untrustworthiness."[7][40]In the United Kingdom, guidelines such as those issued byACPOare followed to help document the authenticity and integrity of evidence.
Digital investigators, particularly in criminal investigations, have to ensure that conclusions are based upon factual evidence and their own expert knowledge.[7]In the US, for example, Federal Rules of Evidence state that a qualified expert may testify “in the form of an opinion or otherwise” so long as:
(1) the testimony is based upon sufficient facts or data, (2) the testimony is the product of reliable principles and methods, and (3) the witness has applied the principles and methods reliably to the facts of the case.[41]
The sub-branches of digital forensics may each have their own specific guidelines for the conduct of investigations and the handling of evidence. For example, mobile phones may be required to be placed in aFaraday shieldduring seizure or acquisition to prevent further radio traffic to the device. In the UK forensic examination of computers in criminal matters is subject toACPOguidelines.[7]There are also international approaches to providing guidance on how to handleelectronic evidence. The "Electronic Evidence Guide" by theCouncil of Europeoffers a framework for law enforcement and judicial authorities in countries who seek to set up or enhance their own guidelines for the identification and handling of electronic evidence.[42]
The admissibility of digital evidence relies on the tools used to extract it. In the US, forensic tools are subjected to theDaubert standard, where the judge is responsible for ensuring that the processes and software used were acceptable.
In a 2003 paper, Brian Carrier argued that the Daubert guidelines required the code of forensic tools to be published and peer reviewed. He concluded that "open source tools may more clearly and comprehensively meet the guideline requirements than would closed-source tools."[43]
In 2011,Josh Bruntystated that the scientific validation of the technology and software associated with performing a digital forensic examination is critical to any laboratory process. He argued that "the science of digital forensics is founded on the principles of repeatable processes and quality evidence therefore knowing how to design and properly maintain a good validation process is a key requirement for any digital forensic examiner to defend their methods in court."[44]
One of the key issues relating to validating forensic tools is determining a 'baseline' or reference point for tool testing/evaluation. There have been numerous attempts to provide an environment for testing the functionality of forensic tools such as the Computer Forensic Tool Testing (CFTT) programme developed by NIST ".[45]
To allow for the different environments in which practitioners operate there have also been many attempts to create a framework for customizing test/evaluation environments.[46][47][48]These resources focus on a single or limited number of target systems. However, they do not scale well when attempts are made to test/evaluate tools designed for large networks or the cloud which have become more commonplace in investigations over the years. As of 2024 the only framework that addresses the use of remote agents by forensic tools for distributed processing/collection is that developed by Adams[49]
Digital forensics investigation is not restricted to retrieve data merely from the computer, as laws are breached by the criminals and small digital devices (e.g. tablets, smartphones, flash drives) are now extensively used. Some of these devices have volatile memory while some have non-volatile memory. Sufficient methodologies are available to retrieve data from volatile memory, however, there is lack of detailed methodology or a framework for data retrieval from non-volatile memory sources.[50]Depending on the type of devices, media or artifacts, digital forensics investigation is branched into various types.
The goal of computer forensics is to explain the current state of a digital artifact; such as a computer system, storage medium or electronic document.[51]The discipline usually covers computers,embedded systems(digital devices with rudimentary computing power and onboard memory) and static memory (such as USB pen drives).
Computer forensics can deal with a broad range of information; from logs (such as internet history) through to the actual files on the drive. In 2007, prosecutors used aspreadsheetrecovered from the computer ofJoseph Edward Duncanto showpremeditationand secure thedeath penalty.[4]Sharon Lopatka's killer was identified in 2006 after email messages from him detailing torture and death fantasies were found on her computer.[7]
Mobile device forensics is a sub-branch of digital forensics relating to recovery of digital evidence or data from amobile device. It differs from Computer forensics in that a mobile device will have an inbuilt communication system (e.g.GSM) and, usually, proprietary storage mechanisms. Investigations usually focus on simple data such as call data and communications (SMS/Email) rather than in-depth recovery of deleted data.[7][52]SMSdata from a mobile device investigation helped to exonerate Patrick Lumumba in themurder of Meredith Kercher.[4]
Mobile devices are also useful for providing location information; either from inbuilt gps/location tracking or viacell sitelogs, which track the devices within their range. Such information was used to track down the kidnappers of Thomas Onofri in 2006.[4]
Network forensics is concerned with the monitoring and analysis ofcomputer networktraffic, bothlocalandWAN/internet, for the purposes of information gathering, evidence collection, or intrusion detection.[53]Traffic is usually intercepted at thepacketlevel, and either stored for later analysis or filtered in real-time. Unlike other areas of digital forensics network data is often volatile and rarely logged, making the discipline often reactionary.
In 2000, theFBIlured computer hackers Aleksey Ivanov and Gorshkov to the United States for a fake job interview. By monitoring network traffic from the pair's computers, the FBI identified passwords allowing them to collect evidence directly from Russian-based computers.[7][54]
Forensic Data Analysis is a branch of digital forensics. It examines structured data with the aim to discover and analyze patterns of fraudulent activities resulting from financial crime.
Digitalimage forensics(or forensic image analysis) is a branch of digital forensics that deals with examination and verification of an image's authenticity and content.[55]These can range from Stalin-era airbrushed photos to elaboratedeepfakevideos.[56][57]This has broad implications for a wide variety of crimes, for determining the validity of information presented in civil and criminal trials, and for verifying images and information that are circulated through news and social media.[56][58][59][57]
Database forensics is a branch of digital forensics relating to the forensic study ofdatabasesand theirmetadata.[60]Investigations use database contents, log files and in-RAMdata to build a timeline or recover relevant information.
IoT forensics is a branch of Digital forensics that has the goal of identifying and extracting digital information from devices belonging to theInternet of thingsfield, to be used for forensics investigations as potential source of evidence.[61]
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https://en.wikipedia.org/wiki/Digital_forensics
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Hardware-based full disk encryption(FDE) is available from manyhard disk drive(HDD/SSD) vendors, including:Hitachi, Integral Memory, iStorage Limited,Micron,Seagate Technology,Samsung,Toshiba,Viasat UK, andWestern Digital. Thesymmetric encryption keyis maintained independently from the computer'sCPU, thus allowing the complete data store to be encrypted and removing computer memory as a potential attack vector.
Hardware-FDE has two major components: the hardware encryptor and the data store.
There are currently four varieties of hardware-FDE in common use:
Hardware designed for a particular purpose can often achieve better performance thandisk encryption software, and disk encryption hardware can be made more transparent to software than encryption done in software. As soon as the key has been initialised, the hardware should in principle be completely transparent to the OS and thus work with any OS. If the disk encryption hardware is integrated with the media itself the media may be designed for better integration. One example of such design would be through the use of physical sectors slightly larger than the logical sectors.
Usually referred to asself-encrypting drive(SED).
HDD FDE is made by HDD vendors using theOPALand Enterprise standards developed by theTrusted Computing Group.[1]Key managementtakes place within the hard disk controller and encryption keys are 128 or 256bitAdvanced Encryption Standard(AES) keys.Authenticationon power up of the drive must still take place within theCPUvia either asoftwarepre-boot authenticationenvironment (i.e., with asoftware-based full disk encryptioncomponent - hybrid full disk encryption) or with aBIOSpassword. In additions, some SEDs supportIEEE 1667standard.[2]
Hitachi,Micron,Seagate,Samsung, andToshibaare the disk drive manufacturers offeringTrusted Computing GroupOpal Storage SpecificationSerial ATAdrives. HDDs have become a commodity so SED allow drive manufacturers to maintain revenue.[3]Older technologies include the proprietary Seagate DriveTrust, and the older, and less secure,PATASecurity command standard shipped by all drive makers includingWestern Digital. Enterprise SAS versions of the TCG standard are called "TCG Enterprise" drives.
Within a standardhard drive form factorcase the encryptor (BC),keystore and a smaller form factor, commercially available, hard disk drive is enclosed.
Examples includeViasat UK (formerly Stonewood Electronics)with their FlagStone, Eclypt[4]and DARC-ssd[5]drives or GuardDisk[6]with anRFIDtoken.
The insertedhard driveFDE allows a standardform factorhard disk driveto be inserted into it. The concept can be seen on[7]
The encryptor bridge and chipset (BC) is placed between the computer and the standard hard disk drive, encrypting every sector written to it.
Intelannounced the release of the Danbury chipset[9]but has since abandoned this approach.[citation needed]
Hardware-based encryption when built into the drive or within the drive enclosure is notably transparent to the user. The drive, except for bootup authentication, operates just like any drive, with no degradation in performance. There is no complication or performance overhead, unlikedisk encryption software, since all the encryption is invisible to theoperating systemand the hostcomputer's processor.
The two main use cases areData at restprotection, and Cryptographic Disk Erasure.
For Data at rest protection a computer or laptop is simply powered off. The disk now self-protects all the data on it. The data is safe because all of it, even the OS, is now encrypted, with a secure mode ofAES, and locked from reading and writing. The drive requires an authentication code which can be as strong as 32bytes (256bits) to unlock.
Crypto-shreddingis the practice of 'deleting' data by (only) deleting or overwriting the encryption keys.
When a cryptographic disk erasure (or crypto erase) command is given (with proper authentication credentials), the drive self-generates a new media encryption key and goes into a 'new drive' state.[10]Without the old key, the old data becomes irretrievable and therefore an efficient means of providingdisk sanitisationwhich can be a lengthy (and costly) process. For example, an unencrypted and unclassified computer hard drive that requires sanitising to conform withDepartment of DefenseStandards must be overwritten 3+ times;[11]a one Terabyte Enterprise SATA3 disk would take many hours to complete this process. Although the use of fastersolid-state drives(SSD) technologies improves this situation, the take up by enterprise has so far been slow.[12]The problem will worsen as disk sizes increase every year. With encrypted drives a complete and secure data erasure action takes just a few milliseconds with a simple key change, so a drive can be safely repurposed very quickly. This sanitisation activity is protected in SEDs by the drive's own key management system built into the firmware in order to prevent accidental data erasure with confirmation passwords and secure authentications related to the original key required.
Whenkeysare self-generated randomly, generally there is no method to store a copy to allowdata recovery. In this case protecting this data from accidental loss or theft is achieved through a consistent and comprehensive data backup policy. The other method is for user-defined keys, for some Enclosed hard disk drive FDE,[13]to be generated externally and then loaded into the FDE.
Recent hardware models circumventsbootingfrom other devices and allowing access by using a dualMaster Boot Record(MBR) system whereby the MBR for the operating system and data files is all encrypted along with a special MBR which is required to boot theoperating system. In SEDs, all data requests are intercepted by theirfirmware, that does not allow decryption to take place unless the system has beenbootedfrom the special SEDoperating systemwhich then loads theMBRof the encrypted part of the drive. This works by having a separatepartition, hidden from view, which contains the proprietaryoperating systemfor the encryption management system. This means no other boot methods will allow access to the drive.[citation needed]
Typically FDE, once unlocked, will remain unlocked as long as power is provided.[14]Researchers atUniversität Erlangen-Nürnberghave demonstrated a number of attacks based on moving the drive to another computer without cutting power.[14]Additionally, it may be possible to reboot the computer into an attacker-controlled operating system without cutting power to the drive.
When a computer with a self-encrypting drive is put intosleep mode, the drive is powered down, but the encryption password is retained in memory so that the drive can be quickly resumed without requesting the password. An attacker can take advantage of this to gain easier physical access to the drive, for instance, by inserting extension cables.[14]
The firmware of the drive may be compromised[15][16]and so any data that is sent to it may be at risk. Even if the data is encrypted on the physical medium of the drive, the fact that the firmware is controlled by a malicious third-party means that it can be decrypted by that third-party. If data is encrypted by the operating system, and it is sent in a scrambled form to the drive, then it would not matter if the firmware is malicious or not.
Hardware solutions have gained criticism for being poorly documented. Many aspects of how the encryption is done are not published by the vendor. This leaves the user with little possibility to judge the security of the product and potential attack methods. It also increases the risk of avendor lock-in.
In addition, implementing system wide hardware-based full disk encryption is prohibitive for many companies due to the high cost of replacing existing hardware. This makes migrating to hardware encryption technologies more difficult and would generally require a clear migration and central management solution for both hardware- and software-basedfull disk encryptionsolutions.[17]however Enclosed hard disk drive FDE and Removable Hard Drive FDE are often installed on a single drive basis.
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https://en.wikipedia.org/wiki/Disk_encryption_hardware
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Disk encryption softwareis acomputer securitysoftware that protects the confidentiality of data stored on computer media (e.g., ahard disk,floppy disk, orUSB device) by usingdisk encryption.
Compared to access controls commonly enforced by anoperating system(OS), encryption passively protects data confidentiality even when the OS is not active, for example, if data is read directly from the hardware or by a different OS. In addition,crypto-shreddingsuppresses the need to erase the data at the end of the disk's lifecycle.
Disk encryption generally refers to wholesale encryption that operates on an entirevolumemostly transparently to the user, the system, and applications. This is generally distinguished from file-level encryption that operates by user invocation on a single file or group of files, and which requires the user to decide which specific files should be encrypted. Disk encryption usually includes all aspects of the disk, including directories, so that an adversary cannot determine content, name or size of any file. It is well suited to portable devices such aslaptop computersandthumb driveswhich are particularly susceptible to being lost or stolen. If used properly, someone finding a lost device cannot penetrate actual data, or even know what files might be present.
The disk's data is protected usingsymmetric cryptographywith the key randomly generated when a disk's encryption is first established. This key is itself encrypted in some way using a password or pass-phrase known (ideally) only to the user. Thereafter, in order to access the disk's data, the user must supply the password to make the key available to the software. This must be done sometime after each operating system start-up before the encrypted data can be used.
Done in software,encryptiontypically operates at a level between all applications and most system programs and the low-leveldevice driversby "transparently" (from a user's point of view) encrypting data after it is produced by a program but before it is physically written to the disk. Conversely, it decrypts data immediately after being read but before it is presented to a program. Properly done, programs are unaware of these cryptographic operations.
Some disk encryption software (e.g.,TrueCryptorBestCrypt) provide features that generally cannot be accomplished withdisk hardware encryption: the ability to mount "container" files as encrypted logical disks with their ownfile system; and encrypted logical "inner" volumes which are secretly hidden within the free space of the more obvious "outer" volumes. Such strategies provideplausible deniability.
Well-known examples of disk encryption software include,BitLockerfor Windows;FileVaultfor Apple OS/X;LUKSa standard free software mainly forLinuxandTrueCrypt, a non-commercial freeware application, for Windows, OS/X and Linux.
Some disk encryption systems, such asVeraCrypt,CipherShed(active open source forks of the discontinuedTrueCryptproject),BestCrypt(proprietary trialware), offer levels ofplausible deniability, which might be useful if a user is compelled to reveal the password of an encrypted volume.
Hidden volumes are asteganographicfeature that allows a second, "hidden", volume to reside within the apparent free space of a visible "container" volume (sometimes known as "outer" volume). The hidden volume has its own separate file system, password, and encryption key distinct from the container volume.
The content of the hidden volume is encrypted and resides in the free space of the file system of the outer volume—space which would otherwise be filled with random values if the hidden volume did not exist. When the outer container is brought online through the disk encryption software, whether the inner or outer volume ismounteddepends on the password provided. If the "normal" password/key of the outer volume proves valid, the outer volume is mounted; if the password/key of the hidden volume proves valid, then (and only then) can the existence of the hidden volume even be detected, and it is mounted; otherwise if the password/key does not successfully decrypt either the inner or outer volume descriptors, then neither is mounted.
Once a hidden volume has been created inside the visible container volume, the user will store important-looking information (but which the user does not actually mind revealing) on the outer volume, whereas more sensitive information is stored within the hidden volume.
If the user is forced to reveal a password, the user can reveal the password to the outer volume, without disclosing the existence of the hidden volume. The hidden volume will not be compromised, if the user takes certain precautions in overwriting the free areas of the "host" disk.[2]
Volumes, be they stored in a file or a device/partition, may intentionally not contain any discernible "signatures" or unencrypted headers. As cipher algorithms are designed to be indistinguishable from apseudorandom permutationwithout knowing thekey, the presence of data on the encrypted volume is also undetectable unless there are known weaknesses in the cipher.[3]This means that it is impossible to prove that any file or partition is an encrypted volume (rather than random data) without having the password to mount it. This characteristic also makes it impossible to determine if a volume contains another hidden volume.
A file hosted volume (as opposed to partitions) may look out of place in some cases since it will be entirely random data placed in a file intentionally. However, a partition or device hosted volume will look no different from a partition or device that has been wiped with a common disk wiping tool such asDarik's Boot and Nuke. One can plausibly claim that such a device or partition has been wiped to clear personal data.
Portable or "traveller mode" means the encryption software can be run without installation to the system hard drive. In this mode, the software typically installs a temporarydriverfrom the portable media. Since it is installing a driver (albeit temporarily), administrative privileges are still required.
Some disk encryption software allows encrypted volumes to be resized. Not many systems implement this fully and resort to using "sparse files" to achieve this.[citation needed]
Encrypted volumes contain "header" (or "CDB") data, which may be backed up. Overwriting these data will destroy the volume, so the ability to back them up is useful.
Restoring the backup copy of these data may reset the volume's password to what it was when the backup was taken.
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Disk encryptionis a special case ofdata at restprotection when the storage medium is a sector-addressable device (e.g., a hard disk). This article presents cryptographic aspects of the problem. For an overview, seedisk encryption. For discussion of different software packages and hardware devices devoted to this problem, seedisk encryption softwareanddisk encryption hardware.
Disk encryption methods aim to provide three distinct properties:
The first property requires defining anadversaryfrom whom the data is being kept confidential. The strongest adversaries studied in the field of disk encryption have these abilities:
A method provides good confidentiality if the only information such an adversary can determine over time is whether the data in a sector has or has not changed since the last time they looked.
The second property requires dividing the disk into severalsectors, usually 512 bytes (4096bits) long, which are encrypted and decrypted independently of each other. In turn, if the data is to stay confidential, the encryption method must betweakable; no two sectors should be processed in exactly the same way. Otherwise, the adversary could decrypt any sector of the disk by copying it to an unused sector of the disk and requesting its decryption. Whereas a purpose of a usual block cipherEK{\displaystyle E_{K}}is to mimic a random permutation for any secret keyK{\displaystyle K}, the purpose oftweakableencryptionEKT{\displaystyle E_{K}^{T}}is to mimic a random permutation for any secret keyK{\displaystyle K}and any known tweakT{\displaystyle T}.
The third property is generally non-controversial. However, it indirectly prohibits the use ofstream ciphers, since stream ciphers require, for their security, that the same initial state not be used twice (which would be the case if a sector is updated with different data); thus this would require an encryption method to store separate initial states for every sector on disk—seemingly a waste of space. The alternative, ablock cipher, is limited to a certain block size (usually 128 or 256 bits). Because of this, disk encryption chiefly studieschaining modes, which expand the encryption block length to cover a wholedisk sector. The considerations already listed make several well-known chaining modes unsuitable:ECB mode, which cannot be tweaked, and modes that turn block ciphers into stream ciphers, such as theCTR mode.
These three properties do not provide any assurance of disk integrity; that is, they don't tell you whether an adversary has been modifying your ciphertext. In part, this is because an absolute assurance of disk integrity is impossible: no matter what, an adversary could always revert the entire disk to a prior state, circumventing any such checks. If some non-absolute level of disk integrity is desired, it can be achieved within the encrypted disk on a file-by-file basis usingmessage authentication codes.
Although it used to be commonly accepted that disk encryption should be length-preserving, some additional features do justify the use of extra space. One example isauthenticated encryption, which takes extra space in exchange for guaranteeing the integrity of the sector. One application of this guarantee would be to prevent an attacker from triggering kernel bugs by breaking the filesystem.[1]
Disk encryption methods are also distinguished into "narrow-block" and "wide-block" methods. For a sector-sized plaintext, narrow-block method encrypts it in multiple blocks, while a wide-block methods does it in just one. Narrow-block methods such as LRW, XES, and XTS allow an attacker to exploit the block granularity to perform traffic analysis and replay.[2]A wide-block cipher ideally makes the entire ciphertext unrecognizable for a change anywhere in the plaintext.[3]
Like most encryption schemes, block cipher-based disk encryption makes use ofmodes of operation, which allow encrypting larger amounts of data than the ciphers' block-size (typically 128 bits). Modes are therefore rules on how to repeatedly apply the ciphers' single-block operations.
Cipher-block chaining(CBC) is a common chaining mode in which the previous block's ciphertext isxoredwith the current block's plaintext before encryption:
Since there isn't a "previous block's ciphertext" for the first block, aninitialization vector(IV) must be used asC−1{\displaystyle C_{-1}}. This, in turn, makes CBC tweakable in some ways.
CBC suffers from some problems. For example,ifthe IVs are predictable, then an adversary may leave a "watermark" on the disk, i.e., store a specially created file or combination of files identifiable even after encryption. The exact method of constructing the watermark depends on the exact function providing the IVs, but the general recipe is to create two encrypted sectors with identical first blocksb1{\displaystyle b_{1}}andb2{\displaystyle b_{2}}; these two are then related to each other byb1⊕IV1=b2⊕IV2{\displaystyle b_{1}\oplus IV_{1}=b_{2}\oplus IV_{2}}. Thus the encryption ofb1{\displaystyle b_{1}}is identical to the encryption ofb2{\displaystyle b_{2}}, leaving a watermark on the disk. The exact pattern of "same-different-same-different" on disk can then be altered to make the watermark unique to a given file.
To protect against the watermarking attack, a cipher or a hash function is used to generate the IVs from the key and the current sector number, so that an adversary cannot predict the IVs. In particular, theESSIVapproach uses a block cipher in CTR mode to generate the IVs.
ESSIV is a method for generatinginitialization vectorsforblock encryptionto use in disk encryption. The usual methods for generating IVs are predictable sequences of numbers based on, for example, time stamp or sector number, and permit certain attacks such as awatermarking attack. ESSIV prevents such attacks by generating IVs from a combination of the sector number SN with the hash of the key. It is the combination with the key in form of ahashthat makes the IV unpredictable.[4][5]
ESSIV was designed by Clemens Fruhwirth and has been integrated into theLinux kernelsince version 2.6.10, though a similar scheme has been used to generate IVs for OpenBSD's swap encryption since 2000.[6]
ESSIV is supported as an option by thedm-crypt[7]andFreeOTFEdisk encryption systems.
While CBC (with or without ESSIV) ensures confidentiality, it does not ensure integrity of the encrypted data. If the plaintext is known to the adversary, it is possible to change every second plaintext block to a value chosen by the attacker, while the blocks in between are changed to random values. This can be used for practical attacks on disk encryption in CBC or CBC-ESSIV mode.[8]
The tweakable narrow-block encryption (LRW)[9]is an instantiation of the mode of operations introduced by Liskov, Rivest, and Wagner[10](see Theorem 2). This mode uses two keys:K{\displaystyle K}is the key for the block cipher andF{\displaystyle F}is an additional key of the same size as block. For example, for AES with a 256-bit key,K{\displaystyle K}is a 256-bit number andF{\displaystyle F}is a 128-bit number. Encrypting blockP{\displaystyle P}with logical index (tweak)I{\displaystyle I}uses the following formula:
Here multiplication⊗{\displaystyle \otimes }and addition⊕{\displaystyle \oplus }are performed in thefinite field(GF(2128){\displaystyle {\text{GF}}\left(2^{128}\right)}for AES). With some precomputation, only a single multiplication per sector is required (note that addition in a binary finite field is a simple bitwise addition, also known as xor):F⊗I=F⊗(I0⊕δ)=F⊗I0⊕F⊗δ{\displaystyle F\otimes I=F\otimes (I_{0}\oplus \delta )=F\otimes I_{0}\oplus F\otimes \delta }, whereF⊗δ{\displaystyle F\otimes \delta }are precomputed for all possible values ofδ{\displaystyle \delta }. This mode of operation needs only a single encryption per block and protects against all the above attacks except a minor leak: if the user changes a single plaintext block in a sector then only a single ciphertext block changes. (Note that this is not the same leak the ECB mode has: with LRW mode equal plaintexts in different positions are encrypted to different ciphertexts.)
Somesecurity concerns exist with LRW, and this mode of operation has now been replaced by XTS.
LRW is employed byBestCryptand supported as an option fordm-cryptandFreeOTFEdisk encryption systems.
Another tweakable encryption mode, XEX (xor–encrypt–xor), was designed by Rogaway[11]to allow efficient processing of consecutive blocks (with respect to the cipher used) within one data unit (e.g., a disk sector). The tweak is represented as a combination of the sector address and index of the block within the sector (the original XEX mode proposed by Rogaway[11]allows several indices). The ciphertext,C{\displaystyle C}, is obtained using:
where:
The basic operations of the LRW mode (AES cipher andGalois fieldmultiplication) are the same as the ones used in theGalois/Counter Mode(GCM), thus permitting a compact implementation of the universal LRW/XEX/GCM hardware.
The original XEX has a weakness.[12]
Ciphertext stealingprovides support for sectors with size not divisible by block size, for example, 520-byte sectors and 16-byte blocks. XTS-AES was standardized on December 19, 2007[13]asIEEE P1619.[14]The XTS standard requires using a different key for the IV encryption than for the block encryption; this differs from XEX which uses only a single key.[11][15]: 1–4As a result, users wantingAES-256 and AES-128 encryption must supply 512 bits and 256 bits of key respectively. The two keys (i.e., both halves of the XTS key) must be distinct for XTS to be CCA-secure, since XTS computes the sequenceαj{\displaystyle \alpha ^{j}}starting atj=0{\displaystyle j=0}; this differs from XEX which starts atj=1{\displaystyle j=1}.[11]: 7[15]: 6
On January 27, 2010,NISTreleased Special Publication (SP) 800-38E[16]in final form. SP 800-38E is a recommendation for the XTS-AES mode of operation, as standardized by IEEE Std 1619-2007, for cryptographic modules. The publication approves the XTS-AES mode of theAESalgorithm by reference to the IEEE Std 1619-2007, subject to one additional requirement, which limits the maximum size of each encrypted data unit (typically asectorordisk block) to 220AES blocks. According to SP 800-38E, "In the absence of authentication or access control, XTS-AES provides more protection than the other approved confidentiality-only modes against unauthorized manipulation of the encrypted data."
XTS is supported byBestCrypt,Botan,NetBSD's cgd,[17]dm-crypt,FreeOTFE,TrueCrypt,VeraCrypt,[18]DiskCryptor,FreeBSD'sgeli,OpenBSDsoftraid disk encryption software,OpenSSL,Mac OS X Lion'sFileVault2,Windows 10'sBitLocker[19]andwolfCrypt.
XTS mode is susceptible to data manipulation and tampering, and applications must employ measures to detect modifications of data if manipulation and tampering is a concern: "...since there are noauthentication tagsthen any ciphertext (original or modified by attacker) will be decrypted as some plaintext and there is no built-in mechanism to detect alterations. The best that can be done is to ensure that any alteration of the ciphertext will completely randomize the plaintext, and rely on the application that uses this transform to include sufficient redundancy in its plaintext to detect and discard such random plaintexts." This would require maintaining checksums for all data and metadata on disk, as done inZFSorBtrfs. However, in commonly used file systems such asext4andNTFSonly metadata is protected against tampering, while the detection of data tampering is non-existent.[20]
The mode is susceptible to traffic analysis, replay and randomization attacks on sectors and 16-byte blocks. As a given sector is rewritten, attackers can collect fine-grained (16 byte) ciphertexts, which can be used for analysis or replay attacks (at a 16-byte granularity). It would be possible to define sector-wide block ciphers, unfortunately with degraded performance (see below).[2]
CMC and EME protect even against the minor leak mentioned above for LRW. Unfortunately, the price is a twofold degradation of performance: each block must be encrypted twice; many consider this to be too high a cost, since the same leak on a sector level is unavoidable anyway.
CMC, introduced by Halevi and Rogaway, stands for CBC–mask–CBC: the whole sector encrypted in CBC mode (withC−1=EA(I){\displaystyle C_{-1}=E_{A}(I)}), the ciphertext is masked by xoring with2(C0′⊕Ck−1′){\displaystyle 2(C'_{0}\oplus C'_{k-1})}, and re-encrypted in CBC mode starting from the last block. When the underlying block cipher is a strongpseudorandom permutation(PRP) then on the sector level the scheme is a tweakable PRP. One problem is that in order to decryptP0{\displaystyle P_{0}}one must sequentially pass over all the data twice.
In order to solve this problem, Halevi and Rogaway introduced a parallelizable variant called EME (ECB–mask–ECB). It works in the following way:
Note that unlike LRW and CMC there is only a single keyK{\displaystyle K}.
CMC and EME were considered for standardization bySISWG. EME is patented, and so is not favored to be a primary supported mode.[21]
HCTR (2005) is mode of operation for block ciphers that is length-preserving, wide-block, and tweakable.[22]It, however, has a bug in the specification and another in its security proof, rendering its claimedsecurity levelinvalid. HCTR2 (2021) is a variant that fixes these issues and improves on security, performance, and flexibility.[23]HCTR2 is available in the Linux kernel since version 6.0.
HCTR and HCTR2 uses a custom block cipher mode of operation called XCTR; AES-128-XCTR is usually used for HCTR2. HCTR2 uses a polynomial hash function called POLYVAL. HCTR2 is efficient on modern processors with anAES instructionsandcarry-less multiplication instructions.[23]
The HBSH (hash, block cipher, stream cipher, hash) construction, published by Google employees in 2018, allows a fast stream cipher to be used in disk encryption. TheAdiantumscheme used in low-end Android devices specifically choosesNH, 256-bitAdvanced Encryption Standard(AES-256),ChaCha12, andPoly1305. The construction is tweakable and wide-block. It requires three passes over the data, but is still faster than AES-128-XTS on a ARM Cortex-A7 (which has noAES instruction set).[24]It is available in the Linux kernel since version 5.0.
In 2023, Aldo Gunsing, Joan Daemen and Bart Mennink presented the "double-decker" construction, which also uses a stream cipher. It is again tweakable and wide-block.[3]
While theauthenticated encryptionschemeIAPMprovides encryption as well as an authentication tag, the encryption component of the IAPM mode completely describes the LRW and XEX schemes above, and henceXTSwithout theciphertext stealingaspect. This is described in
detail in Figures 8 and 5 of the US patent 6,963,976.[25]
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Filesystem-level encryption,[1]often calledfile-based encryption,FBE, orfile/folder encryption, is a form ofdisk encryptionwhere individual files or directories areencryptedby thefile systemitself.
This is in contrast to thefull disk encryptionwhere the entire partition or disk, in which the file system resides, is encrypted.
Types of filesystem-level encryption include:
The advantages of filesystem-level encryption include:
Unlike cryptographic file systems orfull disk encryption, general-purpose file systems that include filesystem-level encryption do not typically encrypt file systemmetadata, such as the directory structure, file names, sizes or modification timestamps. This can be problematic if the metadata itself needs to be kept confidential. In other words, if files are stored with identifying file names, anyone who has access to the physical disk can know which documents are stored on the disk, although not the contents of the documents.
One exception to this is the encryption support being added to theZFSfilesystem. Filesystem metadata such as filenames, ownership, ACLs, extended attributes are all stored encrypted on disk. The ZFS metadata relating to the storage pool is stored inplaintext, so it is possible to determine how many filesystems (datasets) are available in the pool, including which ones are encrypted. The content of the stored files and directories remain encrypted.
Another exception isCryFSreplacement forEncFS.
Cryptographic file systems are specialized (not general-purpose) file systems that are specifically designed with encryption and security in mind. They usually encrypt all the data they contain – including metadata. Instead of implementing an on-disk format and their ownblock allocation, these file systems are often layered on top of existing file systems e.g. residing in a directory on a host file system. Many such file systems also offer advanced features, such asdeniable encryption, cryptographically secure read-onlyfile system permissionsand different views of the directory structure depending on the key or user ...
One use for a cryptographic file system is when part of an existing file system issynchronizedwith 'cloud storage'. In such cases the cryptographic file system could be 'stacked' on top, to help protect data confidentiality.
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Hardware-based full disk encryption(FDE) is available from manyhard disk drive(HDD/SSD) vendors, including:Hitachi, Integral Memory, iStorage Limited,Micron,Seagate Technology,Samsung,Toshiba,Viasat UK, andWestern Digital. Thesymmetric encryption keyis maintained independently from the computer'sCPU, thus allowing the complete data store to be encrypted and removing computer memory as a potential attack vector.
Hardware-FDE has two major components: the hardware encryptor and the data store.
There are currently four varieties of hardware-FDE in common use:
Hardware designed for a particular purpose can often achieve better performance thandisk encryption software, and disk encryption hardware can be made more transparent to software than encryption done in software. As soon as the key has been initialised, the hardware should in principle be completely transparent to the OS and thus work with any OS. If the disk encryption hardware is integrated with the media itself the media may be designed for better integration. One example of such design would be through the use of physical sectors slightly larger than the logical sectors.
Usually referred to asself-encrypting drive(SED).
HDD FDE is made by HDD vendors using theOPALand Enterprise standards developed by theTrusted Computing Group.[1]Key managementtakes place within the hard disk controller and encryption keys are 128 or 256bitAdvanced Encryption Standard(AES) keys.Authenticationon power up of the drive must still take place within theCPUvia either asoftwarepre-boot authenticationenvironment (i.e., with asoftware-based full disk encryptioncomponent - hybrid full disk encryption) or with aBIOSpassword. In additions, some SEDs supportIEEE 1667standard.[2]
Hitachi,Micron,Seagate,Samsung, andToshibaare the disk drive manufacturers offeringTrusted Computing GroupOpal Storage SpecificationSerial ATAdrives. HDDs have become a commodity so SED allow drive manufacturers to maintain revenue.[3]Older technologies include the proprietary Seagate DriveTrust, and the older, and less secure,PATASecurity command standard shipped by all drive makers includingWestern Digital. Enterprise SAS versions of the TCG standard are called "TCG Enterprise" drives.
Within a standardhard drive form factorcase the encryptor (BC),keystore and a smaller form factor, commercially available, hard disk drive is enclosed.
Examples includeViasat UK (formerly Stonewood Electronics)with their FlagStone, Eclypt[4]and DARC-ssd[5]drives or GuardDisk[6]with anRFIDtoken.
The insertedhard driveFDE allows a standardform factorhard disk driveto be inserted into it. The concept can be seen on[7]
The encryptor bridge and chipset (BC) is placed between the computer and the standard hard disk drive, encrypting every sector written to it.
Intelannounced the release of the Danbury chipset[9]but has since abandoned this approach.[citation needed]
Hardware-based encryption when built into the drive or within the drive enclosure is notably transparent to the user. The drive, except for bootup authentication, operates just like any drive, with no degradation in performance. There is no complication or performance overhead, unlikedisk encryption software, since all the encryption is invisible to theoperating systemand the hostcomputer's processor.
The two main use cases areData at restprotection, and Cryptographic Disk Erasure.
For Data at rest protection a computer or laptop is simply powered off. The disk now self-protects all the data on it. The data is safe because all of it, even the OS, is now encrypted, with a secure mode ofAES, and locked from reading and writing. The drive requires an authentication code which can be as strong as 32bytes (256bits) to unlock.
Crypto-shreddingis the practice of 'deleting' data by (only) deleting or overwriting the encryption keys.
When a cryptographic disk erasure (or crypto erase) command is given (with proper authentication credentials), the drive self-generates a new media encryption key and goes into a 'new drive' state.[10]Without the old key, the old data becomes irretrievable and therefore an efficient means of providingdisk sanitisationwhich can be a lengthy (and costly) process. For example, an unencrypted and unclassified computer hard drive that requires sanitising to conform withDepartment of DefenseStandards must be overwritten 3+ times;[11]a one Terabyte Enterprise SATA3 disk would take many hours to complete this process. Although the use of fastersolid-state drives(SSD) technologies improves this situation, the take up by enterprise has so far been slow.[12]The problem will worsen as disk sizes increase every year. With encrypted drives a complete and secure data erasure action takes just a few milliseconds with a simple key change, so a drive can be safely repurposed very quickly. This sanitisation activity is protected in SEDs by the drive's own key management system built into the firmware in order to prevent accidental data erasure with confirmation passwords and secure authentications related to the original key required.
Whenkeysare self-generated randomly, generally there is no method to store a copy to allowdata recovery. In this case protecting this data from accidental loss or theft is achieved through a consistent and comprehensive data backup policy. The other method is for user-defined keys, for some Enclosed hard disk drive FDE,[13]to be generated externally and then loaded into the FDE.
Recent hardware models circumventsbootingfrom other devices and allowing access by using a dualMaster Boot Record(MBR) system whereby the MBR for the operating system and data files is all encrypted along with a special MBR which is required to boot theoperating system. In SEDs, all data requests are intercepted by theirfirmware, that does not allow decryption to take place unless the system has beenbootedfrom the special SEDoperating systemwhich then loads theMBRof the encrypted part of the drive. This works by having a separatepartition, hidden from view, which contains the proprietaryoperating systemfor the encryption management system. This means no other boot methods will allow access to the drive.[citation needed]
Typically FDE, once unlocked, will remain unlocked as long as power is provided.[14]Researchers atUniversität Erlangen-Nürnberghave demonstrated a number of attacks based on moving the drive to another computer without cutting power.[14]Additionally, it may be possible to reboot the computer into an attacker-controlled operating system without cutting power to the drive.
When a computer with a self-encrypting drive is put intosleep mode, the drive is powered down, but the encryption password is retained in memory so that the drive can be quickly resumed without requesting the password. An attacker can take advantage of this to gain easier physical access to the drive, for instance, by inserting extension cables.[14]
The firmware of the drive may be compromised[15][16]and so any data that is sent to it may be at risk. Even if the data is encrypted on the physical medium of the drive, the fact that the firmware is controlled by a malicious third-party means that it can be decrypted by that third-party. If data is encrypted by the operating system, and it is sent in a scrambled form to the drive, then it would not matter if the firmware is malicious or not.
Hardware solutions have gained criticism for being poorly documented. Many aspects of how the encryption is done are not published by the vendor. This leaves the user with little possibility to judge the security of the product and potential attack methods. It also increases the risk of avendor lock-in.
In addition, implementing system wide hardware-based full disk encryption is prohibitive for many companies due to the high cost of replacing existing hardware. This makes migrating to hardware encryption technologies more difficult and would generally require a clear migration and central management solution for both hardware- and software-basedfull disk encryptionsolutions.[17]however Enclosed hard disk drive FDE and Removable Hard Drive FDE are often installed on a single drive basis.
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In re Boucher(case citation: No. 2:06-mJ-91, 2009 WL 424718) is afederalcriminalcaseinVermont, which was the first to directly address the question of whether investigators can compel a suspect to reveal theirencryptionpassphraseorpassword, despite theU.S. Constitution'sFifth Amendmentprotection againstself-incrimination. Amagistrate judgeheld that producing the passphrase would constitute self-incrimination. In its submission on appeal to the District Court, the Government stated that it does not seek the password for the encrypted hard drive, but only sought to force Boucher to produce the contents of his encrypted hard drive in an unencrypted format by opening the drive before the grand jury. A District Court judge agreed with the government, holding that, given Boucher's initial cooperation in showing some of the content of his computer to border agents, producing the complete contents would not constitute self-incrimination.
In late 2009, Boucher finally gave up his password and investigators found numerous images and videosdepicting sexual abuse of children. In January 2010, Boucher was sentenced to 3 years in prison and deported.[1]
On 17 December 2006, thelaptop computerofdefendantSebastien D. Boucher (born in 1977)[2][3]was inspected when he crossed the border from Canada into the United States atDerby Line, Vermont. The laptop was powered-up when the border was crossed, which allowed its contents to be browsed. Images containingchild pornographywere allegedly seen by Immigration and Customs Enforcement (ICE) border agents who seized the laptop, questioned Boucher and then arrested him on a complaint
charging him with transportation of child pornography in violation of 18 U.S.C. 2252A(a)(1). The laptop was subsequently powered-down. When the laptop was switched on and booted on 29 December 2006, it was not possible to access its entire storage capability. This was because the laptop had been protected byPGP Diskencryption.[4]As a result, investigators working for the US government were unable to view the contents of drive "Z:", which allegedly contained the illegal content. A grand jury thensubpoenaedthe defendant to provide the password to theencryption keyprotecting the data.
On November 29, 2007, U.S. Magistrate JudgeJerome Niedermeierof theUnited States District Court for the District of Vermontstated "Compelling Boucher to enter the password forces him to produce evidence that could be used to incriminate him."[4]Accordingly, Niedermeier quashed the subpoena.
On January 2, 2008, the United States appealed the magistrate's opinion to the District Court in a sealed motion (court docket, case #: 2:06-mJ-00091-wks-jjn-1).[5]The appeal was heard by U.S. District JudgeWilliam K. Sessions.[6]Oral arguments were scheduled for April 30, 2008.[7]
On February 19, 2009, Judge Sessions reversed the magistrate's ruling and directed Boucher "to provide an unencrypted version of the Z drive viewed by the ICE agent."
Boucher accessed the Z drive of his laptop at the ICE agent's request. The ICE agent viewed the contents of some of the Z drive's files, and ascertained that they may consist of images or videos of child pornography. The Government thus knows of the existence and location of the Z drive and its files. Again providing access to the unencrypted Z drive 'adds little or nothing to the sum total of the Government's information about the existence and location of files that may contain incriminating information. Fisher, 425 U.S. at 411.
Boucher's act of producing an unencrypted version of the Z drive likewise is not necessary to authenticate it. He has already admitted to possession of the computer, and provided the Government with access to the Z drive. The Government has submitted that it can link Boucher with the files on his computer without making use of his production of an unencrypted version of the Z drive, and that it will not use his act of production as evidence of authentication.[8]
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Single sign-on(SSO) is an authentication scheme that allows a user to log in with a single ID to any of several related, yet independent, software systems.
True single sign-on allows the user to log in once and access services without re-entering authentication factors.
It should not be confused with same-sign on (Directory Server Authentication), often accomplished by using theLightweight Directory Access Protocol(LDAP) and stored LDAP databases on (directory) servers.[1][2]
A simple version of single sign-on can be achieved overIP networksusingcookiesbut only if the sites share a common DNS parent domain.[3]
For clarity, a distinction is made between Directory Server Authentication (same-sign on) and single sign-on: Directory Server Authentication refers to systems requiring authentication for each application but using the same credentials from a directory server, whereas single sign-on refers to systems where a single authentication provides access to multiple applications by passing the authentication token seamlessly to configured applications.
Conversely,single sign-offorsingle log-out(SLO) is the property whereby a single action of signing out terminates access to multiple software systems.
As different applications and resources support differentauthenticationmechanisms, single sign-on must internally store the credentials used for initial authentication and translate them to the credentials required for the different mechanisms.
Other shared authentication schemes, such asOpenIDandOpenID Connect, offer other services that may require users to make choices during a sign-on to a resource, but can be configured for single sign-on if those other services (such as user consent) are disabled. An increasing number of federated social logons, likeFacebook Connect, do require the user to enter consent choices upon first registration with a new resource, and so are not always single sign-on in the strictest sense.
Benefits of using single sign-on include:
SSO shares centralizedauthentication serversthat all other applications and systems use for authentication purposes and combines this with techniques to ensure that users do not have to actively enter their credentials more than once.
The termreduced sign-on(RSO) has been used by some to reflect the fact thatsingle sign-onis impractical in addressing the need for different levels of secure access in the enterprise, and as such more than one authentication server may be necessary.[6]
As single sign-on provides access to many resources once the user is initially authenticated ("keys to the castle"), it increases the negative impact in case the credentials are available to other people and misused. Therefore, single sign-on requires an increased focus on the protection of the user credentials, and should ideally be combined with strong authentication methods likesmart cardsandone-time passwordtokens.[6]
Single sign-on also increases dependence on highly-available authentication systems; a loss of their availability can result in denial of access to all systems unified under the SSO. SSO can be configured with session failover capabilities in order to maintain the system operation.[7]Nonetheless, the risk of system failure may make single sign-on undesirable for systems to which access must be guaranteed at all times, such as security or plant-floor systems.
Furthermore, the use of single-sign-on techniques utilizingsocial networking servicessuch asFacebookmay render third party websites unusable within libraries, schools, or workplaces that block social media sites for productivity reasons. It can also cause difficulties in countries with activecensorshipregimes, such asChinaand its "Golden Shield Project", where the third party website may not be actively censored, but is effectively blocked if a user's social login is blocked.[8][9]
In March 2012,[10]a research paper reported an extensive study on the security ofsocial loginmechanisms. The authors found 8 serious logic flaws in high-profile ID providers and relying party websites, such asOpenID(includingGoogle IDand PayPal Access),Facebook,Janrain,Freelancer,FarmVille, andSears.com. Because the researchers informed ID providers and relying party websites prior to public announcement of the discovery of the flaws, the vulnerabilities were corrected, and no security breaches have been reported.[11]
In May 2014, a vulnerability namedCovert Redirectwas disclosed.[12]It was first reported "Covert Redirect Vulnerability Related toOAuth 2.0and OpenID" by its discoverer Wang Jing, a Mathematical PhD student fromNanyang Technological University, Singapore.[13][14][15]In fact, almost all[weasel words]Single sign-on protocols are affected. Covert Redirect takes advantage of third-party clients susceptible tocross-site scripting(XSS) oropen redirect.[16]
In December 2020, flaws in federated authentication systems were discovered to have been utilized by attackers during the2020 United States federal government data breach.[17][18]
Due to how single sign-on works, by sending a request to the logged-in website to get a SSO token and sending a request with the token to the logged-out website, the token cannot be protected with theHttpOnlycookie flag and thus can be stolen by an attacker if there is an XSS vulnerability on the logged-out website, in order to dosession hijacking. Another security issue is that if the session used for SSO is stolen (which can be protected with the HttpOnly cookie flag unlike the SSO token), the attacker can access all the websites that are using the SSO system.
As originally implemented in Kerberos andSAML, single sign-on did not give users any choices about releasing their personal information to each new resource that the user visited. This worked well enough within a single enterprise, like MIT where Kerberos was invented, or major corporations where all of the resources were internal sites. However, as federated services likeActive Directory Federation Servicesproliferated, the user'sprivate informationwas sent out to affiliated sites not under control of the enterprise that collected the data from the user. Sinceprivacy regulationsare now tightening with legislation like theGDPR, the newer methods likeOpenID Connecthave started to become more attractive; for example MIT, the originator of Kerberos, now supportsOpenID Connect.[19]
Single sign-on in theory can work without revealing identifying information such as email addresses to the relying party (credential consumer), but many credential providers do not allow users to configure what information is passed on to the credential consumer. As of 2019, Google and Facebook sign-in do not require users to share email addresses with the credential consumer. "Sign in with Apple" introduced iniOS 13allows a user to request a unique relay email address each time the user signs up for a new service, thus reducing the likelihood of account linking by the credential consumer.[20]
Windowsenvironment - Windows login fetches TGT.Active Directory-aware applications fetch service tickets, so the user is not prompted to re-authenticate.
Unix/Linuxenvironment - Login via KerberosPAMmodules fetches TGT. Kerberized client applications such asEvolution,Firefox, andSVNuse service tickets, so the user is not prompted to re-authenticate.
Initial sign-on prompts the user for thesmart card. Additionalsoftware applicationsalso use the smart card, without prompting the user to re-enter credentials. Smart-card-based single sign-on can either use certificates or passwords stored on the smart card.
Integrated Windows Authenticationis a term associated withMicrosoftproducts and refers to theSPNEGO,Kerberos, andNTLMSSPauthentication protocols with respect toSSPIfunctionality introduced with MicrosoftWindows 2000and included with laterWindows NT-based operating systems. The term is most commonly used to refer to the automatically authenticated connections between MicrosoftInternet Information ServicesandInternet Explorer. Cross-platformActive Directoryintegration vendors have extended the Integrated Windows Authentication paradigm to Unix (including Mac) and Linux systems.
Security Assertion Markup Language(SAML) is anXML-based method for exchanging user security information between anSAML identity providerand aSAML service provider.SAML 2.0supportsW3CXML encryption and service-provider–initiated web browser single sign-on exchanges.[21]A user wielding a user agent (usually a web browser) is called the subject in SAML-based single sign-on. The user requests a web resource protected by a SAML service provider. The service provider, wishing to know the identity of the user, issues an authentication request to a SAML identity provider through the user agent. The identity provider is the one that provides the user credentials. The service provider trusts theuser informationfrom the identity provider to provide access to its services or resources.
A newer variation of single-sign-on authentication has been developed using mobile devices as access credentials. Users' mobile devices can be used to automatically log them onto multiple systems, such as building-access-control systems and computer systems, through the use of authentication methods which includeOpenID Connectand SAML,[22]in conjunction with anX.509ITU-Tcryptographycertificate used to identify the mobile device to an access server.
A mobile device is "something you have", as opposed to a password which is "something you know", or biometrics (fingerprint, retinal scan, facial recognition, etc.) which is "something you are". Security experts recommend using at least two out of these three factors (multi-factor authentication) for best protection.
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CCM mode(counter with cipher block chaining message authentication code;counter withCBC-MAC) is amode of operationfor cryptographicblock ciphers. It is anauthenticated encryptionalgorithm designed to provide bothauthenticationandconfidentiality. CCM mode is only defined for block ciphers with a block length of 128 bits.[1][2]
Thenonceof CCM must be carefully chosen to never be used more than once for a givenkey.
This is because CCM is a derivation ofcounter (CTR) modeand the latter is effectively astream cipher.[3]
As the name suggests, CCM mode combinescounter (CTR) modefor confidentiality withcipher block chaining message authentication code (CBC-MAC)for authentication. These two primitives are applied in an "authenticate-then-encrypt" manner: CBC-MAC is first computed on the message to obtain amessage authentication code (MAC), then the message and the MAC are encrypted using counter mode. The main insight is that the same encryption key can be used for both, provided that the counter values used in the encryption do not collide with the (pre-)initialization vectorused in the authentication. Aproof of security[4]exists for this combination, based on the security of the underlying block cipher. The proof also applies to a generalization of CCM for anyblock size, and for any size ofcryptographically strongpseudo-random function(since in both counter mode and CBC-MAC, the block cipher is only ever used in one direction).
CCM mode was designed byRuss Housley, Doug Whiting andNiels Ferguson. At the time CCM mode was developed, Russ Housley was employed byRSA Laboratories.
A minor variation of CCM, called CCM*, is used in theIEEE 802.15.4standard, used as theMAClayer inZigbee. CCM* includes all of the features of CCM. It allows a choice of MAC lengths down to 0 (which disables authentication and becomes encryption-only).[5]
CCM requires two block cipher encryption operations on each block of an encrypted-and-authenticated message, and one encryption on each block of associated authenticated data.
According toCrypto++benchmarks, AES CCM requires 28.6cycles per byteon anIntel Core 2 processorin 32-bit mode.[6]
Notable inefficiencies:
The catalyst for the development of CCM mode was the submission ofoffset codebook (OCB) modefor inclusion in theIEEE 802.11istandard. Opposition was voiced to the inclusion of OCB mode because of a pendingpatentapplication on thealgorithm. Inclusion of a patented algorithm meant significant licensing complications for implementors of the standard.
While the inclusion of OCB mode was disputed based on theseintellectual propertyissues, it was agreed that the simplification provided by an authenticated encryption system was desirable. Therefore, Housley, et al. developed CCM mode as a potential alternative that was not encumbered by patents.
Even though CCM mode is less efficient than OCB mode, a patent free solution was preferable to one complicated by patent licensing issues. Therefore, CCM mode went on to become a mandatory component of the IEEE 802.11i standard, and OCB mode was relegated to optional component status, before eventually being removed altogether.
CCM mode is used inIEEE 802.11i(asCCMP, the CCM encryption protocol forWPA2),IPsec,[7]andTLS1.2,[8]as well asBluetooth Low Energy(as ofBluetooth 4.0).[9]It is available for TLS 1.3, but not enabled by default inOpenSSL.[10]
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Incryptography,CWC Mode(Carter–Wegman +CTRmode) is anAEAD block cipher mode of operationthat provides both encryption and built-in message integrity, similar to CCM and OCB modes. It combines the use of CTR mode with a 128-bit block cipher for encryption with an efficient polynomialCarter–Wegman MACwith a tag length of at most 128 bits and is designed byTadayoshi Kohno,John ViegaandDoug Whiting.[1]
CWC mode was submitted toNIST[2]for standardization, but NIST opted for the similarGCM modeinstead.[3]
Although GCM has weaknesses compared to CWC,[4]the GCM authors successfully argued for GCM.[5]
CWC allows the payload and associated data to be at most 232- 1 blocks or nearly 550 GB.[1]
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https://en.wikipedia.org/wiki/CWC_mode
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Offset codebook mode(OCB mode) is anauthenticated encryptionmode of operationfor cryptographicblock ciphers.[1][2]OCB mode was designed byPhillip Rogaway, who creditsMihir Bellare,John Black, and Ted Krovetz with assistance and comments on the designs. It is based on theintegrity-aware parallelizeable mode(IAPM) of authenticated encryption by Charanjit S. Jutla. TheOCB2version was proven insecure, while the originalOCB1as well asOCB3from 2011 are still considered secure.
OCB mode was designed to provide bothmessage authenticationandprivacy. It is essentially a scheme for integrating amessage authentication code(MAC) into the operation of ablock cipher. In this way, OCB mode avoids the need to use two systems: a MAC for authentication andencryptionfor confidentiality. This results in lower computational cost compared to using separate encryption and authentication functions.
There are three versions of OCB: OCB1, OCB2 and OCB3. OCB1 was published in 2001. OCB2 improves on OCB1 by allowing associated data to be included with the message, providingauthenticated encryption with associated data(AEAD; that is, data that are not encrypted but should be authenticated) and a new method for generating a sequence of offsets. OCB2 was first published in 2003, originally namedauthenticated-encryption mode, oradvanced encryption mode(AEM) andwas shown to be completely insecure in 2019. OCB3, published in 2011, changes again the way offsets are computed and introduces minor performance improvements.
OCB2 was standardized in ISO/IEC19772:2009[3](although it was removed from the standard following the publication of the attack) and a modified OCB3 in RFC7253.[4]The RFC encodes the tag length into the internally formatted nonce.
OCB performance overhead is minimal compared to classical, non-authenticating modes likecipher block chaining. OCB requires one block cipher operation per block of encrypted and authenticated message, and one block cipher operation per block of associated data. There is also one extra block cipher operation required at the end of process.
For comparison,CCM modeoffering similar functionality requires twice as many block cipher operations per message block (associated data requires one, as in OCB).
While OCB is now public domain, Rogaway initially patented OCB mode so that they could charge for commercial licenses and in attempt to stop their work showing up in military-related projects.[5]Rogaway intentionally abandoned their OCB patents in 2021.[6]
Two U.S. patents were issued for OCB mode.[7]The patents have hindered approval by theNational Institute of Standards and Technology.[citation needed]
While OCB mode was patented, Rogaway made three licenses available to allow OCB mode to be freely used in software licensed under theGNU General Public License(later any open source license certified by theOpen Source Initiative[8]), non-commercial non-military projects, and inOpenSSL.
Since Rogaway only applied for patent protection in the U.S., the algorithm has always been free to use in software not developed and not sold inside the U.S.[9]
Niels Fergusonpointed outcollision attackson OCB, which limits the amount of data that can be securely processed under a single key to about 280 terabytes.[10][11]
In October 2018, Inoue and Minematsu presented anexistential forgery attackagainst OCB2 that requires only a single prior encryption query and almost no computational power or storage.[12]The attack does not extend to OCB1 or OCB3, and it requires that the associated data field of the forged ciphertext be empty. Poettering[13]and Iwata[14]improved the forgery attack to a full plaintext recovery attack just a couple of days later. The four authors later produced a joint report.[15]
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EAX mode(encrypt-then-authenticate-then-translate[1]) is amode of operationfor cryptographic block ciphers. It is an Authenticated Encryption with Associated Data (AEAD) algorithm designed to simultaneously provide bothauthenticationandprivacyof the message (authenticated encryption) with a two-pass scheme, one pass for achieving privacy and one for authenticity for each block.
EAX mode was submitted on October 3, 2003, to the attention of NIST in order to replaceCCMas standard AEAD mode of operation, since CCM mode lacks some desirable attributes of EAX and is more complex.
EAX is a flexiblenonce-using two-pass AEAD scheme with no restrictions on block cipher primitive to be used, nor on block size, and supports arbitrary-length messages.Authentication taglength is arbitrarily sizeable up to the used cipher's block size.
The block cipher primitive is used inCTR modefor encryption and asOMACfor authentication over each block through the EAX composition method, that may be seen as a particular case of a more general algorithm called EAX2 and described inThe EAX Mode of Operation[2]
The reference implementation in the aforementioned paper uses AES in CTR mode for encryption combined with AES OMAC for authentication.
Being a two-pass scheme, EAX mode is slower than a well designed one-pass scheme based on the same primitives.
EAX mode has several desirable attributes, notably:
Notably, CCM mode lacks the last 2 attributes (CCM can process Associated Data, it can't pre-process it).
The authors of EAX mode,Mihir Bellare,Phillip Rogaway, andDavid Wagnerplaced the work under public domain and have stated that they were unaware of any patents covering this technology. Thus, EAX mode of operation is believed to be free and unencumbered for any use.
A modification of the EAX mode, so calledEAX′or EAXprime, is used in theANSI C12.22standard for transport of meter-based data over a network. In 2012 Kazuhiko Minematsu,Stefan Lucks, Hiraku Morita and Tetsu Iwata published a paper that proves the security of the mode with messages longer than the key, but demonstrates a trivial attack against short messages using this mode. The authors stated that they did not know whether the ANSI C12.22 protocols were vulnerable to the attack.[3][4]
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ChaCha20-Poly1305is anauthenticated encryption with associated data (AEAD)algorithm, that combines theChaCha20stream cipher with thePoly1305message authentication code.[1]It has fast software performance, and without hardware acceleration, is usually faster thanAES-GCM.[1]: §B
The two building blocks of the construction, the algorithms Poly1305 and ChaCha20, were both independently designed, in 2005 and 2008, byDaniel J. Bernstein.[2][3]
In March 2013, a proposal was made to the IETF TLS working group to includeSalsa20, a winner of theeSTREAMcompetition[4]to replace the aging RC4-based ciphersuites. A discussion followed in the IETF TLS mailing list with various enhancement suggestions, including using Chacha20 instead of Salsa20 and using a universal hashing based MAC for performance. The outcome of this process was the adoption of Adam Langley's proposal for a variant of the original ChaCha20 algorithm (using 32-bit counter and 96-bit nonce) and a variant of the original Poly1305 (authenticating 2 strings) being combined in an IETF draft[5][6]to be used inTLSandDTLS,[7]and chosen, for security and performance reasons, as a newly supported cipher.[8]Shortly after IETF's adoption for TLS, ChaCha20, Poly1305 and the combined AEAD mode are added toOpenSSHvia thechacha20-poly1305@openssh.comauthenticated encryption cipher[9][10]but kept the original 64-bit counter and 64-bit nonce for the ChaCha20 algorithm.
In 2015, the AEAD algorithm was standardized in RFC 7539[11]and in RFC 7634[12]to be used in IPsec. The same year, it was integrated by Cloudflare as an alternative ciphersuite.[13]
In 2016 RFC 7905[14]describes how to use it in the TLS 1.2 and DTLS 1.2 protocols.
In June 2018, RFC 7539 was updated and replaced by RFC 8439.[1]
The ChaCha20-Poly1305 algorithm takes as input a 256-bit key and a 96-bitnonceto encrypt a plaintext,[1]with a ciphertext expansion of 128-bit (the tag size). In the ChaCha20-Poly1305 construction, ChaCha20 is used in counter mode to derive a key stream that isXORedwith the plaintext. The ciphertext and the associated data is then authenticated using a variant of Poly1305 that first encodes the two strings into one. The way that a cipher and a one time authenticator are combined is precisely identical toAES-GCMconstruction in how the first block is used to seed the authenticator and how the ciphertext is then authenticated with a 16-byte tag.
The main external difference with ChaCha20 is its 64 byte (512 bit) block size, in comparison to 16 bytes (128 bit) with both AES-128 and AES-256. The larger block size enables higher performance on modern CPUs and allows for larger streams before the 32 bit counter overflows.
The XChaCha20-Poly1305 construction is an extended 192-bit nonce variant of the ChaCha20-Poly1305 construction, usingXChaCha20instead ofChaCha20. When choosing nonces at random, the XChaCha20-Poly1305 construction allows for better security than the original construction. The draft attempt to standardize the construction expired in July 2020.[15]
Salsa20-Poly1305 and XSalsa20-Poly1305 are variants of the ChaCha20-Poly1305 andXChaCha20-Poly1305algorithms, usingSalsa20andXSalsa20in place of ChaCha20 and XChaCha20. They are implemented inNaCl[16]and libsodium[17]but not standardized. The variants using ChaCha are preferred in practice as they provide betterdiffusionper round than Salsa.[2]
ChaCha20 can be replaced with its reduced-round variants ChaCha12 and ChaCha8, yielding ChaCha12-Poly1305 and ChaCha8-Poly1305. The same modification can be applied to XChaCha20-Poly1305. These are implemented by the RustCrypto team and not standardized.[18]
ChaCha20-Poly1305 is used inIPsec,[1]SSH,[19]TLS 1.2,DTLS1.2,TLS 1.3,[14][19]WireGuard,[20]S/MIME 4.0,[21]OTRv4[22]and multiple other protocols and implemented inOpenSSLandlibsodium. Additionally, the algorithm is used in the backup softwareBorg[23]in order to provide standard data encryption and in thecopy-on-writefilesystemBcachefsfor the purpose of optional whole filesystem encryption.[24]
ChaCha20-Poly1305 usually offers better performance than the more prevalentAES-GCMalgorithm, except on systems where the CPU(s) have theAES-NI instruction setextension[1]. As a result, ChaCha20-Poly1305 is sometimes preferred over AES-GCM due to its similar levels of security and in certain use cases involvingmobile devices, which mostly useARM-based CPUs. Because ChaCha20-Poly1305 has less overhead than AES-GCM, ChaCha20-Poly1305 on mobile devices may consume less power than AES-GCM.
The ChaCha20-Poly1305 construction is generally secure in thestandard modeland theideal permutation model, for the single- and multi-user setting.[25]However, similarly toGCM, the security relies on choosing a uniquenoncefor every message encrypted. Compared to AES-GCM, implementations of ChaCha20-Poly1305 are less vulnerable totiming attacks.
To be noted, when theSSHprotocol uses ChaCha20-Poly1305 as underlying primitive, it is vulnerable to theTerrapin attack.
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A new mode calledSophie Germain Counter Mode (SGCM)has been proposed as a variant of theGalois/Counter Modeof operation for block ciphers. Instead of the binary field GF(2128), it uses modular arithmetic in GF(p) wherepis asafe prime2128+ 12451with correspondingSophie Germain primep− 1/2= 2127+ 6225.[1]SGCM does prevent the specific "weak key" attack described in its paper, however there are other ways of modifying the message that will achieve the same forgery probability against SGCM as is possible against GCM: by modifying a validn-word message, you can create a SGCM forgery with probability circan/2128.[2]That is, its authentication bounds are no better than those ofGalois/Counter Mode. SGCM when implemented in hardware has a higher gate count[clarification needed]than GCM.[citation needed]However, its authors expect software implementations of SGCM to have similar or superior performance to GCM on most software platforms.[citation needed]
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Incryptography,signcryptionis a public-key primitive that simultaneously performs the functions of bothdigital signatureandencryption.
Encryption and digital signature are two fundamental cryptographic tools that can guarantee theconfidentiality,integrity, andnon-repudiation. Until 1997, they were viewed as important but distinct building blocks of various cryptographic systems. In public key schemes, a traditional method is to digitally sign a message then followed by an encryption (signature-then-encryption) that can have two problems: Low efficiency and high cost of such summation, and the case that any arbitrary scheme cannot guarantee security. Signcryption is a relatively new cryptographic technique that is supposed to perform the functions of digital signature and encryption in a single logical step and can effectively decrease the computational costs and communication overheads in comparison with the traditional signature-then-encryption schemes.
Signcryption provides the properties of both digital signatures and encryption schemes in a way that is more efficient than signing and encrypting separately. This means that at least some aspect of its efficiency (for example the computation time) is better than any hybrid of digital signature and encryption schemes, under a particular model of security. Note that sometimeshybrid encryptioncan be employed instead of simple encryption, and a single session-key reused for several encryptions to achieve better overall efficiency across many signature-encryptions than a signcryption scheme but the session-key reuse causes the system to lose security under even the relatively weakCPAmodel. This is the reason why a random session key is used for each message in a hybrid encryption scheme but for a givenlevel of security(i.e., a given model, say CPA), a signcryption scheme should be more efficient than any simple signature-hybrid encryption combination.
The first signcryption scheme was introduced byYuliang Zhengin 1997.[1]Zheng also proposed anelliptic curve-based signcryption scheme that saves 58% of computational and 40% of communication costs when it is compared with the traditional elliptic curve-based signature-then-encryption schemes.[2]There are also many other signcryption schemes that have been proposed throughout the years, each of them having its own problems and limitations, while offering different levels of security and computational costs.
A signcryption scheme typically consists of three algorithms: Key Generation (Gen), Signcryption (SC), and Unsigncryption (USC). Gen generates a pair of keys for any user, SC is generally a probabilistic algorithm, and USC is most likely deterministic. Any signcryption scheme should have the following properties:[3]
Example signcryption schemes include:
Signcryption is seen[citation needed]to have several applications including the following:
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Incomputer scienceandcryptography,Whirlpool(sometimes styledWHIRLPOOL) is acryptographic hash function. It was designed byVincent Rijmen(co-creator of theAdvanced Encryption Standard) andPaulo S. L. M. Barreto, who first described it in 2000.
The hash has been recommended by theNESSIEproject. It has also been adopted by theInternational Organization for Standardization(ISO) and theInternational Electrotechnical Commission(IEC) as part of the joint ISO/IEC 10118-3international standard.
Whirlpool is a hash designed after theSquareblock cipher, and is considered to be in that family of block cipher functions.
Whirlpool is aMiyaguchi-Preneelconstruction based on a substantially modifiedAdvanced Encryption Standard(AES).
Whirlpool takes a message of any length less than 2256bits and returns a 512-bitmessage digest.[3]
The authors have declared that
The original Whirlpool will be calledWhirlpool-0, the first revision of Whirlpool will be calledWhirlpool-Tand the latest version will be calledWhirlpoolin the following test vectors.
The Whirlpool hash function is aMerkle–Damgård constructionbased on anAES-likeblock cipherW inMiyaguchi–Preneelmode.[2]
Theblock cipherW consists of an 8×8 state matrixS{\displaystyle S}of bytes, for a total of 512 bits.
The encryption process consists of updating the state with four round functions over 10 rounds. The four round functions are SubBytes (SB), ShiftColumns (SC), MixRows (MR) and AddRoundKey (AK). During each round the new state is computed asS=AK∘MR∘SC∘SB(S){\displaystyle S=AK\circ MR\circ SC\circ SB(S)}.
TheSubBytesoperation applies a non-linear permutation (the S-box) to each byte of the state independently. The 8-bit S-box is composed of 3 smaller 4-bit S-boxes.
TheShiftColumnsoperation cyclically shifts each byte in each column of the state. Columnjhas its bytes shifted downwards byjpositions.
TheMixRowsoperation is a right-multiplication of each row by an 8×8 matrix overGF(28){\displaystyle GF({2^{8}})}. The matrix is chosen such that thebranch number(an important property when looking at resistance todifferential cryptanalysis) is 9, which is maximal.
TheAddRoundKeyoperation uses bitwisexorto add a key calculated by the key schedule to the current state. The key schedule is identical to the encryption itself, except the AddRoundKey function is replaced by anAddRoundConstantfunction that adds a predetermined constant in each round.
The Whirlpool algorithm has undergone two revisions since its original 2000 specification.
People incorporating Whirlpool will most likely use the most recent revision of Whirlpool; while there are no known security weaknesses in earlier versions of Whirlpool, the most recent revision has better hardware implementation efficiency characteristics, and is also likely to be more secure. As mentioned earlier, it is also the version adopted in the ISO/IEC 10118-3international standard.
The 512-bit (64-byte) Whirlpool hashes (also termedmessage digests) are typically represented as 128-digithexadecimalnumbers.The following demonstrates a 43-byteASCIIinput (not including quotes) and the corresponding Whirlpool hashes:
The authors providereference implementationsof the Whirlpool algorithm, including a version written inCand a version written inJava.[2]These reference implementations have been released into the public domain.[2]
Research on the security analysis of the Whirlpool function however, has revealed that on average, the introduction of 8 random faults is sufficient to compromise the 512-bit Whirlpool hash message being processed and the secret key of HMAC-Whirlpool within the context of Cloud of Things (CoTs). This emphasizes the need for increased security measures in its implementation.[5]
Two of the first widely used mainstream cryptographic programs that started using Whirlpool wereFreeOTFE, followed byTrueCryptin 2005.[citation needed]
VeraCrypt(a fork ofTrueCrypt) included Whirlpool (the final version) as one of its supported hash algorithms.[6]
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Incryptography,Square(sometimes writtenSQUARE) is ablock cipherinvented byJoan DaemenandVincent Rijmen. The design, published in 1997, is a forerunner toRijndael, which has been adopted as theAdvanced Encryption Standard. Square was introduced together with a new form ofcryptanalysisdiscovered byLars Knudsen, called the "Square attack".
The structure of Square is asubstitution–permutation networkwith eight rounds, operating on 128-bit blocks and using a 128-bitkey.
Square is not patented.
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https://en.wikipedia.org/wiki/Square_(cipher)
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Incryptography, acipher block chaining message authentication code(CBC-MAC) is a technique for constructing amessage authentication code (MAC)from ablock cipher. The message is encrypted with some block cipher algorithm incipher block chaining (CBC) modeto create a chain of blocks such that each block depends on the proper encryption of the previous block. This interdependence ensures that a change to any of the plaintext bits will cause the final encrypted block to change in a way that cannot be predicted or counteracted without knowing the key to the block cipher.
To calculate the CBC-MAC of messagem, one encryptsmin CBC mode with zeroinitialization vectorand keeps the last block. The following figure sketches the computation of the CBC-MAC of a message comprising blocksm1‖m2‖⋯‖mx{\displaystyle m_{1}\|m_{2}\|\cdots \|m_{x}}using a secret keykand a block cipherE:
CBC-MAC on its own is not secure for variable-length messages[1](see the discussion below) and is currently used to construct apseudorandom function family[2]and as a component of theCCM mode.
The CBC-MAC construct is used as part of theCCM modeutilized inIEEE 802.11iandNIST SP 800-97(asCCMP, the CCM encryption protocol forWPA2),IPsec,[3]andTLS1.2,[4]as well asBluetooth Low Energy(as ofBluetooth 4.0, seeNIST SP 800-121Rev2).[5]It is available for TLS 1.3, but not enabled by default inOpenSSL.[6]
CBC-MAC is also used as a "conditioning component" (a.k.a. randomness extractor,[2]a method to generate bitstrings withfull entropy) inNIST SP 800-90B.
FIPS PUB 113Computer Data Authenticationis a (now obsolete)U.S. government standardthat specified the CBC-MAC algorithm usingDESas the block cipher.
The CBC-MAC algorithm is also included into ANSI X9.9, ANSI X9.19, ISO 8731-1, andISO/IEC 9797-1MAC (Algorithm 1).[7]
If the block cipher used is secure (meaning that it is apseudorandom permutation), then CBC-MAC is secure for fixed-length messages.[1]However, by itself, it is not secure for variable-length messages. Thus, any single key must only be used for messages of a fixed and known length. This is because an attacker who knows the correctauthentication tag(i.e. CBC-MAC) pairs for two messages(m,t){\displaystyle (m,t)}and(m′,t′){\displaystyle (m',t')}can generate a third messagem″{\displaystyle m''}whose CBC-MAC will also bet′{\displaystyle t'}. This is simply done by XORing the first block ofm′{\displaystyle m'}withtand then concatenatingmwith this modifiedm′{\displaystyle m'}; i.e., by makingm″=m‖[(m1′⊕t)‖m2′‖…‖mx′]{\displaystyle m''=m\|[(m_{1}'\oplus t)\|m_{2}'\|\dots \|m_{x}']}. When computing the MAC for the messagem″{\displaystyle m''}, it follows that we compute the MAC formin the usual manner ast, but when this value is chained forwards to the stage computingEKMAC(m1′⊕t){\displaystyle E_{K_{\text{MAC}}}(m_{1}'\oplus t)}we will perform an exclusive OR operation with the value derived for the MAC of the first message. The presence of that tag in the new message means it will cancel, leaving no contribution to the MAC from the blocks of plain text in the first messagem:EKMAC(m1′⊕t⊕t)=EKMAC(m1′){\displaystyle E_{K_{\text{MAC}}}(m_{1}'\oplus t\oplus t)=E_{K_{\text{MAC}}}(m_{1}')}and thus the tag form″{\displaystyle m''}ist′{\displaystyle t'}.
This problem cannot be solved by adding a message-size block to the end.[8]There are three main ways of modifying CBC-MAC so that it is secure for variable length messages: 1) Input-length key separation; 2) Length-prepending; 3) Encrypt last block.[8]In such a case, it may also be recommended to use a different mode of operation, for example,CMACorHMACto protect the integrity of variable-length messages.
One solution is to include the length of the message in the first block;[9]in fact CBC-MAC has been proven secure as long as no two messages that are prefixes of each other are ever used and prepending the length is a special case of this.[10]This can be problematic if the message length may not be known when processing begins.
Encrypt-last-block CBC-MAC (ECBC-MAC)[11]is defined asCBC-MAC-ELB(m, (k1,k2)) =E(k2, CBC-MAC(k1,m)).[8]Compared to the other discussed methods of extending CBC-MAC to variable-length messages, encrypt-last-block has the advantage of not needing to know the length of the message until the end of the computation.
As with many cryptographic schemes, naïve use of ciphers and other protocols may lead to attacks being possible, reducing the effectiveness of the cryptographic protection (or even rendering it useless). We present attacks which are possible due to using the CBC-MAC incorrectly.[12]
One common mistake is to reuse the same keykfor CBC encryption and CBC-MAC. Although a reuse of a key for different purposes is a bad practice in general, in this particular case the mistake leads to a spectacular attack:
Suppose Alice has sent to Bob the cipher text blocksC=C1‖C2‖…‖Cn{\displaystyle C=C_{1}\|C_{2}\|\dots \|C_{n}}. During the transmission process, Eve can tamper with any of theC1,…,Cn−1{\displaystyle C_{1},\dots ,C_{n-1}}cipher-text blocks and adjust any of the bits therein as she chooses, provided that the final block,Cn{\displaystyle C_{n}}, remains the same. We assume, for the purposes of this example and without loss of generality, that the initialization vector used for the encryption process is a vector of zeroes.
When Bob receives the message, he will first decrypt the message by reversing the encryption process which Alice applied, using the cipher text blocksC=C1‖C2‖⋯‖Cn{\displaystyle C=C_{1}\|C_{2}\|\cdots \|C_{n}}. The tampered message, delivered to Bob in replacement of Alice's original, isC′=C1′‖…‖Cn−1′‖Cn{\displaystyle C'=C_{1}'\|\dots \|C_{n-1}'\|C_{n}}.
Bob first decrypts the message received using the shared secret keyKto obtain corresponding plain text. Note that all plain text produced will be different from that which Alice originally sent, because Eve has modified all but the last cipher text block. In particular, the final plain text,Pn′{\displaystyle P_{n}'}, differs from the original,Pn{\displaystyle P_{n}}, which Alice sent; althoughCn{\displaystyle C_{n}}is the same,Cn−1′≠Cn−1{\displaystyle C_{n-1}'\not =C_{n-1}}, so a different plain textPn′{\displaystyle P_{n}'}is produced when chaining the previous cipher text block into the exclusive-OR after decryption ofCn{\displaystyle C_{n}}:Pn′=Cn−1′⊕EK−1(Cn){\displaystyle P_{n}'=C_{n-1}'\oplus E_{K}^{-1}(C_{n})}.
It follows that Bob will now compute the authentication tag using CBC-MAC over all the values of plain text which he decoded. The tag for the new message,t′{\displaystyle t'}, is given by:
Notice that this expression is equal to
which is exactlyCn{\displaystyle C_{n}}:
and it follows thatt′=Cn=t{\displaystyle t'=C_{n}=t}.
Therefore, Eve was able to modify the cipher text in transit (without necessarily knowing what plain text it corresponds to) such that an entirely different message,P′{\displaystyle P'}, was produced, but the tag for this message matched the tag of the original, and Bob was unaware that the contents had been modified in transit. By definition, a Message Authentication Code isbrokenif we can find a different message (a sequence of plain-text pairsP′{\displaystyle P'}) which produces the same tag as the previous message,P, withP≠P′{\displaystyle P\not =P'}. It follows that the message authentication protocol, in this usage scenario, has been broken, and Bob has been deceived into believing Alice sent him a message which she did not produce.
If, instead, we use different keys for the encryption and authentication stages, sayK1{\displaystyle K_{1}}andK2{\displaystyle K_{2}}, respectively, this attack is foiled. The decryption of the modified cipher-text blocksCi′{\displaystyle C_{i}'}obtains some plain text stringPi′{\displaystyle P_{i}'}. However, due to the MAC's usage of a different keyK2{\displaystyle K_{2}}, we cannot "undo" the decryption process in the forward step of the computation of the message authentication code so as to produce the same tag; each modifiedPi′{\displaystyle P_{i}'}will now be encrypted byK2{\displaystyle K_{2}}in the CBC-MAC process to some valueMACi≠Ci′{\displaystyle \mathrm {MAC} _{i}\not =C_{i}'}.
This example also shows that a CBC-MAC cannot be used as a collision-resistant one-way function: given a key it is trivial to create a different message which "hashes" to the same tag.
When encrypting data using a block cipher incipher block chaining(or another) mode, it is common to introduce aninitialization vectorto the first stage of the encryption process. It is typically required that this vector be chosen randomly (anonce) and that it is not repeated for any given secret key under which the block cipher operates. This provides semantic security, by means of ensuring the same plain text is not encrypted to the same cipher text, allowing an attacker to infer a relationship exists.
When computing a message authentication code, such as by CBC-MAC, the use of an initialization vector is a possible attack vector.
In the operation of a ciphertext block chaining cipher, the first block of plain text is mixed with the initialization vector using an exclusive OR (P1⊕IV{\displaystyle P_{1}\oplus IV}). The result of this operation is the input to the block cipher for encryption.
However, when performing encryption and decryption, we are required to send the initialization vector in plain text - typically as the block immediately preceding the first block of cipher text - such that the first block of plain text can be decrypted and recovered successfully. If computing a MAC, we will also need to transmit the initialization vector to the other party in plain text so that they can verify the tag on the message matches the value they have computed.
If we allow the initialization vector to be selected arbitrarily, it follows that the first block of plain text can potentially be modified (transmitting a different message) while producing the same message tag.
Consider a messageM1=P1|P2|…{\displaystyle M_{1}=P_{1}|P_{2}|\dots }. In particular, when computing the message tag for CBC-MAC, suppose we choose an initialization vectorIV1{\displaystyle IV_{1}}such that computation of the MAC begins withEK(IV1⊕P1){\displaystyle E_{K}(IV_{1}\oplus P_{1})}. This produces a (message, tag) pair(M1,T1){\displaystyle (M_{1},T_{1})}.
Now produce the messageM2=P1′|P2|…{\displaystyle M_{2}=P_{1}'|P_{2}|\dots }. For each bit modified inP1′{\displaystyle P_{1}'}, flip the corresponding bit in the initialization vector to produce the initialization vectorIV1′{\displaystyle IV_{1}'}. It follows that to compute the MAC for this message, we begin the computation byEK(P1′⊕IV1′){\displaystyle E_{K}(P_{1}'\oplus IV_{1}')}. As bits in both the plain text and initialization vector have been flipped in the same places, the modification is cancelled in this first stage, meaning the input to the block cipher is identical to that forM1{\displaystyle M_{1}}. If no further changes are made to the plain text, the same tag will be derived despite a different message being transmitted.
If the freedom to select an initialization vector is removed and all implementations of CBC-MAC fix themselves on a particular initialization vector (often the vector of zeroes, but in theory, it could be anything provided all implementations agree), this attack cannot proceed.
To sum up, if the attacker is able to set the IV that will be used for MAC verification, he can perform arbitrary modification of the first data block without invalidating the MAC.
Sometimes IV is used as a counter to prevent message replay attacks.
However, if the attacker can predict what IV will be used for MAC verification,
he or she can replay previously observed message by modifying the first data block to compensate for the change in the IV that will be used for the verification.
For example, if the attacker has observed messageM1=P1|P2|…{\displaystyle M_{1}=P_{1}|P_{2}|\dots }withIV1{\displaystyle IV_{1}}and knowsIV2{\displaystyle IV_{2}}, he can produceM1′=(P1⊕IV1⊕IV2)|P2|…{\displaystyle M_{1}'=(P_{1}\oplus IV_{1}\oplus IV_{2})|P_{2}|\dots }that will pass MAC verification withIV2{\displaystyle IV_{2}}.
The simplest countermeasure is to encrypt the IV before using it (i.e., prepending IV to the data). Alternatively MAC in CFB mode can be used, because in CFB mode the IV is encrypted before it is XORed with the data.
Another solution (in case protection against message replay attacks is not required) is to always use a zero vector IV.[13]Note that the above formula forM1′{\displaystyle M_{1}'}becomesM1′=(P1⊕0⊕0)|P2|⋯=P1|P2|⋯=M1{\displaystyle M_{1}'=(P_{1}\oplus 0\oplus 0)|P_{2}|\dots =P_{1}|P_{2}|\dots =M_{1}}. So sinceM1{\displaystyle M_{1}}andM1′{\displaystyle M_{1}'}are the same message, by definition they will have the same tag. This is not a forgery, rather the intended use of CBC-MAC.
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One-key MAC(OMAC) is a family ofmessage authentication codesconstructed from ablock ciphermuch like theCBC-MACalgorithm. It may be used to provide assurance of the authenticity and, hence, the integrity of data. Two versions are defined:
OMAC is free for all uses: it is not covered by any patents.[4]
The core of the CMAC algorithm is a variation ofCBC-MACthatBlackandRogawayproposed and analyzed under the name "XCBC"[5]and submitted toNIST.[6]The XCBC algorithm efficiently addresses the security deficiencies of CBC-MAC, but requires three keys.
Iwata and Kurosawa proposed an improvement of XCBC that requires less key material (just one key) and named the resulting algorithmOne-Key CBC-MAC(OMAC) in their papers.[1]They later submitted the OMAC1 (= CMAC),[2]a refinement of OMAC, and additional security analysis.[7]
To generate anℓ-bit CMAC tag (t) of a message (m) using ab-bit block cipher (E) and a secret key (k), one first generates twob-bit sub-keys (k1andk2) using the following algorithm (this is equivalent to multiplication byxandx2in afinite fieldGF(2b)). Let ≪ denote the standard left-shift operator and ⊕ denote bit-wiseexclusive or:
As a small example, supposeb= 4,C= 00112, andk0=Ek(0) = 01012. Thenk1= 10102andk2= 0100 ⊕ 0011 = 01112.
The CMAC tag generation process is as follows:
The verification process is as follows:
CMAC-C1[8]is a variant of CMAC that provides additionalcommitment and context-discovery securityguarantees.
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PMAC, which stands forparallelizable MAC, is amessage authentication codealgorithm. It was created byPhillip Rogaway.
PMAC is a method of taking ablock cipherand creating an efficient message authentication code that is reducible in security to the underlying block cipher.
PMAC is similar in functionality to theOMACalgorithm.
PMAC is no longer patented and can be used royalty-free. It was originally patented byPhillip Rogaway, but he has since abandoned his patent filings.[1]
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Cicada 3301is the name given to eight sets ofpuzzlesposted under the name "3301" online between 2012 and 2014. The first puzzle started on January 4, 2012,[1]on4chan[2]and ran for nearly a month. A second round of puzzles began one year later on January 4, 2013, and then a third round following the confirmation of a fresh clue posted onTwitteron January 4, 2014.[3][4]The third puzzle remains unsolved. The stated intent was to recruit "intelligent individuals" by presenting a series of puzzles to be solved; no new puzzles were published on January 4, 2015. A new clue was posted on Twitter on January 5, 2016.[5][6]Cicada 3301 posted their last verifiedOpenPGP-signed message in April 2017, denying the validity of any unsigned puzzle.[7]
The puzzles focused heavily ondata security,cryptography,steganography, andInternetanonymity.[8][9][10]It has been called "the most elaborate and mysterious puzzle of the Internet age",[11]and is listed as one of the "top 5 eeriest, unsolved mysteries of the Internet" byThe Washington Post,[12]and much speculation exists as to its function. Many have speculated that the puzzles are a recruitment tool for theNSA,CIA,[13]MI6, a "Masonic conspiracy",[14]or a cyber mercenary group.[2][8]Others have stated Cicada 3301 is analternate reality game, although no company or individual has attempted to monetize it.[11]
The stated purpose of the puzzles each year was to recruit "highly intelligent individuals", although the ultimate purpose remains unknown.[2]Theories have included claims that Cicada 3301 is a secret society with the goal of improvingcryptography,privacy, andanonymityor that it is a cult or religion.[15][16][17]According to statements of several people who won the 2012 puzzle, 3301 typically uses non-puzzle-based recruiting methods, but created the Cicada puzzles because they were looking for potential members with cryptography and computer security skills.[15]
The first puzzle, of 2013, was solved by Marcus Wanner.[18]According to him, those who solved the puzzles were asked questions about their support of information freedom, online privacy and freedom, and rejection of censorship. Those who answered satisfactorily at this stage were invited to a private forum, where they were instructed to devise and complete a project intended to further the ideals of the group.[15]He did not finish his work on a method of general decryption and the website was removed.[citation needed]
The Cicada 3301 clues spanned many different forms of communication media, including but not limited to the Internet, telephone, original music, bootableLinuxCDs,digital images, physical paper signs, and pages of unpublished cryptic books written in runes. In total, there were two pieces of music, titled "The Instar Emergence" and "Interconnectedness", accompanying the Cicada clues. However, neither of them were part ofa standard repertoire, and neither the composers nor performers have been identified. Cicada 3301 also wrote a book, titledLiber Primus(Latin forFirst Book), which contains many pages, only some of which have been decrypted. In addition to using many varying techniques to encrypt, encode, or hide data, these clues also referenced a wide variety of books, poetry, artwork, and music.[2]Each clue was signed by the sameOpenPGPprivate key to confirm authenticity.[10][19]
Authorities from theLos Andes ProvinceofChileclaimed that Cicada 3301 is a "hacker group" and engaged in illegal activities. Cicada 3301 responded to this claim by issuing aPGP-signed statement denying any involvement in illegal activity.[20][21]
In July 2015, a group calling themselves "3301" hacked intoPlanned Parenthood's database;[22]however, the group appeared to have had no association with Cicada 3301.[23]Cicada 3301 later issued a PGP-signed statement stating they "are not associated with this group in any way" and also stated that Cicada 3301 did not "condone their use of our name, number, or symbolism".[24]The hacker group later confirmed that they were not affiliated with Cicada 3301.[25]
TheUnited States Navyreleased a cryptographic challenge based on the Cicada 3301 recruitment puzzles in 2014 calling it Project Architeuthis.[26][27]
The plot of "Nautilus", a 2014 episode ofPerson of Interest, featured a large-scale game very similar to the Cicada 3301 puzzles. Both feature a series of worldwide cryptographic puzzles, but as the title implies, these feature the image of anautilusshell instead of a cicada logo.[28]Person of InterestcreatorJonathan Nolanand producer Greg Plageman stated in an interview that Cicada 3301 was the inspiration for the episode: "Episode 2, I'm particularly fascinated by the subject underneath it. Look up Cicada 3301 on the Internet. It's a very interesting concept out there that we then put into a larger story that connects to our show".[29]
In the video gameAssassin's Creed Originsreleased in 2017, a member of the Isu civilization references Cicada when listing off various mysteries of history.[30]
The organization is the subject of the 2021 comedy-thriller filmDark Web: Cicada 3301.[31][32]Directed byAlan Ritchson, who co-wrote the script with Joshua Montcalm, it starsJack Kesy,Conor Leslie,Ron Funches,Kris Holden-Ried, Andreas Apergis, and director Ritchson. The film follows a hacker who participates in Cicada's recruitment game while evading theNational Security Agency(NSA).[33][34]
The Cicada 3301 puzzles play a major role in thevisual novelAnonymous;Code.
The Cicada 3301 puzzles have seemingly inspired another competition named Cicada Detroit focused on "decoding ciphers, cryptography and hidden messages"[35]
As of June 2024, aransomwaregroup calling themselvesCicada 3301started spreading ransomware. There is no evidence the group is affiliated with the original puzzles.[36]
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Inherent viceis the tendency in physical objects to deteriorate because of the fundamental instability of the components of which they are made, as opposed to deterioration caused by external forces.[1][2]All objects have some kind of inherent vice as a result of the baseline law ofentropy.
The term is broadly used inarchivalpractice to recognize the material constraints ofpreservationactivities. For example, many kinds of paper have acid in them that makes them chemically unstable. Over time, the acid will eat away the text on the page and cause paper to turn yellow or brown and become brittle. As the acid continues to break down thecellulosefibers, the paper disintegrates.[3]In the world ofphilately, the adhesive on the back of stamps is both an inherent vice—any exposure to moisture will compromise their ability to be preserved—as well as the purpose for which the stamps were made.[3]In the case of film, an example of inherent vice is the innate chemical instability ofcellulose acetate film, which can result in the degradation known as "vinegar syndrome" due to the distinctive vinegar odor it produces.[4]
Slowing this tendency of objects to self-destruct requires an understanding of how materials interact. This includes not just an understanding of the intrinsic qualities of the materials themselves, but also the way that they affect and are affected by the other materials that they come into contact with.[5]For example, leather and metal are two materials which are frequently used in combination with each other, but react to each other over time to causecorrosionon the metal.[3]
The presence of deteriorating agents is a problem which can be tempered by selecting archival quality materials, such as acid free paper.[5]However, frequently the objective of manufacturers is to make a process (i.e. papermaking, book binding, etc.) faster and easier; the longevity of the items they produce is not their primary concern.[2]
The term inherent vice is used in law as well as in library and archival science. One legal definition of inherent vice is "an exclusion found in most property insurance policies eliminating coverage for loss caused by a quality in property that causes it to damage or destroy itself."[6]
Inherent vice can be used as a justification for refusing to insure an item, as its intrinsically self-destructive nature may make it unacceptable risk to a carrier or insurer.[7]
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Acryptographickeyis calledephemeralif it is generated for each execution of a key establishment process.[1]In some cases ephemeral keys are used more than once, within a single session (e.g., in broadcast applications) where the sender generates only one ephemeral key pair per message and theprivate keyis combined separately with each recipient'spublic key. Contrast with astatic key.
Private (resp. public) ephemeral key agreement keys are the private (resp. public) keys of asymmetric key pairs that are used a single key establishment transaction to establish one or more keys (e.g., key wrapping keys, data encryption keys, orMACkeys) and, optionally, other keying material (e.g.,initialization vectors).
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Random number generationis a process by which, often by means of arandom number generator(RNG), a sequence ofnumbersorsymbolsis generated that cannot be reasonably predicted better than byrandomchance. This means that the particular outcome sequence will contain some patterns detectable in hindsight but impossible to foresee. True random number generators can behardware random-number generators(HRNGs), wherein each generation is a function of the current value of a physical environment's attribute that is constantly changing in a manner that is practically impossible to model. This would be in contrast to so-called "random number generations" done bypseudorandom number generators(PRNGs), which generate numbers that only look random but are in fact predetermined—these generations can be reproduced simply by knowing the state of the PRNG.[1]
Variousapplications of randomnesshave led to the development of different methods for generatingrandomdata. Some of these have existed since ancient times, including well-known examples like the rolling ofdice,coin flipping, theshufflingofplaying cards, the use ofyarrowstalks (fordivination) in theI Ching, as well as countless other techniques. Because of the mechanical nature of these techniques, generating large quantities of sufficiently random numbers (important in statistics) required much work and time. Thus, results would sometimes be collected and distributed asrandom number tables.
Several computational methods for pseudorandom number generation exist. All fall short of the goal of true randomness, although they may meet, with varying success, some of thestatistical tests for randomnessintended to measure how unpredictable their results are (that is, to what degree their patterns are discernible). This generally makes them unusable for applications such ascryptography. However, carefully designedcryptographically secure pseudorandom number generators(CSPRNGS) also exist, with special features specifically designed for use in cryptography.
Random number generators have applications ingambling,statistical sampling,computer simulation,cryptography,completely randomized design, and other areas where producing an unpredictable result is desirable. Generally, in applications having unpredictability as the paramount feature, such as in security applications,hardware generatorsare generally preferred over pseudorandom algorithms, where feasible.
Pseudorandom number generators are very useful in developingMonte Carlo-methodsimulations, asdebuggingis facilitated by the ability to run the same sequence of random numbers again by starting from the samerandom seed. They are also used in cryptography – so long as theseedis secret. The sender and receiver can generate the same set of numbers automatically to use as keys.
The generation ofpseudorandom numbersis an important and common task in computer programming. While cryptography and certain numerical algorithms require a very high degree ofapparentrandomness, many other operations only need a modest amount of unpredictability. Some simple examples might be presenting a user with a "random quote of the day", or determining which way a computer-controlled adversary might move in a computer game. Weaker forms ofrandomnessare used inhash algorithmsand in creatingamortizedsearchingandsorting algorithms.
Some applications that appear at first sight to be suitable forrandomizationare in fact not quite so simple. For instance, a system that "randomly" selects music tracks for a background music system must onlyappearrandom, and may even have ways to control the selection of music: a truly random system would have no restriction on the same item appearing two or three times in succession.
There are two principal methods used to generate random numbers. The first method measures some physical phenomenon that is expected to be random and then compensates for possible biases in the measurement process. Example sources include measuringatmospheric noise, thermal noise, and other external electromagnetic and quantum phenomena. For example, cosmic background radiation or radioactive decay as measured over short timescales represent sources of naturalentropy(as a measure of unpredictability or surprise of the number generation process).
The speed at which entropy can be obtained from natural sources is dependent on the underlying physical phenomena being measured. Thus, sources of naturally occurringtrueentropy are said to beblocking– they are rate-limited until enough entropy is harvested to meet the demand. On some Unix-like systems, including mostLinux distributions, the pseudo device file/dev/randomwill block until sufficient entropy is harvested from the environment.[2]Due to this blocking behavior, large bulk reads from/dev/random, such as filling ahard disk drivewith random bits, can often be slow on systems that use this type of entropy source.
The second method uses computationalalgorithmsthat can produce long sequences of apparently random results, which are in fact completely determined by a shorter initial value, known as a seed value orkey. As a result, the entire seemingly random sequence can be reproduced if the seed value is known. This type of random number generator is often called apseudorandom number generator. This type of generator typically does not rely on sources of naturally occurring entropy, though it may be periodically seeded by natural sources. This generator type is non-blocking, so they are not rate-limited by an external event, making large bulk reads a possibility.
Some systems take a hybrid approach, providing randomness harvested from natural sources when available, and falling back to periodically re-seeded software-basedcryptographically secure pseudorandom number generators(CSPRNGs). The fallback occurs when the desired read rate of randomness exceeds the ability of the natural harvesting approach to keep up with the demand. This approach avoids the rate-limited blocking behavior of random number generators based on slower and purely environmental methods.
While a pseudorandom number generator based solely on deterministic logic can never be regarded as atruerandom number source in the purest sense of the word, in practice they are generally sufficient even for demanding security-critical applications. Carefully designed and implemented pseudorandom number generators can be certified for security-critical cryptographic purposes, as is the case with theyarrow algorithmandfortuna. The former is the basis of the/dev/randomsource of entropy onFreeBSD,AIX,macOS,NetBSD, and others.OpenBSDuses a pseudorandom number algorithm known asarc4random.[dubious–discuss][3]
The earliest methods for generating random numbers, such as dice, coin flipping and roulette wheels, are still used today, mainly in games and gambling as they tend to be too slow for most applications in statistics and cryptography.
Ahardware random number generatorcan be based on an essentially random atomic or subatomic physical phenomenon whose unpredictability can be traced to the laws ofquantum mechanics.[4][5]Sources ofentropyincluderadioactive decay,thermal noise,shot noise, avalanche noise inZener diodes,clock drift, the timing of actual movements of ahard diskread-write head, andradio noise. However, physical phenomena and tools used to measure them generally feature asymmetries andsystematic biasesthat make their outcomes not uniformly random. Arandomness extractor, such as acryptographic hash function, can be used to approach a uniform distribution of bits from a non-uniformly random source, though at a lower bit rate.
The appearance of wideband photonic entropy sources, such asoptical chaosandamplified spontaneous emissionnoise, greatly aid the development of the physical random number generator. Among them, optical chaos[6][7]has a high potential to physically produce high-speed random numbers due to its high bandwidth and large amplitude. A prototype of a high-speed, real-time physical random bit generator based on a chaotic laser was built in 2013.[8]
Various imaginative ways of collecting this entropic information have been devised. One technique is to run a hash function against a frame of a video stream from an unpredictable source.Lavarandused this technique with images of a number oflava lamps.HotBitsmeasured radioactive decay withGeiger–Muller tubes,[9]whileRandom.orguses variations in the amplitude of atmospheric noise recorded with a normal radio.
Another common entropy source is the behavior of human users of the system. While people are not considered good randomness generators upon request, they generate random behavior quite well in the context of playingmixed strategygames.[10]Some security-related computer software requires the user to make a lengthy series of mouse movements or keyboard inputs to create sufficient entropy needed to generate randomkeysor to initialize pseudorandom number generators.[11]
Most computer-generated random numbers use PRNGs which are algorithms that can automatically create long runs of numbers with good random properties but eventually the sequence repeats (or the memory usage grows without bound). These random numbers are fine in many situations but are not as random as numbers generated from electromagnetic atmospheric noise used as a source of entropy.[12]The series of values generated by such algorithms is generally determined by a fixed number called aseed. One of the most commonPRNGis thelinear congruential generator, which uses the recurrence
to generate numbers, wherea,bandmare large integers, andXn+1{\displaystyle X_{n+1}}is the next inXas a series of pseudorandom numbers. The maximum number of numbers the formula can produce is themodulus,m. The recurrence relation can be extended to matrices to have much longer periods and better statistical properties
.[13]To avoid certain non-random properties of a single linear congruential generator, several such random number generators with slightly different values of the multiplier coefficient,a, can be used in parallel, with amasterrandom number generator that selects from among the several different generators.
A simple pen-and-paper method for generating random numbers is the so-calledmiddle-square methodsuggested byJohn von Neumann. While simple to implement, its output is of poor quality. It has a very short period and severe weaknesses, such as the output sequence almost always converging to zero. A recent innovation is to combine the middle square with aWeyl sequence. This method produces high-quality output through a long period.[14]
Most computer programming languages include functions or library routines that provide random number generators. They are often designed to provide a random byte or word, or afloating pointnumberuniformly distributedbetween 0 and 1.
The quality i.e. randomness of such library functions varies widely from completely predictable output, to cryptographically secure. The default random number generator in many languages, including Python, Ruby, R, IDL and PHP is based on theMersenne Twisteralgorithm and isnotsufficient for cryptography purposes, as is explicitly stated in the language documentation. Such library functions often have poor statistical properties, and some will repeat patterns after only tens of thousands of trials. They are often initialized using a computer'sreal-time clockas the seed, since such a clock is 64 bit and measures in nanoseconds, far beyond the person'sprecision. These functions may provide enough randomness for certain tasks (for example video games) but are unsuitable where high-quality randomness is required, such as in cryptography applications, or statistics.[15]
Much higher quality random number sources are available on most operating systems; for example/dev/randomon various BSD flavors, Linux, Mac OS X, IRIX, and Solaris, orCryptGenRandomfor Microsoft Windows. Most programming languages, including those mentioned above, provide a means to access these higher-quality sources.
Random number generation may also be performed by humans, in the form of collecting various inputs fromend usersand using them as a randomization source. However, most studies find that human subjects have some degree of non-randomness when attempting to produce a random sequence of e.g. digits or letters. They may alternate too much between choices when compared to a good random generator;[16]thus, this approach is not widely used. However, for the very reason that humans perform poorly in this task, human random number generation can be used as a tool to gain insights into brain functions otherwise not accessible.[17]
Even given a source of plausible random numbers (perhaps from a quantum mechanically based hardware generator), obtaining numbers which are completely unbiased takes care. In addition, behavior of these generators often changes with temperature, power supply voltage, the age of the device, or other outside interference.
Generated random numbers are sometimes subjected to statistical tests before use to ensure that the underlying source is still working, and then post-processed to improve their statistical properties. An example would be the TRNG9803[18]hardware random number generator, which uses an entropy measurement as a hardware test, and then post-processes the random sequence with a shift register stream cipher. It is generally hard to use statistical tests to validate the generated random numbers. Wang and Nicol[19]proposed a distance-based statistical testing technique that is used to identify the weaknesses of several random generators. Li and Wang[20]proposed a method of testing random numbers based on laser chaotic entropy sources using Brownian motion properties.
Statistical tests are also used to give confidence that the post-processed final output from a random number generator is truly unbiased, with numerousrandomness testsuites being developed.
Most random number generators natively work with integers or individual bits, so an extra step is required to arrive at thecanonicaluniform distribution between 0 and 1. The implementation is not as trivial as dividing the integer by its maximum possible value. Specifically:[21][22]
The mainstream algorithm, used byOpenJDK,Rust, andNumPy, is described in a proposal forC++'s STL. It does not use the extra precision and suffers from bias only in the last bit due to round-to-even.[23]Other numeric concerns are warranted when shifting thiscanonicaluniform distribution to a different range.[24]A proposed method for theSwift programming languageclaims to use the full precision everywhere.[25]
Uniformly distributed integers are commonly used in algorithms such as theFisher–Yates shuffle. Again, a naive implementation may induce a modulo bias into the result, so more involved algorithms must be used. A method that nearly never performs division was described in 2018 by Daniel Lemire,[26]with the current state-of-the-art being the arithmetic encoding-inspired 2021 "optimal algorithm" by Stephen Canon ofApple Inc.[27]
Most 0 to 1 RNGs include 0 but exclude 1, while others include or exclude both.
Given a source of uniform random numbers, there are a couple of methods to create a new random source that corresponds to aprobability density function. One method called theinversion method, involves integrating up to an area greater than or equal to the random number (which should be generated between 0 and 1 for proper distributions). A second method called theacceptance-rejection method, involves choosing an x and y value and testing whether the function of x is greater than the y value. If it is, the x value is accepted. Otherwise, the x value is rejected and the algorithm tries again.[28][29]
As an example for rejection sampling, to generate a pair ofstatistically independentstandard normally distributedrandom numbers (x,y), one may first generate thepolar coordinates(r,θ), wherer2~χ22andθ~UNIFORM(0,2π)(seeBox–Muller transform).
The outputs of multiple independent RNGs can be combined (for example, using a bit-wiseXORoperation) to provide a combined RNG at least as good as the best RNG used. This is referred to assoftware whitening.
Computational and hardware random number generators are sometimes combined to reflect the benefits of both kinds. Computational random number generators can typically generate pseudorandom numbers much faster than physical generators, while physical generators can generate true randomness.
Some computations making use of a random number generator can be summarized as the computation of a total or average value, such as the computation of integrals by theMonte Carlo method. For such problems, it may be possible to find a more accurate solution by the use of so-calledlow-discrepancy sequences, also calledquasirandomnumbers. Such sequences have a definite pattern that fills in gaps evenly, qualitatively speaking; a truly random sequence may, and usually does, leave larger gaps.
The following sites make available random number samples:
Since much cryptography depends on a cryptographically secure random number generator for key andcryptographic noncegeneration, if a random number generator can be made predictable, it can be used asbackdoorby an attacker to break the encryption.
The NSA is reported to have inserted a backdoor into theNISTcertifiedcryptographically secure pseudorandom number generatorDual EC DRBG. If for example an SSL connection is created using this random number generator, then according toMatthew Greenit would allow NSA to determine the state of the random number generator, and thereby eventually be able to read all data sent over the SSL connection.[30]Even though it was apparent that Dual_EC_DRBG was a very poor and possibly backdoored pseudorandom number generator long before the NSA backdoor was confirmed in 2013, it had seen significant usage in practice until 2013, for example by the prominent security companyRSA Security.[31]There have subsequently been accusations that RSA Security knowingly inserted a NSA backdoor into its products, possibly as part of theBullrun program. RSA has denied knowingly inserting a backdoor into its products.[32]
It has also been theorized that hardware RNGs could be secretly modified to have less entropy than stated, which would make encryption using the hardware RNG susceptible to attack. One such method that has been published works by modifying the dopant mask of the chip, which would be undetectable to optical reverse-engineering.[33]For example, for random number generation in Linux, it is seen as unacceptable to use Intel'sRDRANDhardware RNG without mixing in the RDRAND output with other sources of entropy to counteract any backdoors in the hardware RNG, especially after the revelation of the NSA Bullrun program.[34][35]
In 2010,a U.S. lottery draw was riggedby the information security director of theMulti-State Lottery Association(MUSL), who surreptitiously installed backdoormalwareon the MUSL's secure RNG computer during routine maintenance.[36]During the hacks the man won a total amount of $16,500,000 over multiple years.
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Incryptography,forward secrecy(FS), also known asperfect forward secrecy(PFS), is a feature of specifickey-agreement protocolsthat gives assurances thatsession keyswill not be compromised even if long-term secrets used in the session key exchange are compromised, limiting damage.[1][2][3]ForTLS, the long-term secret is typically theprivate keyof the server. Forward secrecy protects past sessions against future compromises of keys or passwords. By generating a unique session key for every session a user initiates, the compromise of a single session key will not affect any data other than that exchanged in the specific session protected by that particular key. This by itself is not sufficient for forward secrecy which additionally requires that a long-term secret compromise does not affect the security of past session keys.
Forward secrecy protects data on thetransport layerof a network that uses common transport layer security protocols, includingOpenSSL,[4]when its long-term secret keys are compromised, as with theHeartbleedsecurity bug. If forward secrecy is used, encrypted communications and sessions recorded in the past cannot be retrieved and decrypted should long-term secret keys or passwords be compromised in the future, even if the adversary actively interfered, for example via aman-in-the-middle (MITM) attack.
The value of forward secrecy is that it protects past communication. This reduces the motivation for attackers to compromise keys. For instance, if an attacker learns a long-term key, but the compromise is detected and the long-term key is revoked and updated, relatively little information is leaked in a forward secure system.
The value of forward secrecy depends on the assumed capabilities of an adversary. Forward secrecy has value if an adversary is assumed to be able to obtain secret keys from a device (read access) but is either detected or unable to modify the way session keys are generated in the device (full compromise). In some cases an adversary who can read long-term keys from a device may also be able to modify the functioning of the session key generator, as in the backdooredDual Elliptic Curve Deterministic Random Bit Generator. If an adversary can make the random number generator predictable, then past traffic will be protected but all future traffic will be compromised.
The value of forward secrecy is limited not only by the assumption that an adversary will attack a server by only stealing keys and not modifying the random number generator used by the server but it is also limited by the assumption that the adversary will only passively collect traffic on the communications link and not be active using a man-in-the-middle attack. Forward secrecy typically uses an ephemeralDiffie–Hellman key exchangeto prevent reading past traffic. The ephemeral Diffie–Hellman key exchange is often signed by the server using a static signing key. If an adversary can steal (or obtain through a court order) this static (long term) signing key, the adversary can masquerade as the server to the client and as the client to the server and implement a classic man-in-the-middle attack.[5]
The term "perfect forward secrecy" was coined by C. G. Günther in 1990[6]and further discussed byWhitfield Diffie,Paul van Oorschot, and Michael James Wiener in 1992,[7]where it was used to describe a property of the Station-to-Station protocol.[8]
Forward secrecy has also been used to describe the analogous property ofpassword-authenticated key agreementprotocols where the long-term secret is a (shared)password.[9]
In 2000 theIEEEfirst ratifiedIEEE 1363, which establishes the related one-party and two-party forward secrecy properties of various standard key agreement schemes.[10]
An encryption system has the property of forward secrecy if plain-text (decrypted) inspection of the data exchange that occurs during key agreement phase of session initiation does not reveal the key that was used to encrypt the remainder of the session.
The following is a hypothetical example of a simple instant messaging protocol that employs forward secrecy:
Forward secrecy (achieved by generating new session keys for each message) ensures that past communications cannot be decrypted if one of the keys generated in an iteration of step 2 is compromised, since such a key is only used to encrypt a single message. Forward secrecy also ensures that past communications cannot be decrypted if the long-term private keys from step 1 are compromised. However, masquerading as Alice or Bob would be possible going forward if this occurred, possibly compromising all future messages.
Forward secrecy is designed to prevent the compromise of a long-term secret key from affecting the confidentiality of past conversations. However, forward secrecy cannot defend against a successfulcryptanalysisof the underlyingciphersbeing used, since a cryptanalysis consists of finding a way to decrypt an encrypted message without the key, and forward secrecy only protects keys, not the ciphers themselves.[11]A patient attacker can capture a conversation whose confidentiality is protected through the use ofpublic-key cryptographyand wait until the underlying cipher is broken (e.g. largequantum computerscould be created which allow thediscrete logarithm problemto be computed quickly), a.k.a.harvest now, decrypt laterattacks. This would allow the recovery of old plaintexts even in a system employing forward secrecy.
Non-interactive forward-secure key exchange protocols face additional threats that are not relevant to interactive protocols. In amessage suppressionattack, an attacker in control of the network may itself store messages while preventing them from reaching the intended recipient; as the messages are never received, the corresponding private keys may not be destroyed or punctured, so a compromise of the private key can lead to successful decryption. Proactively retiring private keys on a schedule mitigates, but does not eliminate, this attack. In amalicious key exhaustionattack, the attacker sends many messages to the recipient and exhausts the private key material, forcing a protocol to choose between failing closed (and enablingdenial of serviceattacks) or failing open (and giving up some amount of forward secrecy).[12]
Most key exchange protocols areinteractive, requiring bidirectional communication between the parties. A protocol that permits the sender to transmit data without first needing to receive any replies from the recipient may be callednon-interactive, orasynchronous, orzero round trip(0-RTT).[13][14]
Interactivity is onerous for some applications—for example, in a secure messaging system, it may be desirable to have astore-and-forwardimplementation, rather than requiring sender and recipient to be online at the same time; loosening the bidirectionality requirement can also improve performance even where it is not a strict requirement, for example at connection establishment or resumption. These use cases have stimulated interest in non-interactive key exchange, and, as forward security is a desirable property in a key exchange protocol, in non-interactive forward secrecy.[15][16]This combination has been identified as desirable since at least 1996.[17]However, combining forward secrecy and non-interactivity has proven challenging;[18]it had been suspected that forward secrecy with protection againstreplay attackswas impossible non-interactively, but it has been shown to be possible to achieve all three desiderata.[14]
Broadly, two approaches to non-interactive forward secrecy have been explored,pre-computed keysandpuncturable encryption.[16]
With pre-computed keys, many key pairs are created and the public keys shared, with the private keys destroyed after a message has been received using the corresponding public key. This approach has been deployed as part of theSignal protocol.[19]
In puncturable encryption, the recipient modifies their private key after receiving a message in such a way that the new private key cannot read the message but the public key is unchanged.Ross J. Andersoninformally described a puncturable encryption scheme for forward secure key exchange in 1997,[20]andGreen & Miers (2015)formally described such a system,[21]building on the related scheme ofCanetti, Halevi & Katz (2003), which modifies the private key according to a schedule so that messages sent in previous periods cannot be read with the private key from a later period.[18]Green & Miers (2015)make use ofhierarchical identity-based encryptionandattribute-based encryption, whileGünther et al. (2017)use a different construction that can be based on any hierarchical identity-based scheme.[22]Dallmeier et al. (2020)experimentally found that modifyingQUICto use a 0-RTT forward secure and replay-resistant key exchange implemented with puncturable encryption incurred significantly increased resource usage, but not so much as to make practical use infeasible.[23]
Weak perfect forward secrecy (Wpfs) is the weaker property whereby when agents' long-term keys are compromised, the secrecy of previously established session-keys is guaranteed, but only for sessions in which the adversary did not actively interfere. This new notion, and the distinction between this and forward secrecy was introduced by Hugo Krawczyk in 2005.[24][25]This weaker definition implicitly requires that full (perfect) forward secrecy maintains the secrecy of previously established session keys even in sessions where the adversarydidactively interfere, or attempted to act as a man in the middle.
Forward secrecy is present in several protocol implementations, such asSSHand as an optional feature inIPsec(RFC 2412).Off-the-Record Messaging, a cryptography protocol and library for many instant messaging clients, as well asOMEMOwhich provides additional features such as multi-user functionality in such clients, both provide forward secrecy as well asdeniable encryption.
InTransport Layer Security(TLS),cipher suitesbased onDiffie–Hellmankey exchange (DHE-RSA, DHE-DSA) andelliptic curve Diffie–Hellmankey exchange (ECDHE-RSA, ECDHE-ECDSA) are available. In theory, TLS can use forward secrecy since SSLv3, but many implementations do not offer forward secrecy or provided it with lower grade encryption.[26]TLS 1.3 removed support for RSA for key exchange, leaving Diffie-Hellman (with forward-secrecy) as the sole algorithm for key exchange.[27]
OpenSSLsupports forward secrecy usingelliptic curve Diffie–Hellmansince version 1.0,[28]with a computational overhead of approximately 15% for the initial handshake.[29]
TheSignal Protocoluses theDouble Ratchet Algorithmto provide forward secrecy.[30]
On the other hand, among popular protocols currently in use,WPA Personaldid not support forward secrecy before WPA3.[31]
Since late 2011, Google provided forward secrecy with TLS by default to users of itsGmailservice,Google Docsservice, and encrypted search services.[28]Since November 2013,Twitterprovided forward secrecy with TLS to its users.[32]Wikishosted by theWikimedia Foundationhave all provided forward secrecy to users since July 2014[33]and are requiring the use of forward secrecy since August 2018.
Facebook reported as part of an investigation into email encryption that, as of May 2014, 74% of hosts that supportSTARTTLSalso provide forward secrecy.[34]TLS 1.3, published in August 2018, dropped support for ciphers without forward secrecy. As of February 2019[update], 96.6% of web servers surveyed support some form of forward secrecy, and 52.1% will use forward secrecy with most browsers.[35]
At WWDC 2016, Apple announced that all iOS apps would need to use App Transport Security (ATS), a feature which enforces the use of HTTPS transmission. Specifically, ATS requires the use of an encryption cipher that provides forward secrecy.[36]ATS became mandatory for apps on January 1, 2017.[37]
TheSignalmessaging application employs forward secrecy in its protocol, notably differentiating it from messaging protocols based onPGP.[38]
Forward secrecy is supported on 92.6% of websites on modern browsers, while 0.3% of websites do not support forward secrecy at all as of May 2024.[39]
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40-track modeis asteganographictechnique that allows for hidden data on a 3.5 inchfloppy diskette.
A 3.5 inch 1.44MB mini-floppy diskette contains 80 tracks, 18 sectors per track, and 512 bytes per sector. A 3.5 inch 720k diskette contains 80 tracks, 9 sectors per track, and 512 bytes per sector. This technique refers to formatting an 80-track 1.44MB diskette as a special 40-track 720KB diskette.
In doing so one may create, in effect, two 40-track partitions. The former of these partitions is visible and usable as in a normal diskette, and the latter hidden.
One may then fill the unallocated (hidden) 40 tracks with up to 720KB of secret or encrypted data, that will not be superficially visible to a user.
Writing a 1.44MB floppy in 40-track mode causes the allocated tracks to be written
to the actual even numbered tracks, thus causing drives attempting to read a 1.44MB diskette as a 720KB diskette to become confused because of the strange data in between even numbered tracks. The hidden data then resides on the odd numbered tracks.
Generally, device drivers only copy allocated data, and thus traditional copies of such a disk would generally not contain the hidden data.
Equivalents of this technique can easily be done on almost any media.
This technique is different than the "40th track" copy protection schemes used during the 80s and 90s.
TheKGBand seniorFBIagentRobert Hanssenused this technique to communicate with one another between 1985 and 2001.[1]
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Anacrosticis apoemor other word composition in which thefirstletter (or syllable, or word) of each new line (orparagraph, or other recurring feature in the text) spells out a word, message or the alphabet.[1]The term comes from the Frenchacrostichefrom post-classicalLatinacrostichis, fromKoine Greekἀκροστιχίς, fromAncient Greekἄκρος"highest, topmost" andστίχος"verse".[2]As a form ofconstrained writing, an acrostic can be used as amnemonicdevice to aid memory retrieval. When thelastletter of each new line (or other recurring feature) forms a word it is called atelestich(ortelestic); the combination of an acrostic and a telestich in the same composition is called adouble acrostic(e.g. the first-century LatinSator Square).
Acrostics are common in medieval literature, where they usually serve to highlight the name of the poet or his patron, or to make a prayer to a saint. They are most frequent in verse works but can also appear in prose. The Middle High German poetRudolf von Emsfor example opens all his great works with an acrostic of his name, and his world chronicle marks the beginning of each age with an acrostic of the key figure (Moses, David, etc.). In chronicles, acrostics are common in German and English but rare in other languages.[3]
Relatively simple acrostics may merely spell out the letters of the alphabet in order; such an acrostic may be called an 'alphabetical acrostic' orabecedarius. These acrostics occur in theHebrew Biblein the first four of the five chapters of theBook of Lamentations, in the praise of the good wife inProverbs 31:10-31, and inPsalms9-10,25,34,37,111,112,119and145.[4]Notable among the acrostic Psalms is the longPsalm 119, which typically is printed in subsections named after the 22 letters of theHebrew alphabet, each section consisting of 8 verses, each of which begins with the same letter of the alphabet and the entire psalm consisting of 22 x 8 = 176 verses; andPsalm 145, which is recited three times a day in theJewish services. Some acrostic psalms are technically imperfect. For example,Psalm 9andPsalm 10appear to constitute a single acrostic psalm together, but the length assigned to each letter is unequal and five of the 22 letters of the Hebrew alphabet are not represented and the sequence of two letters is reversed. In Psalm 25 one Hebrew letter is not represented, the following letter (Resh) repeated. In Psalm 34 the current final verse, 23, does fit verse 22 in content, but adds an additional line to the poem. In Psalms 37 and 111 the numbering of verses and the division into lines are interfering with each other; as a result in Psalm 37, for the lettersDalethandKaphthere is only one verse, and the letterAyinis not represented. Psalm 111 and 112 have 22 lines, but 10 verses. Psalm 145 does not represent the letterNun, having 21 one verses, but one Qumran manuscript of this Psalm does have that missing line, which agrees with theSeptuagint. Some, like O Palmer Robertson, see the acrostic Psalms of book 1 and book 5 of Psalms as teaching and memory devices as well as transitions between subjects in the structure of the Psalms.[5]
Often the ease of detectability of an acrostic can depend on the intention of its creator. In some cases an author may desire an acrostic to have a better chance of being perceived by an observant reader, such as the acrostic contained in theHypnerotomachia Poliphili(where the key capital letters are decorated with ornate embellishments). However, acrostics may also be used as a form ofsteganography, where the author seeks to conceal the message rather than proclaim it. This might be achieved by making the key letters uniform in appearance with the surrounding text, or by aligning the words in such a way that the relationship between the key letters is less obvious. These are referred to asnull ciphersin steganography, using the first letter of each word to form a hidden message in an otherwise innocuous text.[6]Using letters to hide a message, as in acrostic ciphers, was popular during theRenaissance, and could employ various methods of enciphering, such as selecting other letters than initials based on a repeating pattern (equidistant letter sequences), or even concealing the message by starting at the end of the text and working backwards.[7]
A well-known acrostic in Greek is for the phraseJESUS CHRIST, GOD’S SON, SAVIOUR, the initial letters of which spellΙΧΘΥΣ(ICHTHYS), which meansfish:
According toCicero, acrostics were a regular feature ofSibylline prophecies(which were written in Greekhexameters. The type of acrostic is that known as a “gamma acrostic” (from the shape of the Greek letterΓ), where the same words are found both horizontally and vertically.[8]Cicero refers to an acrostic in this passage using the Greek wordἀκροστιχίς.
The 3rd-century BC didactic poetAratus, who was much admired and imitated by Cicero, Virgil and other Latin writers, appears to have started a fashion for using acrostics. One example is the famous passage inPhaenomena783–7 where the wordλεπτή'slender, subtle'occurs as a gamma acrostic and also twice in the text, as well as diagonally in the text and even cryptically taking the initial letters of certain words in lines 2 and 1:[9]
Several acrostics have recently been discovered in Roman poets, especially inVirgil. Among others, inEclogue9the acrosticVNDIS'in the waves'(lines 34–38) immediately precedes the wordsquis est nam ludus in undis?'for what is your game in the waves?'', andDEA DIO(i.e.dea Dione'the goddess Dione') (lines 46–51) in a passage which mentions the goddessDione(another name forVenus).[10]InEclogue8, alongside a passage dedicating the poem to an unnamed person and asking him to accept it, Neil Adkin reads the wordsTV SI ES ACI(i.e.accipe) ('if you are the one, accept!').[10]
InAeneid7.601–4, a passage which mentionsMarsand war, describing the custom of opening the gates of theTemple of Janus, the nameMARS(the god of war) appears in acrostic form as well as in the text as follows:[11]
InGeorgics1 429–433, next to a passage which contains the wordsnamque is certissimus auctor'for he is the most certain author', the double-letter reverse acrosticMA VE PV(i.e. Publius Vergilius Maro) is found on alternate lines.[10]
InEclogue 6, 13–24 Virgil uses a double acrostic, with the same wordLAESIS'for those who have been harmed'going both upwards and downwards starting from the same letter L in line 19.[12]Another double acrostic is found inAeneid 2, where the wordPITHI(i.e.πείθει, Greek for he ‘persuades’ or ‘he deceives’) is found first backwards at 103–107, then forwards at 142–146, at the beginning and end of a speech by Sinon persuading the Trojans to bring the wooden horse into the city.[13]The discoverer of this acrostic, Neil Adkin, points out that the same wordπείθειoccurs at more or less exactly the same line-numbers in a repeated line describing how Odysseus’ wife Penelope deceived the suitors inOdyssey2.106 and 24.141.
Another transliterated Greek word used as an acrostic in a pseudo-Sibylline prophecy has recently been noticed in the syllablesDE CA TE(i.e. Greekδεκάτη'tenth') inEclogue 4, 9–11, with the sameDEC A TErepeated cryptically both forwards and backwards in line 11.[14]
In another pseudo-Sibylline prophecy in poem 5 ofTibullus book 2the wordsAVDI ME‘hear me!’ are picked out in the first letters of alternate lines at the beginning of the prophecy.[15]
Virgil’s friendHoracealso made occasional use of acrostics, but apparently much less than Virgil. Examples areDISCE‘learn!’ (Odes1.18.11–15) (forming a gamma acrostic with the worddiscernunt'they discern'in line 18) andOTIA'leisure'inSatires1.2.7–10, which appears just after Horace has been advised to take a rest from writing satire. The acrosticOTIAalso occurs inOvid,Metamorphoses15.478–81, a passage describing the return of the peace-loving kingNuma Pompiliusto Rome.[16]Odes4.2, which starts with the wordPindarum'(the poet) Pindar' has next to it the truncated acrostic PIN in a gamma formation.[17]In the first poem of Horace'sEpodes(which were also known asIambi'iambics'), the first two lines beginibis ... amice, and it has been suggested that these words were deliberately chosen so that their initial letters IBI ... AM could be rearranged to read IAMBI.[18]
Towards the end of the 2nd century AD[19]a verse-summary of the plot was added to each of the plays ofPlautus. Each of these has an acrostic of the name of the play, for example:
The 3rd century AD poetCommodianwrote a series of 80 short poems on Christian themes calledInstructiones. Each of these is fully acrostic (with the exception of poem 60, where the initial letters are in alphabetical order), starting withPRAEFATIO‘preface’ andINDIGNATIO DEI‘the wrath of God’. The initials of poem 80, read backwards, giveCOMMODIANUS MENDICUS CHRISTI‘Commodian, Christ’s beggar’.
Chapters 2–5 of Book 12 in theRight Ginza, aMandaic text, are acrostic hymns, with each stanza ordered according to a letter of theMandaic alphabet.[20]
There is an acrostic secreted in the Dutch national anthemWilhelmus[21](William): the first letters of its fifteen stanzas spell WILLEM VAN NASSAU. This was one of the hereditary titles of William of Orange (William the Silent), who introduces himself in the poem to the Dutch people. This title also returned in the 2010speech from the throne, during theDutch State Opening of Parliament, whose first 15 lines also formed WILLEM VAN NASSOV.
Vladimir Nabokov's short story "The Vane Sisters" is known for its acrostic final paragraph, which contains a message from beyond the grave.
In 1829,Edgar Allan Poewrote an acrostic and simply titled itAn Acrostic, possibly dedicated to his cousin Elizabeth Rebecca Herring (though the initials L.E.L. refer toLetitia Elizabeth Landon):
Elizabeth it is in vain you say"Love not" — thou sayest it in so sweet a way:In vain those words from thee or L.E.L.Zantippe's talents had enforced so well:Ah! if that language from thy heart arise,Breath it less gently forth — and veil thine eyes.Endymion, recollect, when Luna triedTo cure his love — was cured of all beside —His folly — pride — and passion — for he died.
In 1939,Rolfe Humphriesreceived a lifelong ban from contributing toPoetrymagazineafter he penned and attempted to publish "a poem containing a concealed scurrilous phrase aimed at a well-known person", namelyNicholas Murray Butler. The poem, entitled "An ode for a Phi Beta Kappa affair", was inunrhymed iambic pentameter, contained oneclassicalreferenceper line, and ran as follows:
Niobe's daughters yearn to the womb again,Ioniansbright and fair, to the chill stone;Chaos in cry,Actaeon's angry pack,Hounds ofMolossus, shaggy wolves drivenOverAmpsanctus' vale andPentheus' glade,LaelapsandLadon, Dromas,Canace,As these in fury harry brake and hillSo the great dogs of evil bay the world.Memory, Mother ofMuses, be resignedUntil KingSaturncomes to rule again!Remember now no more the golden dayRemember now no more the fading gold,Astraeafled,Proserpinainhell;You searchers of the earth be reconciled!Because, through all the blight of human woe,UnderRobigo's rust, andClotho's shears,The mind of man still keeps its argosies,Lacedaemonian Helen wakes her tower,Echoreplies, and lamentation loudReverberates fromThracetoDelosIsle;Itylusgrieves, for whom thenightingaleSweetly as ever tunes her Daulian strain.And overTenedosthe flagship burns.How shall men loiter when the great moon shinesOpaque upon the sail, andArgiveseasRear like blue dolphins their cerulean curves?Samosis fallen,Lesbosstreams with fire,Etna in rage,Canopuscold in hate,Summon the Orphic bard to stranger dreams.And so for us who raiseAthene's torch.Sufficient to her message in this hour:Sons ofColumbia, awake, arise!
Acrostic: Nicholas Murray Butler is a horse's ass.
In October 2009,CaliforniagovernorArnold Schwarzeneggersent anoteto assemblymanTom Ammianoin which the first letters of lines 3-9 spell "Fuck You"; Schwarzenegger claimed that the acrostic message was coincidental, which mathematicians Stephen Devlin and Philip Stark disputed as statistically implausible.[22][23][24]
In January 2010,Jonathan I. Schwartz, the CEO ofSun Microsystems, sent an email to Sun employees on the completion of the acquisition of Sun byOracle Corporation. The initial letters of the first seven paragraphs spelled "BeatIBM".[25]
James May, former presenter on the BBC programTop Gear, was fired from the publicationAutocarfor spelling out a message using the large redinitialat the beginning of each review in the publication'sRoad Test Yearbook Issuefor 1992. Properly punctuated, the message reads: "So you think it's really good, yeah? You should try making the bloody thing up; it's a real pain in the arse."[26]
In the 2012 third novel of hisCaged Flower[27]series, author Cullman Wallace used acrostics as a plot device. The parents of a protagonist send e-mails where the first letters of the lines reveal their situation in a concealed message.
On 19 August 2017, the members of presidentDonald Trump'sCommittee on Arts and Humanitiesresigned in protest over his response to theUnite the Right rallyincident in Charlottesville, Virginia. The members' letter of resignation contained the acrostic "RESIST" formed from the first letter of each paragraph.[28]
On 23 August 2017,University of California, Berkeleyenergy professor Daniel Kammen resigned from his position as a State Department science envoy with a resignation letter in which the word "IMPEACH" was spelled out by the first letters of each paragraph.[29]
In the video gameZorkthe first letters of sentences in a prayer spelled "Odysseus" which was a possible solution to aCyclopsencounter in another room.[30]
On 4 May 2024,Noelia Voigtresigned asMiss USA 2023with a resignation letter containing an acrostic spelling out "I am silenced".[31]
Adouble acrostic, may have words at the beginning and end of its lines, as in this example, on the name ofStroud, by Paul Hansford:
The first letters make up the acrostic and the last letters the telestich; in this case they are identical.
Another example of a double acrostic is the first-century LatinSator Square.[32]
As well as being a double acrostic, the square contains severalpalindromes, and it can be read as a 25-letter palindromic sentence (of an obscure meaning).[33][34]
The poemBehold, O God!, by William Browne,[35]can be considered a complex kind of acrostic.
In the manuscript, some letters are capitalized and written extra-large, non-italic, and in red, and the lines are shifted left or right and internally spaced out as necessary to position the red letters within three crosses that extend through all the lines of the poem.
The letters within each cross spell out a verse from theNew Testament:
The "INRI" at the top of the middle cross stands forIēsus Nazarēnus,Rēx Iūdaeōrum, Latin for "Jesus of Nazareth, King of the Jews" (John 19:19). The three quotes represent the three figures crucified on Golgotha, as recorded in the gospels of Matthew and Luke.
(The text of the manuscript shown differs significantly from the text usually published, including in the reference.[35]Many of the lines have somewhat different wording; and while the acrostics are the same as far as they go, the published text is missing the last four lines, truncating the acrostics to "Lord, remember me when thou comest into thy kin", "O God, my God, why hast thou forsak", and "If thou art the Christ, save thyself". The manuscript text is printed below, first as normal poetry, then spaced and bolded to bring out the acrostics. The word "Thou" in line 8 is not visible in this photograph, but is in the published version and is included in a cross-stitch sampler of the poem from 1793.[36])
Behold, O God! In rivers of my tearsI come to thee! bow down thy blessed earsTo hear my Plaint; and let thine eyes which keepContinual watch behold a Sinner weep:Let not, O God my God my Sins, tho' great,And numberless, between thy Mercy's-SeatAnd my poor Soul have place; since we are taught,[Thou]Lord, remember'st thyne, if Thou art sought.I come not, Lord, with any other meritThan what I by my Saviour Christ inherit:Be then his wounds my balm— his stripes my Bliss;His thorns my crown; my death be blest in his.And thou, my blest Redeemer, Saviour, God,Quit my accounts, withhold thy vengeful rod!O beg for me, my hopes on Thee are set;And Christ forgive me, since thou'st paid my debtThe living font, the Life, the Way, I know,And but to thee, O whither shall I go?All other helps are vain: grant thine to me,For in thy cross my saving health I see.O hearken then, that I with faith implore,Lest Sin and Death sink me to rise no more.Lastly, O God, my course direct and guide,In Death defend me, that I never slide;And at Doomsday let me be rais'd again,To live with thee sweet Jesus say, Amen.
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TheBible code(Hebrew:הצופן התנ"כי,hatzofen hatanachi), also known as theTorah code, is a purported set ofencodedwords within a Hebrew text of theTorahthat, according to proponents, has predicted significant historical events. The statistical likelihood of the Bible code arising by chance has been thoroughly researched, and it is now widely considered to bestatistically insignificant, as similar phenomena can be observed in any sufficiently lengthy text.[1]Although Bible codes have been postulated and studied for centuries, the subject has been popularized in modern times byMichael Drosnin's bookThe Bible Code(1997) and the movieThe Omega Code(1999).
Some tests purportedly showing statistically significant codes in the Bible were published as a "challenging puzzle" in a peer-reviewed academic journal in 1994,[2]which was pronounced "solved" in a subsequent 1999 paper published in the same journal.[3]
Discussion around one specificsteganographicmethod became widespread in 1994 when Doron Witztum,Eliyahu Ripsand Yoav Rosenberg published a paper, "Equidistant Letter Sequences in the Book of Genesis", in the scientific journalStatistical Science.[4][5]The paper, which was presented by the journal as a "challenging puzzle", presented what appeared to be strong statistical evidence that biographical information about famous rabbis was encoded in the text of the Book of Genesis, centuries before those rabbis lived.[2]
The primary method by which purportedly meaningful messages have been extracted is theEquidistant Letter Sequence(ELS), also referred to asdilug[6][7](דילוג,'skipping [of letters]'). Letters are selected based on a starting point and counting every nth letter based on a given 'skip number' in a given direction. For example, taking every fourth letter in the phrase "thissentencefitsan ELS", when read backwards and ignoring spaces, derives the word 'Safest'.
In some cases, multiple terms may be derived from an 'ELS letter array' (text in a grid, with the same number of letters in each line). In the example provided, part of theKing James Version's rendering ofGenesis(26:5–10) is shown with 21 letters per line, showing ELSs for "Bible" and "code".[5]
Once a specific word has been found using the ELS method, other words are sought based on the same letter spacing.[8]Code proponents Haralick and Rips have published an example of a longer, extended ELS, which reads, "Destruction I will call you; cursed is Bin Laden and revenge is to the Messiah".[9]
Proponents claim that such ELS extensions that form phrases or sentences have statistical significance, maintaining that the longer the extended ELS, the less likely it is to be the result of chance.[10]Critics reply, as in theSkeptical Inquirerdeconstruction of 1997,[11]that the longer ELS is in fact effectively nothing more than further increased number of permutations, employing a massive application of thelook-elsewhere effect.
The 13th-century SpanishrabbiBachya ben Asherdescribed an ELS in the Bible. His four-letter example related to the traditional zero-point of theHebrew calendar. Over the following centuries there are hints that the ELS technique was known, e.g. inPardes Rimonimof the 16th century mysticMoshe Cordovero.[12][13]
In the 20th century, many examples were found byMichael Ber Weissmandland published by his students after his death in 1957. In the 1980s, some discoveries of Israeli school teacher Avraham Oren came to the attention of the mathematicianEliyahu Ripsat theHebrew Universityof Jerusalem. Rips then took up the study together with his religious studies partnersDoron Witztumand Alexander Rotenberg, among several others.
Rips and Witztum and Yoav Rosenberg designed computer software for the ELS technique and subsequently found many examples. About 1985, they decided to carry out a formal test, and the "Great rabbis experiment" was born. This experiment tested the hypothesis that ELS's of the names of famous rabbinic personalities and their respective birth and death dates form a more compact arrangement than could be explained by chance. Their definition of "compact" was complex but, roughly, two ELSs were compactly arranged if they can be displayed together in a small window. When Ripset al.carried out the experiment, the data was measured and found to be statistically significant, supporting their hypothesis.
The "great rabbis experiment" went through several iterations, and was eventually published in 1994, in thepeer-reviewed journalStatistical Science. The editorial board was highly skeptical due to the fact that computers can be used to "mine" data for patterns that intuitively seem surprising but upon careful analysis are found to be statistically insignificant. While they did find a number of possible sources of error, they were unable to find anyone willing to put in the substantial time and energy required to properly reanalyze the data. However, they did find it intriguing, and therefore decided to offer it as a "challenging puzzle" for anyone interested in doing so. An unintended result of this was that outsiders mistook this as a confirmation of the paper's claims.[14]
Another experiment, in which the names of the famous rabbis were matched against the places of their births and deaths (rather than the dates), was conducted in 1997 by Harold Gans, former SeniorCryptologicMathematician for the United StatesNational Security Agency.[15]
Again, the results were interpreted as being meaningful and thus suggestive of a more than chance result.[16]These Bible codes became known to the public primarily due to the American journalistMichael Drosnin, whose bookThe Bible Code(1997) was a best-seller in many countries. Rips issued a public statement that he did not support Drosnin's work or conclusions;[17][18]even Gans has stated that, although the book says the codes in the Torah can be used to predict future events, "This is absolutely unfounded. There is no scientific or mathematical basis for such a statement, and the reasoning used to come to such a conclusion in the book is logically flawed."[19][18]In 2002, Drosnin published a second book on the same subject, calledBible Code II: the Countdown.
TheJewish outreachgroupAish HaTorahemploys Bible codes in their Discovery Seminars to persuade secular Jews of the divinity of the Torah, and to encourage them to trust in traditional Orthodox Jewish teachings.[20]Use of Bible code techniques also spread into certain Christian circles, especially in theUnited States. The main early proponents wereYakov Rambsel, who is aMessianic Jew, andGrant Jeffrey. Another Bible code technique was developed in 1997 by Dean Coombs (also Christian). Variouspictogramsare claimed to be formed by words and sentences using ELS.[21]
Since 2000, physicist Nathan Jacobi, an agnostic Jew, and engineer Moshe Aharon Shak, an orthodox Jew, claim to have discovered hundreds of examples of lengthy, extended ELSs.[22]The number of extended ELSs at various lengths is compared with those expected from a non-encoded text, as determined by a formula fromMarkov chaintheory.[23]
The precise order of consonantal letters represented in the HebrewMasoretic Textis not consistent across manuscripts in any period. It is known from earlier versions, such as theDead Sea Scrolls, that the number of letters was not constant even in the first centuries CE. The Bible code theory thus does not seem to account for these variations.[24]
In 1999, Australian mathematicianBrendan McKay, Israeli mathematiciansDror Bar-NatanandGil Kalai, and Israeli psychologistMaya Bar-Hillel(collectively known as "MBBK") published a paper inStatistical Science, in which they argued that the case of Witztum, Rips and Rosenberg (WRR) was "fatally defective, and that their result merely reflects on the choices made in designing their experiment and collecting the data for it."[25]The MBBK paper was reviewed anonymously by four professional statisticians prior to publication. In the introduction to the paper, Robert Kass, the Editor of the Journal who previously had described the WRR paper as a "challenging puzzle" wrote that "considering the work of McKay, Bar-Natan, Kalai and Bar-Hillel as a whole it indeed appears, as they conclude, that the puzzle has been solved".[14]
From their observations, MBBK created analternative hypothesisto explain the "puzzle" of how the codes were discovered. MBBK's argument was not strictly mathematical, rather it asserted that the WRR authors and contributors had intentionally:
The MBBK paper argued that the ELS experiment is extraordinarily sensitive to very small changes in the spellings of appellations, and the WRR result "merely reflects on the choices made in designing their experiment and collecting the data for it."
The MBBK paper demonstrated that this "tuning", when combined with what MBBK asserted was available "wiggle" room, was capable of generating a result similar to WRR's Genesis result in a Hebrew translation ofWar and Peace. Bar-Hillel subsequently summarized the MBBK view that the WRR paper was a hoax, an intentionally and carefully designed "magic trick".[26]
Harold Gans, a formercryptanalystat theNational Security Agency, argued that MBBK's hypothesis implies a conspiracy between WRR and their co-contributors to fraudulently tune the appellations in advance. Gans argues that the conspiracy must include Doron Witztum, Eliyahu Rips, and S. Z. Havlin, because they all say Havlin compiled the appellations independently. Gans argues further that such a conspiracy must include the multiple rabbis who have written a letter confirming the accuracy of Havlin's list. Finally, argues Gans, such a conspiracy must also include the multiple participants of the cities experiment conducted by Gans (which includes Gans himself). Gans concludes that "the number of people necessarily involved in [the conspiracy] will stretch the credulity of any reasonable person."[27]Gans further argued that while "the mathematical issues are difficult for non-mathematicians to comprehend, I can summarize as follows: Professor McKay and his colleagues never claimed to have discovered real codes in those non-Torah texts. Their only "successful" results were obtained by deliberately rigging the experiment in such a way that the layman wouldn't recognize the mathematical flaws."[28]
Brendan McKay has replied that he and his colleagues have never accused Havlin or Gans of participating in a conspiracy. Instead, says McKay, Havlin likely did what WRR's early preprints stated he did, in providing "valuable advices". Similarly, McKay accepts Gans's statements that Gans did not prepare the data for his cities experiment himself. McKay concludes that "there is only ONE person who needs to have been involved in knowing fakery, and a handful of his disciples who must be involved in the cover-up (perhaps with good intent)."[29]
The WRR authors issued a series of responses regarding the claims of MBBK,[30]including the claim that no such tuning did or even could have taken place.[31]An earlier WRR response to a request by MBBK authors presented results from additional experiments that used the specific "alternate" name and date formats which MBBK suggested had been intentionally avoided by WRR.[32]Using MBBK's alternates, the results WRR returned showed equivalent or better support for the existence of the codes, and so challenged the "wiggle room" assertion of MBBK. In the wake of the WRR response, author Bar-Natan issued a formal statement of non-response.[33]After a series of exchanges with McKay and Bar-Hillel, WRR author Witztum responded in a new paper[34]claiming that McKay had used smoke screen tactics in creating severalstraw manarguments, and thereby avoided the points made by WRR authors refuting MBBK.[35]Witztum also claimed that, upon interviewing a key independent expert contracted by McKay for the MBBK paper, some experiments performed for MBBK had validated, rather than refuted, the original WRR findings. Witzum questioned why MBBK had expunged these results. McKay replied to these claims.[36]
No publication in a peer reviewed scientific journal has appeared refuting MBBK's paper. In 2006, four new Torah Codes papers were published at theIEEE Computer Society's 18th International Conference on Pattern Recognition (ICPR'06).[37]
Robert Aumann, agame theoristand winner of theNobel Prize in Economicsin 2005, has followed the Bible code research and controversy for many years. He wrote:[38]
Though the basic thesis of the research seems wildly improbable, for many years I thought that an ironclad case had been made for the codes; I did not see how 'cheating' could have been possible. Then came the work of the 'opponents' (see, for example, McKay, Bar-Natan, Bar-Hillel and Kalai, Statistical Science 14 (1999), 149–173). Though this work did not convince me that the data had been manipulated, it did convince me that it could have been; that manipulation was technically possible.
Following an analysis of the experiment and the dynamics of the controversy, stating for example that "almost everybody included [in the controversy] made up their mind early in the game", Aumann concluded:
A priori, the thesis of the Codes research seems wildly improbable... Research conducted under my own supervision failed to confirm the existence of the codes – though it also did not establish their non-existence. So I must return to my a priori estimate, that the Codes phenomenon is improbable".[39]
Robert Haralick, a Professor of Computer Science at theCity University of New York, has checked the Bible Code for many years and became convinced of its validity. He contributed a new experiment, checking whether, besides the minimal ELS – in which it was known that WRR's list was successful in Genesis and MBBK's list was successful in War and Peace – there were other, non-minimal ELSs where there is convergence between the rabbis' names and their respective dates. This had the effect of checking convergence found at 2nd minimal ELSs, 3rd minimal ELSs and so on. According to Haralick, the results were impressive; WRR's list was successful until the 20th minimal ELS, whereas MBBK's list failed after the 2nd minimal ELS.https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=9884414ff8612a386b9419afc380680c5b3e05c5Haralick lectured on the subject in front of the participants of the International Conference on Pattern Recognition in 2006.[40]
Journalist Drosnin's books[41]have been criticized by some who believe the Bible code is real but that it cannot predict the future.[42]On Drosnin's claim ofYitzhak Rabin's assassination, Drosnin wrote in his book "The Bible Code" (1997) that "Yigal Amircould not be found in advance". Critics have noted a huge error in the "code" Drosnin claimed to have found: Drosnin misused the Biblical verseDeuteronomy 4:42. Scholars note; "For example, citing again the passage intersecting with Rabin: that passage is from Deuteronomy 4:42, but Drosnin ignores the words immediately following "a murderer who will murder." What comes next is the phrase "unwittingly" (biveli da'at). This is because the verse deals with the cities of refuge where accidental killers can find asylum. In this case, then, the message would refer to an accidental killing of (or by) Rabin and it would therefore be wrong. Another message (p. 17) supposedly contains a "complete" description of the terrorist bombing of a bus in Jerusalem on February 25, 1996. It includes the phrase "fire, great noise," but overlooks the fact that the letters which make up those two words are actually part of a larger phrase fromGenesis 35:4which says: "under the terebinth that was near Shechem." If the phrase does tell of a bus bombing, why not take it to indicate that it would be in Nablus, the site of ancient Shechem?"[43]
Drosnin also made a number of claims and alleged predictions that have since failed. Among the most important, Drosnin clearly states in his book "The Bible Code II", published on December 2, 2002, that there was to be a World War involving an "atomic holocaust" that would allegedly be the end of the world.[44]Another claim Drosnin makes in "The Bible Code II" is that the nation of Libya would develop weapons of mass destruction which would then be given to terrorists who would then use them to attack the West (specifically the United States).[45]In reality,Libyaimproved relations with the West in 2003 and gave up all their existing weapons of mass destruction programs.[46]A final claim Drosnin made in "The Bible Code II" was that Palestinian Authority leader Yasser Arafat would allegedly be assassinated by being shot to death by gunmen which Drosnin specifically stated would be from the Palestinian Hamas movement.[47]This prediction by Drosnin also failed, as Yasser Arafat died on November 11, 2004[48]of what was later declared to be natural causes (specifically a stroke brought on by an unknown infection).[49][50]
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BPCS-steganography (Bit-Plane Complexity Segmentation steganography)is a type ofdigital steganography.
Digital steganography can hide confidential data (i.e. secret files) very securely by embedding them into some media data called "vessel data." The vessel data is also referred to as "carrier, cover, or dummy data". In BPCS-steganographytrue colorimages (i.e.,24-bit colorimages) are mostly used for vessel data. The embedding operation in practice is to replace the "complex areas" on thebit planesof the vessel image with the confidential data. The most important aspect of BPCS-steganography is that the embedding capacity is very large. In comparison to simple image based steganography which uses solely the least important bit of data, and thus (for a24-bit colorimage) can only embed data equivalent to 1/8 of the total size, BPCS-steganography uses multiple bit-planes, and so can embed a much higher amount of data, though this is dependent on the individual image. For a 'normal' image, roughly 50% of the data might be replaceable with secret data before image degradation becomes apparent.
TheHuman visual systemhas such a special property that a too-complicated visual pattern can not be perceived as "shape-informative." For example, on a very flat beach shore every single square-foot area looks the same - it is just a sandy area, no shape is observed. However, if you look carefully, two same-looking areas are entirely different in their sand particle shapes. BPCS-steganography makes use of this property. It replaces complex areas on the bit-planes of the vessel image with other complex data patterns (i.e., pieces of secret files). This replacing operation is called "embedding." No one can see any difference between the two vessel images of before and after the embedding operation.
An issue arises where the data to be embedded appears visually as simple information, if this simple information replaces the complex information in the original image it may create spurious 'real image information'. In this case the data is passed through a binary image conjugation transformation, in order to create a reciprocal complex representation.
This form of steganography was proposed jointly by Eiji Kawaguchi and Richard O. Eason in 1998.[1]Their experimental program (titled Qtech Hide & View) is freely available for educational purposes.[2]Recently, many researchers are tackling itsalgorithmimprovement and applications as well as resistibility studies againststeganalysis.[citation needed]
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Hacktivismois an offshoot ofCult of the Dead Cow(cDc), whose beliefs includeaccess to informationas a basichuman right. It was founded in 1999.
The group's beliefs are described fully in The Hacktivismo Declaration, which seeks to apply theUniversal Declaration of Human Rightsand theInternational Covenant on Civil and Political Rightsto theInternet.Oxblood Ruffin, the director of Hacktivismo, has argued forcefully against definitions ofhacktivismthat include web defacements ordenial-of-service attacks. Hacktivismo has also authored its ownsoftware licenseagreement, theHacktivismo Enhanced-Source Software License Agreement(HESSLA). The HESSLA prohibits use or modification that would violate human rights or introduce features that spy on the user.
In 1999 Cult of the Dead Cow (cDc), a loose network of individuals, announced the formation of Hacktivismo. The group set to explore ways of preventing censorship of the Internet. In particular Hacktivismo focused on firewalls or censoring mechanisms of national governments. Press releases made it clear that cDc and Hacktivismo were different groups; however Hacktivismo was also described as "special operations group" of cDc. A press release in early 2002 described Hacktivismo as "an international cadre of hackers founded by the cDc'sOxblood Ruffin".[1]
The group's beliefs are described fully in the "Hacktivismo Declaration" which is a list of "assertions of liberty in support of an uncensored internet" and seeks to apply theUniversal Declaration of Human Rightsand theInternational Covenant on Civil and Political Rights(ICCPR) to theInternet. The Declaration recalls the duty of member states to the ICCPR to protect the right tofreedom of expressionwith regards to the internet and in this context what is called the "freedom of information".[2]The Hacktivismo Declaration states:
The Hacktivismo Declaration recognizes "the importance to fight against human rights abuses with respect to reasonable access to information on the Internet" and calls upon thehackercommunity to "study ways and means of circumventing state sponsored censorship of the internet" and "implement technologies to challenge information rights violations".
The Hacktivismo Declaration does however recognize that the right tofreedom of expressionis subject to limitations, stating "we recognized the right of governments to forbid the publication of properly categorized state secrets, child pornography, and matters related to personal privacy and privilege, among other accepted restrictions." However, the Hacktivismo Declaration states "but we oppose the use of state power to control access to the works of critics, intellectuals, artists, or religious figures."[2]
Camera/Shy was the first Hacktivismo project released. It debuted in 2002 at theH.O.P.E.2k2 convention inNew York City. Written by The Pull, Camera/Shy is asteganographictool that scans for and deliversdecryptedcontent directly from theWorld Wide Web. It is a stand-alone,Internet Explorer-basedweb browser. It interprets and displays hidden information stored in the junk bits inGIFfiles.[3]
The Six/Four System was written byMixter. The software is acensorshipresistantnetworkproxy. It works by using "trusted peers" to relay network connections overSSLencrypted links.[4]As an example, the distribution includes a program which will act as a web proxy, but where all of the connections will be hidden until they reach the far end trusted peer.[5]
Hacktivismo and the cDc further gained notoriety in 2003 when the Six/Four System became the first product of ahacker groupto receive approval from theUnited States Department of Commercefor export of strongencryption.[6]
ScatterChat is an encryptedinstant messagingclient based onGaim. It was written by J. Salvatore Testa II and released at the H.O.P.E. Number Six conference in New York City on July 22, 2006. The source code is available, licensed under the HESSLA. It providesencryptionas well as integratedonion routingwithTor, and secure file transfers. Scatterchat's security features include immunity fromreplay attacksand limited resistance totraffic analysis.[7][8]Various flaws in the software have been elaborated by researchers.[9][10]
XeroBank Browser(formerly known asTorpark) is a variant of thePortable Firefoxweb browserwithTorbuilt into it. XeroBank is intended for use onportable mediasuch as aUSB flash drivebut it can also be used on anyhard disk drive. cDc/Hacktivismo co-released v.1.5.0.7 along with Steve Topletz on September 19, 2006.[11][12]
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Awarrant canaryis a method by which acommunications service provideraims to implicitly inform its users that the provider has been served with a governmentsubpoenadespite legal prohibitions on revealing the existence of the subpoena. The warrant canary typically informs users that there hasnotbeen a court-issued subpoena as of a particular date. If the canary is not updated for the period specified by the host or if the warning is removed, users might assume the host has been served with such a subpoena. The intention is for a provider to passively warn users of the existence of a subpoena, albeit violating the spirit of a court order not to do so, while not violating the letter of the order.
Some subpoenas, such as those covered under 18 U.S.C. §2709(c) (enacted as part of theUSA Patriot Act), provide criminal penalties for disclosing the existence of the subpoena to any third party, including the service provider's users.[1][2]
National Security Letters(NSL) originated in the1986 Electronic Communications Privacy Actand originally targeted those suspected of being agents of a foreign power.[3]Targeting agents of a foreign power was revised in thePatriot Actin 2001 to allow NSLs to target those who may have information thought to be relevant to either counterintelligence activities or terrorists activities directed against the United States.[3]The idea of using negative pronouncements to thwart the nondisclosure requirements ofcourt ordersand served secret warrants was first proposed by Steven Schear on thecypherpunksmailing list,[4]mainly to uncover targeted individuals atISPs. It was also suggested for and used by public libraries in 2002 in response to theUSA Patriot Act, which could have forced librarians to disclose the circulation history of library patrons.[5][6]
The term is an allusion to the practice ofcoal minersbringingcanariesinto mines to use as an early-warning signal for toxic gases, primarilycarbon monoxideandmethane.[7]The birds are more sensitive to these gases thanhumans, and became sick before the miners, who would then have a chance to escape or put on protectiverespirators.[8]
The first commercial use of a warrant canary was by the UScloud storageprovider rsync.net, which began publishing its canary in 2006.[9]In addition to adigital signature, it provides a recent news headline as proof that the warrant canary was recently posted[10]as well as mirroring the posting internationally.[11]
On November 5, 2013, Apple became the most prominent company to publicly state that it had never received an order for user data under Section 215 of the Patriot Act.[12][13]On September 18, 2014, GigaOm reported that the warrant canary statement did not appear anymore in the next two Apple Transparency Reports, covering July–December 2013 and January–June 2014.[14]Tumblralso included a warrant canary in the transparency report that it issued on February 3, 2014.[15]In August 2014, the online cloud serviceSpider Oakimplemented an encrypted warrant canary that publishes an "All Clear!" message every 6 months. Three PGP signatures from geographically distributed signers must sign each message—so if a government agency forced SpiderOak to update the page, they would need to enlist the help of all three signers.[16]
In September 2014, U.S. security researcherMoxie Marlinspikewrote that "every lawyer I've spoken to has indicated that having a 'canary' you remove or choose not to update would likely have the same legal consequences as simply posting something that explicitly says you've received something."[17][18]
In March 2015 it was reported thatAustraliaoutlawed the use of a certain kind of warrant canary, making it illegal to "disclose information about the existence or non-existence" of a Journalist Information Warrant issued under new mandatory data retention laws.[19]Afterwards, computer security and privacy specialistBruce Schneierwrote in a blog post that "[p]ersonally, I have never believed [warrant canaries] would work. It relies on the fact that a prohibition against speaking doesn't prevent someone from not speaking. But courts generally aren't impressed by this sort of thing, and I can easily imagine a secret warrant that includes a prohibition against triggering the warrant canary. And for all I know, there are right now secret legal proceedings on this very issue."[20]This is not the first Australian law to outlaw warrant canaries. The "Telecommunications (Interception) Amendment Act 1995" was probably the first, making it illegal to "disclose information about the existence or non-existence" of Interception Warrants.[21]
That said, case law specific to theUnited Stateswould render the covert continuance of warrant canaries subject to constitutionality challenges.[citation needed]West Virginia State Board of Education v. BarnetteandWooley v. Maynardrule the Free Speech Clause prohibitscompelling someone to speakagainst one's wishes; this can easily be extended to prevent someone from being compelled to lie.New York Times Co. v. United Statesprotects one exercising the First Amendment to publish government information, even if it is against the wishes of the government, except under grave and exceptional circumstances previously set by act and precedent. This may also have implications in regards to acting against a direct government intervention, similar to a government intervention against a warrant canary.[citation needed]
The following is a non-exhaustive list of companies and organizations whose warrant canaries no longer appear in transparency reports:
In 2015, a coalition of organizations consisting of theEFF,Freedom of the Press Foundation,NYU Law, theCalyx Institute, and theBerkman Centercreated a website called Canary Watch in order to provide a compiled list of all companies providing warrant canaries. Its mission was to provide prompt updates of any changes in a canary's state. It is often difficult for users to ascertain a canary's validity on their own and thus Canary Watch aimed to provide a simple display of all active canaries and any blocks of time that they were not active.[25]In May 2016, it was announced that Canary Watch "will no longer accept submissions of new canaries or monitor the existing canaries for changes or take downs".[26]The coalition of organizations which created Canary Watch explained their decision to discontinue the project by stating that it has achieved its goals to raise awareness about "illegal and unconstitutional national security process, including National Security Letters and other secret court processes." The Electronic Frontier Foundation also noted that "the fact that canaries are non-standard makes it difficult to automatically monitor them for changes or takedowns." They explained that the project had run its course, that ample attention had been brought to canaries, and detailed warrant canary strengths and weaknesses they observed.[26]
In 2016, theRiseuptech collectivefailed to update their warrant canary, due to sealed warrants from a court.[27][28]The canary has since been updated, but no longer states the absence of gag orders.[29]
In February 2024, theEthereumFoundation removed the warrant canary from their website[30]citing "[a] voluntary enquiry from a state authority that included a requirement for confidentiality" in the commit message.
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Incryptographyandsteganography, plausiblydeniable encryptiondescribesencryptiontechniques where the existence of an encrypted file or message is deniable in the sense that an adversary cannot prove that theplaintextdata exists.[1]
The users mayconvincingly denythat a given piece of data is encrypted, or that they are able to decrypt a given piece of encrypted data, or that some specific encrypted data exists.[2]Such denials may or may not be genuine. For example, it may be impossible to prove that the data is encrypted without the cooperation of the users. If the data is encrypted, the users genuinely may not be able to decrypt it. Deniable encryption serves to undermine an attacker's confidence either that data is encrypted, or that the person in possession of it can decrypt it and provide the associated plaintext.
In their pivotal 1996 paper,Ran Canetti,Cynthia Dwork,Moni Naor, andRafail Ostrovskyintroduced the concept of deniable encryption, a cryptographic breakthrough that ensures privacy even under coercion. This concept allows encrypted communication participants to plausibly deny the true content of their messages. Their work lays the foundational principles of deniable encryption, illustrating its critical role in protecting privacy against forced disclosures. This research has become a cornerstone for future advancements in cryptography, emphasizing the importance of deniable encryption in maintaining communication security.[3]The notion of deniable encryption was used byJulian AssangeandRalf Weinmannin the Rubberhose filesystem.[4][2]
Deniable encryption makes it impossible to prove the origin or existence of the plaintext message without the proper decryption key. This may be done by allowing an encrypted message to be decrypted to different sensible plaintexts, depending on thekeyused. This allows the sender to haveplausible deniabilityif compelled to give up their encryption key.
In some jurisdictions, statutes assume that human operators have access to such things as encryption keys. An example is the United Kingdom'sRegulation of Investigatory Powers Act,[5][6]which makes it a crime not to surrenderencryption keyson demand from a government official authorized by the act. According to theHome Office, the burden of proof that an accused person is in possession of a key rests on the prosecution; moreover, the act contains a defense for operators who have lost or forgotten a key, and they are not liable if they are judged to have done what they can to recover a key.[5][6]
Incryptography,rubber-hose cryptanalysisis aeuphemismfor the extraction of cryptographic secrets (e.g. the password to an encrypted file) from a person bycoercionortorture[7]—such as beating that person with a rubberhose, hence the name—in contrast to a mathematical or technicalcryptanalytic attack.
An early use of the term was on thesci.cryptnewsgroup, in a message posted 16 October 1990 byMarcus J. Ranum, alluding tocorporal punishment:
...the rubber-hose technique of cryptanalysis. (in which a rubber hose is applied forcefully and frequently to the soles of the feet until the key to the cryptosystem is discovered, a process that can take a surprisingly short time and is quite computationally inexpensive).[8]
Deniable encryption allows the sender of an encrypted message to deny sending that message. This requires atrusted third party. A possible scenario works like this:
Another scenario involves Alice sending the same ciphertext (some secret instructions) to Bob and Carl, to whom she has handed different keys. Bob and Carl are to receive different instructions and must not be able to read each other's instructions. Bob will receive the message first and then forward it to Carl.
Normally, ciphertexts decrypt to a single plaintext that is intended to be kept secret. However, one form of deniable encryption allows its users to decrypt the ciphertext to produce a different (innocuous but plausible) plaintext and plausibly claim that it is what they encrypted. The holder of the ciphertext will not be able to differentiate between the true plaintext, and the bogus-claim plaintext. In general, oneciphertextcannot be decrypted to all possibleplaintextsunless the key is as large as theplaintext, so it is not practical in most cases for a ciphertext to reveal no information whatsoever about its plaintext.[9]However, some schemes allow decryption to decoy plaintexts that are close to the original in some metric (such asedit distance).[10]
Modern deniable encryption techniques exploit the fact that without the key, it is infeasible to distinguish between ciphertext fromblock ciphersand data generated by acryptographically secure pseudorandom number generator(the cipher'spseudorandom permutationproperties).[11]
This is used in combination with somedecoydata that the user would plausibly want to keep confidential that will be revealed to the attacker, claiming that this is all there is. This is a form ofsteganography.[citation needed]
If the user does not supply the correct key for the truly secret data, decrypting it will result in apparently random data, indistinguishable from not having stored any particular data there.[citation needed]
One example of deniable encryption is acryptographic filesystemthat employs a concept of abstract "layers", where each layer can be decrypted with a different encryption key.[citation needed]Additionally, special "chafflayers" are filled with random data in order to haveplausible deniabilityof the existence of real layers and their encryption keys.[citation needed]The user can store decoy files on one or more layers while denying the existence of others, claiming that the rest of space is taken up by chaff layers.[citation needed]Physically, these types of filesystems are typically stored in a single directory consisting of equal-length files with filenames that are eitherrandomized(in case they belong to chaff layers), orcryptographic hashesof strings identifying the blocks.[citation needed]Thetimestampsof these files are always randomized.[citation needed]Examples of this approach include Rubberhose filesystem.
Rubberhose (also known by its development codename Marutukku)[12]is a deniable encryption program which encrypts data on a storage device and hides the encrypted data. The existence of the encrypted data can only be verified using the appropriate cryptographic key. It was created byJulian Assangeas a tool for human rights workers who needed to protect sensitive data in the field and was initially released in 1997.[12]
The name Rubberhose is a joking reference to thecypherpunksterm rubber-hose cryptanalysis, in which encryption keys are obtained by means of violence.[citation needed]
It was written forLinux kernel2.2,NetBSDandFreeBSDin 1997–2000 byJulian Assange,Suelette Dreyfus, and Ralf Weinmann. The latest version available, still in alpha stage, is v0.8.3.[13]
Another approach used by some conventionaldisk encryption softwaresuites is creating a second encryptedvolumewithin a container volume. The container volume is first formatted by filling it with encrypted random data,[14]and then initializing a filesystem on it. The user then fills some of the filesystem with legitimate, but plausible-looking decoy files that the user would seem to have an incentive to hide. Next, a new encrypted volume (the hidden volume) is allocated within the free space of the container filesystem which will be used for data the user actually wants to hide. Since an adversary cannot differentiate between encrypted data and the random data used to initialize the outer volume, this inner volume is now undetectable.LibreCrypt[15]andBestCryptcan have many hidden volumes in a container;TrueCryptis limited to one hidden volume.[16]
The existence of hidden encrypted data may be revealed by flaws in the implementation.[19][self-published source]It may also be revealed by a so-calledwatermarking attackif an inappropriate cipher mode is used.[20]The existence of the data may be revealed by it 'leaking' into non-encrypted disk space[21]where it can be detected byforensictools.[22][self-published source]
Doubts have been raised about the level of plausible deniability in 'hidden volumes'[23][self-published source]– the contents of the "outer" container filesystem have to be 'frozen' in its initial state to prevent the user from corrupting the hidden volume (this can be detected from the access and modification timestamps), which could raise suspicion. This problem can be eliminated by instructing the system not to protect the hidden volume, although this could result in lost data.[citation needed]
Possession of deniable encryption tools could lead attackers to continue torturing a user even after the user has revealed all their keys, because the attackers could not know whether the user had revealed their last key or not. However, knowledge of this fact can disincentivize users from revealing any keys to begin with, since they will never be able to prove to the attacker that they have revealed their last key.[24]
Some in-transit encrypted messaging suites, such asOff-the-Record Messaging, offerdeniable authenticationwhich gives the participantsplausible deniabilityof their conversations. While deniable authentication is not technically "deniable encryption" in that the encryption of the messages is not denied, its deniability refers to the inability of an adversary to prove that the participants had a conversation or said anything in particular.
This is achieved by the fact that all information necessary to forge messages is appended to the encrypted messages – if an adversary is able to create digitally authentic messages in a conversation (seehash-based message authentication code(HMAC)), they are also able toforgemessages in the conversation. This is used in conjunction withperfect forward secrecyto assure that the compromise of encryption keys of individual messages does not compromise additional conversations or messages.
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Adigital watermarkis a kind of marker covertly embedded in a noise-tolerantsignalsuch as audio, video or image data.[1]It is typically used to identify ownership of the copyright of such a signal. Digital watermarking is the process of hiding digital information in acarrier signal; the hidden information should,[2]but does not need to, contain a relation to the carrier signal. Digital watermarks may be used to verify the authenticity or integrity of the carrier signal or to show the identity of its owners. It is prominently used for tracingcopyright infringementsand forbanknoteauthentication.
Like traditionalphysical watermarks, digital watermarks are often only perceptible under certain conditions, e.g. after using some algorithm.[3]If a digital watermark distorts the carrier signal in a way that it becomes easily perceivable, it may be considered less effective depending on its purpose.[3]Traditional watermarks may be applied to visible media (like images or video), whereas in digital watermarking, the signal may be audio, pictures, video, texts or 3D models. A signal may carry several different watermarks at the same time. Unlikemetadatathat is added to the carrier signal, a digital watermark does not change the size of the carrier signal.
The needed properties of a digital watermark depend on theuse casein which it is applied. For marking media files with copyright information, a digital watermark has to be rather robust against modifications that can be applied to the carrier signal. Instead, if integrity has to be ensured, a fragile watermark would be applied.
Bothsteganographyand digital watermarking employ steganographic techniques to embed data covertly in noisy signals. While steganography aims for imperceptibility to human senses, digital watermarking tries to control the robustness as top priority.
Since a digital copy of data is the same as the original, digital watermarking is a passive protection tool. It just marks data, but does not degrade it or control access to the data.
One application of digital watermarking issource tracking. A watermark is embedded into a digital signal at each point of distribution. If a copy of the work is found later, then the watermark may be retrieved from the copy and the source of the distribution is known. This technique reportedly has been used to detect the source of illegally copied movies.
The termdigital watermarkwas coined by Andrew Tirkel and Charles Osborne in December 1992. The first successful embedding and extraction of asteganographicspread spectrum watermark was demonstrated in 1993 by Andrew Tirkel, Gerard Rankin, Ron Van Schyndel, Charles Osborne, and others.[4]
Watermarks are identification marks produced during the paper-making process. The first watermarks appeared in Italy during the 13th century, but their use rapidly spread across Europe. They were used as a means to identify the paper maker or the trade guild that manufactured the paper. The marks often were created by a wire sewn onto the paper mold. Watermarks continue to be used today as manufacturer's marks and to prevent forgery.
Digital watermarking may be used for a wide range of applications, such as:
The information to be embedded in a signal is called a digital watermark, although in some contexts the phrase digital watermark means the difference between the watermarked signal and the cover signal. The signal where the watermark is to be embedded is called thehostsignal. A watermarking system is usually divided into three distinct steps, embedding, attack, and detection. In embedding, an algorithm accepts the host and the data to be embedded, and produces a watermarked signal.
Then the watermarked digital signal is transmitted or stored, usually transmitted to another person. If this person makes a modification, this is called anattack. While the modification may not be malicious, the term attack arises from copyright protection application, where third parties may attempt to remove the digital watermark through modification. There are many possible modifications, for example, lossy compression of the data (in which resolution is diminished), cropping an image or video, or intentionally adding noise.
Detection(often called extraction) is an algorithm that is applied to the attacked signal to attempt to extract the watermark from it. If the signal was unmodified during transmission, then the watermark still is present and it may be extracted. Inrobustdigital watermarking applications, the extraction algorithm should be able to produce the watermark correctly, even if the modifications were strong. Infragiledigital watermarking, the extraction algorithm should fail if any change is made to the signal.
A digital watermark is calledrobustwith respect to transformations if the embedded information may be detected reliably from the marked signal, even if degraded by any number of transformations. Typical image degradations are JPEG compression, rotation, cropping, additive noise, andquantization.[6]For video content, temporal modifications and MPEG compression often are added to this list. A digital watermark is calledimperceptibleif the watermarked content is perceptually equivalent to the original, unwatermarked content.[7]In general, it is easy to create either robust watermarksorimperceptible watermarks, but the creation of both robustandimperceptible watermarks has proven to be quite challenging.[2]Robust imperceptible watermarks have been proposed as a tool for the protection of digital content, for example as an embeddedno-copy-allowedflag in professional video content.[8]
Digital watermarking techniques may be classified in several ways.
A digital watermark is calledfragileif it fails to be detectable after the slightest modification. Fragile watermarks are commonly used for tamper detection (integrity proof). Modifications to an original work that clearly are noticeable, commonly are not referred to as watermarks, but as generalizedbarcodes.
A digital watermark is calledsemi-fragileif it resists benign transformations, but fails detection after malignant transformations. Semi-fragile watermarks commonly are used to detect malignant transformations.
A digital watermark is calledrobustif it resists a designated class of transformations. Robust watermarks may be used in copy protection applications to carry copy and no access control information.
A digital watermark is calledimperceptibleif the original cover signal and the marked signal are perceptually indistinguishable.
A digital watermark is calledperceptibleif its presence in the marked signal is noticeable (e.g. digital on-screen graphics like a network logo, content bug, codes, opaque images). On videos and images, some are made transparent/translucent for convenience for consumers due to the fact that they block portion of the view; therefore degrading it.
This should not be confused withperceptual, that is, watermarking which uses the limitations of human perception to be imperceptible.
The length of the embedded message determines two different main classes of digital watermarking schemes:
A digital watermarking method is referred to asspread-spectrumif the marked signal is obtained by an additive modification. Spread-spectrum watermarks are known to be modestly robust, but also to have a low information capacity due to hostinterference.
A digital watermarking method is said to be ofquantization typeif the marked signal is obtained by quantization. Quantization watermarks suffer from low robustness, but have a high information capacity due to rejection of host interference.
A digital watermarking method is referred to asamplitude modulationif the marked signal is embedded by additive modification which is similar to spread spectrum method, but is particularly embedded in the spatial domain.
The evaluation of digital watermarking schemes may provide detailed information for a watermark designer or for end-users, therefore, different evaluation strategies exist. Often used by a watermark designer is the evaluation of single properties to show, for example, an improvement. Mostly, end-users are not interested in detailed information. They want to know if a given digital watermarking algorithm may be used for their application scenario, and if so, which parameter sets seems to be the best.
EpsonandKodakhave produced cameras with security features such as the Epson PhotoPC 3000Z and the Kodak DC-290. Both cameras added irremovable features to the pictures which distorted the original image, making them unacceptable for some applications such asforensic evidencein court. According to Blythe and Fridrich, "[n]either camera can provide an undisputable proof of the image origin or its author".[9]A secure digital camera (SDC) was proposed by Saraju Mohanty, et al. in 2003 and published in January 2004. This was not the first time this was proposed.[10]Blythe and Fridrich also have worked on SDC in 2004[9]for adigital camerathat would use lossless watermarking to embed abiometricidentifier together with acryptographic hash.[11]
Reversible data hidingis a technique which enables images to be authenticated and then restored to their original form by removing the digital watermark and replacing the image data that had been overwritten.[12]
Digital watermarking forrelational databaseshas emerged as a candidate solution to provide copyright protection, tamper detection, traitor tracing, and maintaining integrity of relational data. Many watermarking techniques have been proposed in the literature to address these purposes. A survey of the current state-of-the-art and a classification of the different techniques according to their intent, the way they express the watermark, the cover type, granularity level, and verifiability was published in 2010 by Halder et al. in theJournal of Universal Computer Science.[13]
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Invisible ink, also known assecurity inkorsympathetic ink, is a substance used for writing, which is invisible either on application or soon thereafter, and can later be made visible by some means, such asheatorultravioletlight. Invisible ink is one form ofsteganography.
One of the earliest writers to mention an invisible ink isAeneas Tacticus, in the 4th century BC. He mentions it in discussing how to survive under siege but does not indicate the type of ink to be used.[1]This was part of his list of the 20 different methods of secret communications[2]in a book calledOn the Defense of Fortifications. One of the techniques that involvedsteganographyinvolved puncturing a tiny hole above or below letters in a document to spell out a secret message.[3]This did not include an invisible ink but the Germans improved on the method duringWorld War IandWorld War II. They used invisible ink and microdots instead of pinpricks.[3]
Philo of Byzantiummay be the first writer known to describe an invisible ink using a reagent around 217–218 BC, withoak gallsandvitriol.[4]These ingredients were used to makeoak gall ink.[5]People soon discovered that they could write invisibly with one of the ingredients and then cause the writing to appear by adding the other.[6]Pliny the Elderand the Roman poetOvidgave advice on the use of plant juices and milk to write secret messages.[7]
Lemons were also used as organic inks byArabsaround 600AD, and during the 16th century in Europe.[6]
Giovanni Battista della Portais credited with the first recipe for a sympathetic ink, derived fromalumandvinegar, as well as the first book on secret writing and invisible inks,Magia Naturalis(1558, 1589).[8][4]: 24Since then, a wide variety of invisible inks have been used for all sorts of secretive purposes. A formula similar tooak gall inkwas created byJames Jayand used byGeorge Washingtonand theCulper Spy Ringduring theAmerican Revolutionand lemon juice was used by the 'Lemon Juice Spies' (Carl Muller and four other Germans, who all died for their efforts either by suicide or execution, along with John Hahn, an English baker) during World War I.[6][4]In World War II, neutral or acidic solutions ofphenolphthalein, a chemical compound extracted from pills forconstipation, were used as invisible ink.[9]It is colorless but turns pink when exposed to alkali such asammoniaandbicarbonate soda.[9]
Invisible ink can be applied to a writing surface with a specialty purposestylus,stamp,fountain pen,toothpick,calligraphy pen,Cotton swab, or even a finger dipped in the liquid. Once dry, the written surface looks as if it were blank, with a similar texture and reflectivity as the surrounding surface.
The ink can be later made visible by different methods according to the type of invisible ink used. The ink may be revealed by heat or by application of an appropriate chemical, or it may be made visible by viewing underultraviolet light. Inks which are developed by achemical reactionmay depend on anacid-base reaction(likelitmus paper), reactions similar to theblueprintprocess, or any of hundreds of others. Developer fluids may be applied using a spray bottle, but some developers are in the form of vapor, e.g.ammoniafumes used to developphenolphthaleinink.
There are also toy invisible ink pens which have two tips—one tip for invisible ink writing, and another tip for developing the ink. Invisible ink is sometimes used to print parts of pictures or text inbooksfor children to play with, always including a "decoder pen" which is used to show the invisible parts of texts or pictures, thus revealing answers to questions printed in regular ink or completing missing parts of pictures.
Security marker pens orUV markerswith fluorescent ink that glows when illuminated with a UV light is often used to invisibly mark valuable household items in case ofburglary. There are specialty security maker pens formulated for writing on non-porous surfaces such asglass,plastic,metal, etc. The mark can be read by using ablacklightor other UV light source. Security marker pens can be obtained commercially and are widely used as a crime countermeasure.
Some commercially available invisible inks glow very brightly, in a variety of colors, underultravioletlight. This makes them suitable for use in readmission such as hand stamping.
There are some invisible ink types that can only be invisible when applied to certain types of surfaces, but are still visible on others.
Some vendors now offer invisible ink for use in computerinkjet printers. Such inks are usually visible underultravioletlight. Typical uses include printing information on business forms for use by the form processor, without cluttering up the visible contents of the form. For example, someUnited States Postal Servicemail sorting stations use UV-visible ink to print bar codes on mailed envelopes giving routing information for use by mail handling equipment further down the line before delivery.
AnE2Evoting system calledScantegrity IIuses invisible ink to enable the voter to obtain a confirmation code only for the voted selection.[10]
What an "ideal" invisible ink is depends on its intended use. For example, property marking should ideally be done with ink easily read under ultraviolet light, whereas in espionage such an ink would be considered too easily detectable since a large number of letters may be screened relatively quickly using UV light.
Invisible inks are inherently "insecure" against a determined and well-equipped inspector, which must be balanced against the logistical difficulty in carrying out mass-screening of posted mail. It is easier to performlarge-scaleundetected screening of millions of electronic communications, than to mass-screen even a small fraction of conventional mail. Apart from in dictatorships where large numbers of personnel are employed to spy on fellow nationals, screening of posted mail is only feasible in particular situations, such as letters to and from a particular suspect or facility.
The BritishSOEtraining manual used in the Second World War identified the following properties of an "ideal" invisible ink:
From practical experience "6" and "9" were usually incompatible.SOEagents were trained not to risk their lives through reliance on insecure inks, most of which were from World War I. In general,SOEused invisible inks as a back-up method of communication when other, more secure communication techniques were unavailable. The agency was known to supply special inks to its field agents, rather than have them depend upon improvisation from obtainable everyday chemicals. When agents were forced to improvise, they were advised to dilute their invisible ink as much as possible to reduce chances of detection.[11]
Any invisible ink can be made visible by someone who is sufficiently determined, but the limitation is generally time available and the fact that one cannot apply hours of effort to every single piece of paper. Thus successful use of invisible ink depends on not arousing suspicion that invisible ink may be present.
Telltale signs of invisible ink, such as pen scratches from a sharp pen, roughness, or changed reflectivity of the paper (either more dull or more shiny, usually from using undiluted ink), can be obvious to a careful observer who simply makes use of strong light, a magnifying glass and their nose. Also, key words in the visible letter, such as "heat" or any other odd code name, in an out of place context may alert a censor to the presence of invisible ink. Invisible ink is not effective with glossy or very smooth paper types, since thesizingof these papers prevents ink from being absorbed deep into the paper and it is easily visible, especially if the paper is examined under glancing light. There are, however, commercially available inks for non-porous surfaces that are only visible under ultraviolet light and are otherwise virtually invisible on such surfaces.
Using either ultraviolet light or an iodine fume cupboard, messages can be quickly screened for invisible ink and also read without first permanently developing the invisible ink. Thus, if a censor uses this method to intercept messages, the letter may then be sent to the intended recipient, who will be unaware that the secret message has already been intercepted by a third party.
A "screening station" theoretically could involve visual and olfactory inspection, an examination under ultraviolet light and then the heating of all objects in an oven before finally trying exposure to iodine fumes to produce optimal security in optimal time.
For practical reasons, the inks are listed here according to their method of development. It must be understood however that some inks – particularly those of organic origin or those consisting of a mixture of several chemicals – may be made visible by several methods. For example, invisible writing with soap water may be made visible either by heat, reaction with phenolphthalein, viewing under ultraviolet light, or by placing the page inside an iodine fume cupboard.
Some of these are organic substances that oxidize when heated, which usually turns them brown. For this type of "heat fixed" ink, any acidic fluid will work. The most secure way to use any of the following substances for invisible ink is by dilution, usually with water, close to the point when they become difficult to develop.
The writing is rendered visible byheatingthe paper, either on aradiator, byironingit, using a hair dryer, or by placing it in anoven. A 100-wattlight bulbis less likely to damage the paper.
In most cases, these substance changes color when mixed with anacidorbase.
Some inks glow faintly (fluoresce) when under anultravioletlamp. This is a property of many substances, particularly organic substances and body fluids.
Other inks work in a near opposite way byabsorbingultraviolet light but without fluorescing. When these are used onfluorescentpaper, the inked areas fluoresce less than the surrounding paper area when under an ultraviolet lamp. This is especially a property of inks with a yellow tint.
Some UV-visible inks may be detected on a photocopy, due to the relatively strong ultraviolet component in light from the photocopier scanning head.
Examples of inks revealed by ultraviolet light are:
This includes virtually all invisible inks, but pure distilled water can also be used in this way. Application of any fluid will alter the paper surface fibers or sizing.
Fumes created from heating iodine crystals will develop the writing, which will appear brown because the iodine sticks preferentially to the altered areas of the paper. Exposing the paper to strong sunlight will return the writing to its invisible state, as will using a bleach solution.
Slightly dampening paper with a sponge or by steam and then drying it before writing a message will prevent writing from being developed by this method, but overdoing dampening will result in telltale paper cockling.
FormerMI6agentRichard Tomlinsonstated that Pentel Rolling Writer rollerball pens were extensively used by MI-6 agents to produce secret writing in the form of invisible messages while on missions.[16]
In 2002, a gang was indicted for spreading a riot between federal penitentiaries using coded telephone messages, and messages in invisible ink.[17]
In 1995,President Clintonissued an executive order requesting that all agenciesdeclassifyinformation 25 years or older by the year 2000. Six World War I documents referencing the recipes for invisible ink were due to be declassified under this order, including:
In 1999, however, theCentral Intelligence Agency(CIA) successfully exempted these documents, arguing that the recipes provided the basis for more advanced formulas still in use at the time.[19]This exemption made the recipes for invisible ink the oldest classified documents held by theNational Archivesuntil their declassification in 2011. At this time, the CIA no longer considered the documents sensitive due to recent advancements in technology.[20]
Invisible ink is not commonly used in art. Some artists, however, have incorporated invisible ink into their work, either alone or in conjunction with more conventional media.
Jean-Michel Basquiatis known to have worked with invisible ink. In 2012,Sotheby'sLondon discovered Basquiat's signature painted in invisible ink on the 1982 work,Orange Sports Figure. In 2018, analysis by an art conservator revealed invisible ink markings on an untitled Basquiat painting from 1981.[21]
In 2012, theHayward Galleryexhibition,Invisible: Art about the Unseen, 1957-2012, included the 1989 work,Magic Ink, by Gianni Motti. It consisted of two drawings created with undeveloped invisible ink.[22]
In 2015,Aowen Jinexhibited artwork drawn in invisible ink at theHorniman Museumin London. The illustrations, drawn on the walls and floor of the Music Gallery Performance Space, were only visible under UV light.[21]
In addition to traditional chemical methods, modern digital techniques utilize Unicode characters to create invisible text, enabling hidden messages within standard text formats.[23]
Inks that are visible for a period of time without the intention of being made visible again are called disappearing inks. They typically rely on the chemical reaction betweenthymolphthaleinand a basic substance such assodium hydroxide. Thymolphthalein is normally colorless, but turns blue in solution with thebase. As the base reacts with carbon dioxide (always present in the air), thepHdrops below 10.5 and the color disappears. Pens are also available whose inks can be thoroughly erased by swiping a special pen over the original text. Disappearing inks have been used in gag squirtguns, for limited-time secret messages, for security reasons on non-reusable passes, for fraudulent purposes, and for dress-making and other crafts where measurement marks are required to disappear.[24][25][26]
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Incryptography, amusic cipheris analgorithmfor theencryptionof a plaintext into musical symbols or sounds. Music-based ciphers are related to, but not the same asmusical cryptograms. The latter were systems used by composers to create musicalthemesormotifsto represent names based on similarities between letters of the alphabet and musical note names, such as theBACH motif, whereas music ciphers were systems typically used by cryptographers to hide orencodemessages for reasons of secrecy or espionage.
There are a variety of different types of music ciphers as distinguished by both the method of encryption and the musical symbols used. Regarding the former, most are simplesubstitution cipherswith a one-to-one correspondence between individual letters of the alphabet and a specific musical note. There are also historical music ciphers that utilize homophonic substitution (one-to-many),polyphonic substitution(many-to-one), compound cipher symbols, and/orcipher keys; all of which can make the enciphered message more difficult to break.[1]Regarding the type of symbol used for substitution, most music ciphers utilize the pitch of amusical noteas the primary cipher symbol. Since there are fewer notes in a standard musical scale (e.g., seven fordiatonic scalesand twelve forchromatic scales) than there are letters of the alphabet, cryptographers would often combine the note name with additional characteristics––such asoctaveregister,rhythmic duration, orclef––to create a complete set of cipher symbols to match every letter. However, there are some music ciphers which rely exclusively on rhythm instead of pitch[2]or on relativescale degreenames instead of absolute pitches.[3][4][5]
Music ciphers often have both cryptographic and steganographic elements. Simply put, encryption is scrambling a message so that it is unreadable; steganography is hiding a message so no knows it is even there. Most practitioners of music ciphers believed that encrypting text into musical symbols gave it added security because, if intercepted, most people would not even suspect that the sheet music contained a message. However, asFrancesco Lana de Terzinotes, this is usually not because the resulting cipher melody appears to be a normal piece of music, but rather because so few people know enough about music to realize it is not ("ma gl'intelligenti di musica sono poci").[6]A message can also be visually hidden within a page of music without actually being a music cipher.William F. Friedmanembedded a secret message based on FrancisBacon's cipherinto a sheet music arrangement of Stephen Foster's "My Old Kentucky Home" by visually altering the appearance of thenote stems.[7]Another steganographic strategy is to musically encrypt a plaintext, but hide the message-bearing notes within a larger musical score that requires some visual marker that distinguishes them from the meaningless null-symbol notes (e.g., the cipher melody is only in the tenor line or only the notes with stems pointing down).[8][9]
The cipher manuscript from Agostino Amadi there is a musical score in 41v with a pseudo-letter
ciphered in it, which is an imaginary letter that Venice writes to Charles V.Italian historian Paolo Preto.
"...The emperor sent to prince Gritti, with whom he had been familiar for a long time, a music score that looked like a
madrigal....The prince summoned Willaert and the other musicians and asked them to play the melody sent to them by emperor Charles V. When Willaert and the others carefully studied the score, they were unable to play it and confessed they could
not understand it."[10]
Diatonic music ciphers utilize only the seven basic note names of the diatonic scale:A, B, C, D, E, F, andG. While some systems reuse the same seven pitches for multiple letters (e.g., the pitchAcan represent the lettersA,H,O, orV),[11]most algorithms combine these pitches with other musical attributes to achieve a one-to-one mapping. Perhaps the earliest documented music cipher is found in a manuscript from 1432 called "The Sermon Booklets of Friar Nicholas Philip." Philip's cipher uses only five pitches, but each note can appear with one of four different rhythmic durations, thus providing twenty distinct symbols.[12]A similar cipher appears in a 15th-century British anonymous manuscript[13]as well as in a much later treatise byGiambattista della Porta.[14]
In editions of the same treatise (De Furtivis Literarum Notis), Porta also presents a simpler cipher which is much more well-known.Porta's music ciphermaps the lettersAthroughM(omittingJandK) onto a stepwise, ascending, octave-and-a-half scale ofwhole notes(semibreves); with the remainder of the alphabet (omittingVandW) onto a descending scale ofhalf notes(minims).[15]Since alphabetic and scalar sequences are in such close step with each other, this is not a very strong method of encryption, nor are the melodies it produces very natural. Nevertheless, one finds slight variations of this same method employed throughout the 17th and 18th centuries byDaniel Schwenter(1602),[16]John Wilkins(1641),[17]Athanasius Kircher(1650),[18]Kaspar Schott(1655),[19]Philip Thicknesse(1722),[20]and even the British Foreign Office (ca. 1750).[21]
Music ciphers based on thechromatic scaleprovide a larger pool of note names to match with letters of the alphabet. Applyingsharpsandflatsto the seven diatonic pitches yields twenty-one unique cipher symbols. Since this is obviously still less than a standard alphabet, chromatic ciphers also require either a reduced letter set or additional features (e.g., octave register or duration). Most chromatic ciphers were developed by composers in the 20th Century when fully chromatic music itself was more common. A notable exception is a cipher attributed to the composerMichael Haydn(brother of the more famousJoseph Haydn).[22]Haydn's algorithm is one of the most comprehensive with symbols for thirty-one letters of theGerman alphabet, punctuations (usingrest signs), parentheses (usingclefs), and word segmentation (usingbar lines). However, because many of the pitches areenharmonic equivalents, this cipher can only be transmitted as visual steganography, not via musical sound. For example, the notesC-sharpandD-flatare spelled differently, but they sound the same on a piano. As such, if one were listening to an enciphered melody, it would not be possible to hear the difference between the lettersKandL. Furthermore, the purpose of this cipher was clearly not to generate musical themes that could pass for normal music. The use of such an extreme chromatic scale produces wildly dissonant,atonalmelodies that would have been obviously atypical for Haydn's time.
Although chromatic ciphers did not seemed to be favored by cryptographers, there are several 20th-century composers who developed systems for use in their own music:Arthur Honegger,[23]Maurice Duruflé,[24]Norman Cazden,[25]Olivier Messiaen,[26]andJacques Chailley.[27]Similar to Haydn's cipher, most likewise match the alphabet sequentially onto a chromatic scale and rely on octave register to extend to twenty-six letters. Only Messiaen's appears to have been thoughtfully constructed to meet the composer's aesthetic goals. Although he also utilized different octave registers, the letters of the alphabet are not mapped in scalar order and also have distinct rhythmic values. Messiaen called his musical alphabet thelangage communicable, and used it to embed extra-musical text throughout his organ workMéditations sur le Mystère de la Sainte Trinité.
In a compound substitution cipher, each single plaintext letter is replaced by a block of multiple cipher symbols (e.g., 'a' = EN or 'b' = WJU). Similarly, there are compound music ciphers in which each letter is represented by a musical motive with two or more notes. In the case of the former, the compound symbols are to makefrequency analysismore difficult; in the latter, the goal is to make the output more musical.
For example, in 1804, Johann Bücking devised a compound cipher which generates musical compositions in the form of aminuetin thekeyof G Major.[28]Each letter of the alphabet is replaced by a measure of music consisting of a stylistically typical motive with three to six notes. After the plaintext is enciphered, additional pre-composed measures are appended to the beginning and end to provide a suitable musical framing. A few years earlier,Wolfgang Amadeus Mozartappears to have employed a similar technique (with much more sophisticated musical motives), although more likely intended as a parlor game than an actual cipher.[29][30]Since the compound symbols are musically meaningful motives, these ciphers could also be considered similar tocodes.
Friedrich von Öttingen-Wallerstein proposed a different type of compound music cipher modeled after apolybius square cipher.[31]Öttingen-Wallerstein used a 5x5 grid containing the letters of the alphabet (hidden within the names of angels). Instead of indexing the rows and columns with coordinate numbers, he used thesolfegesyllables Ut, Re, Mi Fa, and Sol (i.e., the first five degrees of a diatonic scale). Each letter, therefore, becomes a two-note melodic motive. This same cipher appears in treatises byGustavus Selenus(1624)[32]and Johann Balthasar Friderici (1665)[33](but without credit to the earlier version of Öttingen-Wallerstein).
Because Öttingen-Wallerstein's cipher uses relativescale degrees, rather than fixed note names, it is effectively apolyalphabetic cipher. The same enciphered message could be transposed to a different musical key––with different note names––and still retain the same meaning. The musical key literally becomes a cipher key (orcryptovariable), because the recipient needs that additional information to correctly decipher the melody. Öttingen-Wallerstein insertedrestsas cipherkey markers to indicate when a new musical key was needed to decrypt the message.
Francesco Lana de Terziused a more conventional text-string cryptovariable, to add security to a very straightforward 'Porta-style' music cipher (1670).[34]Similar to aVigenère cipher, a single-letter cipher key shifts the position of the plaintext alphabet in relation to the sequence musical cipher symbols; a multi-letter key word shifts the musical scale for each letter of the text in a repeating cycle.
A more elaborate cipherkey algorithm was found in an anonymous manuscript in Port-Lesney, France, most likely from the mid-18th century.[35]The so-called'Port-Lesney' music cipheruses a mechanical device known as anAlberti cipher disk[36]There are two rotating disks: the outer disk contains two concentric rings (one withtime signaturesand the other with letters of the alphabet); the inner disk has a ring of compound musical symbols, and a small inner circle with three differentclefsigns. The disks are rotated to align the letters of the alphabet with compound musical symbols to encrypt the message. When the melody is written out on a music staff, the corresponding clef and time signature are added to the beginning to indicate the cipher key (which the recipient aligns on their disk to decipher the message). This particular music cipher was apparently very popular, with a dozen variations (in French, German, and English) appearing throughout the 18th and 19th centuries.[37][38][39][40]
The more recent Solfa Cipher[41]combines some of the above cryptovariable techniques. As the name suggests, Solfa Cipher uses relativesolfegedegrees (like Öttingen-Wallerstein) rather than fixed pitches, which allows the same encrypted message to be transposable to different musical keys. Since there are only seven scale degrees, these are combined with a rhythmic component to create enough unique cipher symbols. However, instead of absolute note lengths (e.g., quarter note, half note, etc.) that are employed in most music ciphers, Solfa Cipher uses relativemetricplacement. This type oftonal-metric[42]cipher makes the encrypted melody both harder to break and more musically natural (i.e. similar to common-practice tonal melodies).[43]To decrypt a cipher melody, the recipient needs to know in which musical key and with what rhythmic unit the original message was encrypted, as well as the clef sign and metric location of the first note. The cipher key could also be transmitted as a date by usingSolfalogy, a method of associating each unique date with a tone and modal scale.[44]To further confound interceptors, the transcribed sheet music could be written with a decoy clef, key signature, and time signature. The musical output, however, is a relatively normal, simple, singable tune in comparison to the disjunct, atonal melodies produced by fixed-pitch substitution ciphers.
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ThePolybius square, also known as thePolybius checkerboard, is a device invented by theancient GreeksCleoxenus and Democleitus, and made famous by thehistorianand scholarPolybius.[1]The device is used forfractionatingplaintextcharacters so that they can be represented by a smaller set of symbols, which is useful fortelegraphy,steganography, andcryptography. The device was originally used for fire signalling, allowing for the coded transmission of any message, not just a finite number of predetermined options as was the convention before.[1]
According to Polybius'Histories,the device was invented byCleoxenusandDemocleitus, and further developed by Polybius himself. The device partitioned the alphabet into five tablets with five letters each (except for the last one with only four). There are no surviving tablets from antiquity. Letters are represented by two numbers from one to five, allowing the representation of 25 characters using only 5 numeric symbols.
The original square used theGreek alphabetlaid out as follows:
Modern Greek still uses that same alphabet, as do implementations of the Polybius square in that language.
With theLatin alphabet, this is the typical form:
This alphabet, and this latter form of the Polybius square, is used when implementing the square in other Western European languages such as English, Spanish, French, German, Italian, Portuguese, and Dutch.
Each letter is then represented by its coordinates in the grid. For example, "BAT" becomes "12 11 44". The 26 letters of the Latin/English alphabet do not fit in a 5 × 5 square, two letters must be combined (usually I and J as above, though C and K is an alternative). Alternatively, a 6 × 6 grid may be used to allow numerals or special characters to be included as well as letters.
A 6 × 6 grid is also usually used for theCyrillic alphabet(the most common variant has 33 letters, but some have up to 37)[citation needed]or Japanesehiragana(seecryptography in Japan).
Akeycould be used to reorder the alphabet in the square, with the letters (without duplicates) of the key being placed at the beginning and the remaining letters following it in alphabetical order.[2]For example, the key phrase "polybius cipher"would lead to the reordered square below.
There are several encryption methods using the Polybius square. Three of them are described below.
Let's encrypt the word "SOMETEXT" with aCaesar cipherusing a shift equal to the side of our square (5). To do it, locate the letter of the text and insert the one immediately below it in the same column for the ciphertext. If the letter is in the bottom row, take the one from the top of the same column.
Thus, after encryption, we get:
A more complicated method involves aBifid cipherwithout a key (or, in other words, with a key of plain alphabet):
The message is transformed into coordinates on the Polybius square, and the coordinates are recorded vertically:
Then the coordinates are read row by row:
Next, the coordinates are converted into letters using the same square:
Thus, after encryption, we get:
An advanced variation, which involves the following: the obtained primary ciphertext (result From Method 2) is encrypted again. In this case, it is written out without being split into pairs.
The resulting sequence of digits is cyclically shifted to the left by one step (an odd number of steps (move 3 to the end)):
This sequence is again divided into groups of two:
And is replaced with the final ciphertext according to the table:
Thus, after encryption, we get:
In hisHistories,Polybius outlines the need for effective signalling in warfare, leading to the development of the square. Previously, fire-signalling was useful only for expected, predetermined messages, with no way to convey novel messages about unexpected events.[1]According to Polybius, in the 4th century BCE,Aeneas Tacticusdevised ahydraulic semaphore systemconsisting of matching vessels with sectioned rods labelled with different messages such as "Heavy Infantry", "Ships", and "Corn".[1]This system was slightly better than the basic fire-signalling, but still lacked the ability to convey any needed message. The Polybius square was used to aid in telegraphy, specifically fire-signalling. To send a message, the sender would initially hold up two torches and wait for the recipient to do the same to signal that they were ready to receive the message.[1]The sender would then hold up the first set of torches on his left side to indicate to the recipient which tablet (or row of the square) was to be consulted. The sender would then raise a set of torches on his right side to indicate which letter on the tablet was intended for the message.[1]Both parties would need the same tablets, a telescope (a tube to narrow view, no real magnification), and torches.[1]
The Polybius square has also been used in the form of the "knock code" to signal messages between cells inprisonsby tapping the numbers on pipes or walls.[2]It is said to have been used bynihilistprisoners of theRussianCzarsand also byUSprisoners of warduring theVietnam War.[3]
Arthur Koestlerdescribes the code being used by political prisoners ofStalinin the 1930s in his anti-totalitarian novelDarkness at Noon. (Koestler had been a prisoner-of-war during theSpanish Civil War.)
Indeed, it can be signalled in many simple ways (flashing lamps, blasts of sound,drums,smoke signals) and is much easier to learn than more sophisticated codes like theMorse code. However, it is also somewhat less efficient than more complex codes.
The simple representation also lends itself tosteganography. The figures from one to five can be indicated byknotsin a string, stitches on a quilt, contiguous letters before a wider space or many other ways.[3]
The Polybius square is also used as a basic cipher called the Polybius cipher. This cipher is quite insecure by modern standards, as it is asubstitution cipherwith characters being substituted for pairs of digits, which is easily broken throughfrequency analysis.[2]
The Polybius square and the Polybius cipher can be combined with other cryptographic methods such as theADFGVX cipher,[2]Homophonic cipher[2]and more.
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Steganographic file systemsare a kind offile systemfirst proposed byRoss Anderson,Roger Needham, andAdi Shamir. Their paper proposed two main methods of hiding data: in a series of fixed size files originally consisting of random bits on top of which 'vectors' could be superimposed in such a way as to allow levels of security to decrypt all lower levels but not even know of the existence of any higher levels, or an entire partition is filled with random bits and files hidden in it.
In a steganographic file system using the second scheme,filesare not merely stored, nor storedencrypted, but the entirepartitionis randomized - encrypted files strongly resemble randomized sections of the partition, and so when files are stored on the partition, there is no easy way to discern between meaninglessgibberishand the actual encrypted files. Furthermore, locations of files are derived from the key for the files, and the locations are hidden and available to only programs with the passphrase. This leads to the problem that very quickly files can overwrite each other (because of theBirthday Paradox); this is compensated for by writing all files in multiple places to lessen the chance of data loss.
While there may seem to be no point to a file system which is guaranteed to either be grossly inefficient storage space-wise or to cause data loss and corruption either from data collisions or loss of thekey(in addition to being a complex system, and for having poor read/write performance), performance was not the goal of StegFS. Rather, StegFS is intended to thwart"rubberhose attacks", which usually work because encrypted files are distinguishable from regular files, and authorities can coerce the user until the user gives up the keys and all the files are distinguishable as regular files. However, since in a steganographic file system, the number of files are unknown and every byte looks like an encrypted byte, the authorities cannot know how many files (and hence, keys) are stored. The user hasplausible deniability— he can say there are only a few innocuous files or none at all, and anybody without the keys cannot gainsay the user.
Poul-Henning Kamphas criticized thethreat modelfor steganographic file systems in his paper onGBDE,[1]observing that in certain coercive situations, especially where the searched-for information is in fact not stored in the steganographic file systems, it is not possible for a subject to "get off the hook" by proving that all keys have been surrendered.
Other methods exist; the method laid out before is the one implemented byStegFS, but it is possible tosteganographicallyhide data within image (e.g.PNGDrive) or audio files-ScramDiskor the Linuxloop devicecan do this.[citation needed]
Generally, a steganographic file system is implemented over a steganographic layer, which supplies just the storage mechanism. For example, the steganographic file system layer can be some existing MP3 files, each file contains a chunk of data (or a part of the file system). The final product is a file system that is hardly detected (depending on the steganographic layer) that can store any kind of file in a regular file system hierarchy.
TrueCryptallows for "hidden volumes" - two or more passwords open different volumes in the same file, but only one of the volumes contains secret data.
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Asteganographysoftware tool allows a user to embed hidden data inside a carrier file, such as an image or video, and later extract that data.
It is not necessary to conceal the message in the original file at all. Thus, it is not necessary to modify the original file and thus, it is difficult to detect anything. If a given section is subjected to successive bitwise manipulation to generate the cyphertext, then there is no evidence in the original file to show that it is being used to encrypt a file.
Thecarrieris the signal, stream, or data file into which the hidden data is hidden by making subtle modifications. Examples include audio files, image files, documents, and executable files. In practice, the carrier should look and work the same as the original unmodified carrier, and should appear benign to anyone inspecting it.
Certain properties can raise suspicion that a file is carrying hidden data:
It is a cryptographic requirement that the carrier (e.g. photo) is original, not a copy of something publicly available (e.g., downloaded). This is because the publicly available source data could be compared against the version with a hidden message embedded.
There is a weaker requirement that the embedded message not change the carrier's statistics (or other metrics) such that the presence of a message is detectable. For instance, if the least-significant-bits of the red camera-pixel channel of an image has a Gaussian distribution given a constant colored field, simple image steganography which produces a random distribution of these bits could allow discrimination of stego images from unchanged ones.
The sheer volume of modern (ca 2014) and inane high-bandwidth media (e.g., youtube.com, bittorrent sources. eBay, Facebook, spam, etc.) provides ample opportunity for covert information±.
Hidden data may be split among a set of files, producing acarrier chain, which has the property that all the carriers must be available, unmodified, and processed in the correct order in order to retrieve the hidden data. This additional security feature usually is achieved by:
Steganography tools aim to ensure robustness against modernforensic methods, such as statisticalsteganalysis. Such robustness may be achieved by a balanced mix of:
If the data is detected, cryptography also helps to minimize the resulting damage, since the data is not exposed, only the fact that a secret was transmitted. The sender may be forced to decrypt the data once it is discovered, butdeniable encryptioncan be leveraged to make the decrypted data appear benign.
Strong steganography software relies on amulti-layered architecturewith a deep, documentedobfuscationprocess.
The carrier engine is the core of any steganography tool. Different file formats are modified in different ways, in order to covertly insert hidden data inside them. Processing algorithms include:
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Anaudio watermarkis a unique electronic identifier embedded in an audio signal, typically used to identify ownership of copyright. It is similar to awatermarkon a photograph.
Digital watermarkingis the process of embedding information into a signal (e.g. audio, video or pictures) in a way that is difficult to remove. If the signal is copied, then the information is also carried in the copy. Watermarking has become increasingly important to enable copyright protection and ownership verification.
One technique for audio watermarking is spread spectrum audio watermarking (SSW). In SSW, a narrow-band signal is transmitted over a much larger bandwidth such that the signal energy presented in any signal frequency is undetectable. Thus the watermark is spread over many frequency bands so that the energy in one band is undetectable. An interesting feature of this watermarking technique is that destroying it requires noise of high amplitude to be added to all frequency bands.
Spreading spectrum is done by apseudonoise(PN) sequence. In conventional SSW approaches, the receiver must know the PN sequence used at the transmitter as well as the location of the watermark in the watermarked signal for detecting hidden information.
Although PN sequence detection is possible by usingheuristicapproaches such asevolutionary algorithms, the high computational cost of this task can make it impractical. Much of thecomputational complexityinvolved in the use ofevolutionary algorithmsas an optimization tool is due to thefitness functionevaluation that may either be very difficult to define or be computationally very expensive.
One of the recent proposed approaches—in fast recovering the PN sequence- is the use of fitness granulation as a promising "fitness approximation" scheme. With the use of the fitness granulation approach called "Adaptive Fuzzy Fitness Granulation (AFFG)",[1]the expensive fitness evaluation step is replaced by an approximate model. When evolutionary algorithms are used as a means to extract the hidden information, the process is called Evolutionary Hidden Information Detection, whether fitness approximation approaches are used as a tool to accelerate the process or not.
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Security printingis the field of theprintingindustry that deals with the printing of items such asbanknotes,cheques,passports,tamper-evidentlabels,security tapes, productauthentication,stock certificates,postage stamps, andidentity cards. The main goal of security printing is to preventforgery, tampering, orcounterfeiting. More recently many of the techniques used to protect these high-value documents have become more available to commercial printers, whether they are using the more traditionaloffsetandflexographicpresses or the newer digital platforms. Businesses are protecting their lesser-value documents such as transcripts, coupons and prescription pads by incorporating some of the features listed below to ensure that they cannot be forged or that alteration of the data cannot occur undetected.
A number of technical methods are used in the security printing industry.[1]Security printing is most often done onsecurity paper, but it can also occur on plastic materials.
Secured documents, such as banknotes, usevisibleandtactilefeatures to allow humans to verify their authenticity without tools. TheEuropean Central Bank(ECB) recommends feel, look, and tilt:[2]First check the tactility of the banknote (including the substrate), then look at the optical design and finally the characteristics of certain optical features when tilting the banknote in relation to the incident light.
In general, the introduction of a new banknote series is accompanied by information campaigns describing the design and the security features. Several central banks also providemobile appsexplaining the characteristics by interactive methods and enrich them byanimatedeffects. In general, they use the camera of amobile deviceto explain the features of a presented banknote. As they do not support the direct verification of authenticity they also work with simple printouts or screen displays.
The substrate of most banknotes is made ofpaper, almost always from cotton fibres for strength and durability; in some cases linen or specially coloured or forensic fibres are added to give the paper added individuality and protect against counterfeiting. Paper substrate may also include windows based on laser-cut holes covered by a security foil with holographic elements. All of this makes it difficult to reproduce using common counterfeiting techniques.
Some countries, includingCanada,Nigeria,Romania,Mexico,Hong Kong,New Zealand,Israel,Singapore,Malaysia,United Kingdom, andAustralia, producepolymer (plastic) banknotes, to improve longevity and to make counterfeiting more difficult. Polymer can include transparent windows, diffraction grating, and raised printing.[7]
Mostcurrenciesuse different dimensions of length, width, or both for the differentdenominations, with smaller formats for the lower denominations and larger formats for the higher denominations, to hinder reuse of the substrate with embedded security features for counterfeiting higher denominations.
Blind and visually impaired people may also rely on the format for distinguishing between the denominations.
True watermark
A truewatermarkis a recognizable image or pattern in paper that appears lighter or darker than surrounding paper when viewed with a light from behind the paper, due to paper density variations. A watermark is made by impressing a water coated metal stamp or dandy roll onto the paper during manufacturing. Watermarks were first introduced in Bologna, Italy in 1282; as well as their use in security printing, they have also been used by paper makers to identify their product. For proofing the authenticity, the thinner part of the watermark will shine brighter with a light source in the background and darker with a dark background. The watermark is a proven anti-counterfeit technology because most counterfeits only simulate its appearance by using a printing pattern.
Simulated watermark
Printed with white ink, simulated watermarks have a different reflectance than the base paper and can be seen at an angle. Because the ink is white, it cannot be photocopied or scanned.[8]A similar effect can be achieved by iriodin varnish which creates reflections under certain viewing angles only and is transparent otherwise.
Watermarks are sometimes simulated on polymer currency by printing an according pattern, but with little anti-counterfeiting effect. For example, the Australian dollar has its coat of arms watermarked on all its plastic bills. ADiffractive Optical Element(DOE) within the transparent window can create a comparable effect but requires a laser beam for its verification.
See-through registers are based on complementarypatternson the obverse and reverse of the banknote and constitute a complete pattern underbacklightconditions. Examples are theDof theDeutsche Mark(1989 series, BBk III) and the value number of the first series ofeuro banknotes(ES1). Counterfeiting is difficult because theprinting registrationrequires an extremely high printing accuracy on both sides and minor deviations are easily detectable.
Polymer banknoteswhich are printed on a basically transparent substrate easily provide clear areas by sparing the whitecoating. This window may beoverprintedby patterns. Initially this was the main human security feature for polymer banknotes which cannot use watermark or security threads. It attracted counterfeiting of large volumes when printing technology for polymer substrate became commonly available. Therefore new designs additionallylaminatethis window with an ultra-thin security foil, e.g., on theFrontier seriesof theCanadian dollarwhich was issued from 2011, and theAustralian dollar(2nd series) issued from 2016.
A very similar security feature is achieved with banknotes on paper substrate. For this an area of up to 300 mm² is punched out and sealed with a partially transparent security foil. The ES2 series of euro banknotes is using this feature for the higher denominations (EUR 20 and above) and calls itportrait window. TheEuropean Central Bank(ECB) recommends tolook at the banknote against the light – the window in the hologram becomes transparent and reveals a portrait of Europa on both sides of the note.[9]
Micro-perforationis used asMicroperfin theSwiss francand theRomanian leu. Very small holes are punched or laser-engraved into the substrate or a foil application without generating acrater. In backlight illumination, the holes form a pattern, e.g., the value numeral like in the SFR 20 (eighth series).
Aguillochéis an ornamental pattern formed of two or more curved bands that interlace to repeat a circular design. They are made with ageometric lathe.
This involves the use of extremely small text, and is most often used on currency and bank checks. The text is generally small enough to be indiscernible to the naked eye without either close inspection or the use of a magnifying glass. Cheques, for example, usemicroprintas the signature line.
Optically Variable Ink(OVI) displays different colours depending on the angle at which it is viewed. It usesmica-based glitter.[10]As an example, the euro banknotes use this feature asemerald numberon the ES2 series. The ECB recommends to "tilt the banknote". The shiny number in the bottom left corner displays an effect of the light that moves up and down. The number also changes colour from emerald green to deep blue. The EUR 100 and EUR 200 banknotes also show € symbols inside the number.[11]
Coloured magnetizable inks are prepared by including chromatic pigments of high colour strength. The magnetic pigments’ strong inherent colour generally reduces the spectrum of achievable shades. Generally, pigments should be used at high concentrations to ensure that sufficient magnetizable material is applied even in thin offset coats. Some magnetic pigment are best suited for coloured magnetizable inks due to their lower blackness.
Homogeneous magnetization (no preferred orientation) is easily obtained on pigment made of spherical particles. Best results are achieved when remanence and coercive field strength are very low and the saturating magnetization is high.
When pearlescent pigments are viewed at different angles the angle of the light as it's perceived makes the colour appear to change as the magnetic fields within the particles shift direction.
Ahologrammay be embedded either via hot-stamping foil, wherein an extremely thin layer of only a fewmicrometersof depth is bonded into the paper or a plastic substrate by means of ahot-melt adhesive(called a size coat) and heat from a metal die, or it may be directly embossed as holographic paper, or onto the laminate of a card itself.
When incorporated with a custom design pattern or logo, hologram hot stamping foils become security foils that protect credit cards, passports, bank notes and value documents from counterfeiting. Holograms help in curtailing forging, and duplication of products hence are very essential for security purposes. Once stamped on a product, they cannot be removed or forged, enhancing the product at the same time. Also from a security perspective, if stamped, a hologram is a superior security device as it is virtually impossible to remove from its substrate.[citation needed]
Metal threads and foils, from simple iridescent features to foil colour copying to foils with additional optically variable effects are often used.
There are two kinds of security threads. One is a thin aluminum coated and partly de-metallized polyester film thread with microprinting which is embedded in the security paper as banknote or passport paper. The other kind of security thread is the single or multicolour sewing thread made from cotton or synthetic fibers, mostly UV fluorescent, for thebookbindingof passport booklets. In recent designs the security thread was enhanced with other security features such as holograms or three-dimensional effects when tilted.
On occasion, the banknote designers succumb to the Titanic effect (excess belief in the latest technology), and place too much faith in some particular trick. An example is the forgery of British banknotes in the 1990s. British banknotes in the 1990s featured a "windowed" metal strip through the paper about 1 mm wide that comes to the paper surface every 8 mm. When examined in reflected light, it appears to have a dotted metallic line running across it, but when viewed through transmitted light, the metal strip is dark and solid.
Duplicating this was thought to be difficult, but a criminal gang was able to reproduce it quickly. They used a cheap hot-stamping process to lay down a metal strip on the surface of the paper, then printed a pattern of solid bars over it using white ink to leave the expected metal pattern visible. At their trial, they were found to have forged tens of millions of pounds’ worth of notes over a period of years.[12]
The use of colour can greatly assist the prevention of forgeries. By including a colour on a document a colour photocopier must be used in the attempt to make a copy however the use of these machines also tends to enhance the effectiveness of other technologies such as Void Pantographs and Verification Grids (see Copy-evident below).
By using two or more colours in the background and blending them together a prismatic effect can be created. This can be done on either a traditional or a digital press. When a document using this technique is attempted to be photocopied the scanning and re-creation by a colour copier is inexact usually resulting in banding or blotching and thereby immediate recognition of the document as being a copy.
A frequent example of prismatic colouring is on checks where it is combined with other techniques such as thevoid pantographto increase the difficulty of successful counterfeiting.[13]
Sometimes only the original document has value. An original, signedchequefor example has value but a photocopy of it does not. An original prescription script can be filled but a photocopy of it should not be. Copy-evident technologies provide security to hard copy documents by helping distinguish between the original document and the copy.
The most common technology to help differentiate originals from copies is thevoid pantograph. Void pantographs are essentially invisible to the untrained, naked eye on an original but when scanned or copied the layout of lines, dots and dashes will reveal a word (frequently VOID and hence the name) or symbol that clearly allows the copy to be identified. This technology is available on both traditional presses (offset and flexographic) and on the newer digital platforms. The advantage of a digital press is that in a single pass through the printer a void pantograph with all the variable data can be printed on plain paper.
Copy-evident paper, sometimes marketed as ‘security paper’, is pre-printed void pantograph paper that was usually produced on an offset or flexographic press. The quality of the void pantograph is usually quite good because it was produced on a press with a very high resolution, and, when only a small number of originals are to be printed, it can be a cost-effective solution; however, the advent of the digital printer has rapidly eroded this benefit.
A second technology which complements and enhances the effectiveness of the void pantograph is the Verification Grid. This technology is visible on the original, usually as fine lines or symbols but when photocopied these lines and images disappear; the inverse reaction of the void pantograph. The most common examples of this technology are on the fine lines at the edge of a cheque which will disappear when copied or on a coupon when a symbol, such as a shopping cart, disappears when an unauthorized copy is made. Verification Grid is available for either traditional or digital presses.
Together the void pantograph and the Verification Grid complement each other because the reactions to copying are inverse, resulting in a higher degree of assurance that a hard copy document is an original.
Banknotes are typically printed with fine alignment (so-calledsee-through registration window) between the offset printing on each side of the note. This allows the note to be examined for this feature, and provides opportunities to unambiguously align other features of the note with the printing. Again, this is difficult to imitate accurately enough in most print shops.
Several types of ink are available which change colour with temperature. Security ink with a normal "trigger" temperature of 88 °F (31 °C), which will either disappear or change colours when the ink is rubbed, usually by the fingertips. This is based on a thermochromatic effect.
Serial numbershelp make legitimate documents easier to track and audit. However, they are barely useful as a security feature because duplicates of an existing serial number are not easily detectable, except for a series of identical counterfeits.
To support correct identification serial numbers normally have acheck digitto verify the correct reading of the serial number. In banknote printing the unique serial number provides effective means for the monitoring and verification of the production volume. In some cases the recording of serial numbers may help to track and identify banknotes fromblackmailorrobbery.
In most currencies the serial number is printed on two edges of the banknotes to aggravate the making of so-calledcomposed banknotesby combining parts of different banknotes. Even if made from genuine banknotes, most central banks consider such items as manipulated banknotes without value if the serial numbers do not match.
Security paperfor banknotes is different from standard paper due to special ingredients like fibers fromcotton,linenorabaca. Together with intaglio printing crisp feeling provides an excellent tactile perception (crisp feeling) to reject counterfeits which are based on standard paper withcellulose fibers. Polymer substrates and limp banknotes on paper substrate do not offer this tactile characteristic.
Intaglio printingis a technique in which the image is incised into a surface. Normally, copper or zinc plates are used, and the incisions are created byetchingorengravingthe image, but one may also usemezzotint. In printing, the surface is covered in ink, and then rubbed vigorously with tarlatan cloth or newspaper to remove the ink from the surface, leaving it in the incisions. A damp piece of paper is placed on top, and the plate and paper are run through a printing press that, through pressure, transfers the ink to the paper.
The very sharp printing obtained from the intaglio process is hard to imitate by other means. Intaglio also allows for the creation of latent images which are only visible when the document is viewed at a very shallow angle.
The mobile appValiCashfromKoenig & Bauerevaluates specific characteristics of the intaglio printing ofeuro banknotesprinted on paper substrate.[14]It is available foriOSdevices and takes a picture of the banknote. Within a few seconds it determines abnormality by a message "not successful" but cannot finally identifycounterfeits.
The substrate may beembossedto create raised designs as tactile security feature. It may be combined with intaglio printing. As an example, the euro series ES2 has different pattern of lines at the short edges of the banknote to support blind people in distinguishing thedenominations.
A counterfeit banknote detection pencan be used to quickly determine the starch in wood-based paper substrate. While genuine banknotes hardly change color at all, counterfeits turn black or blue immediately. This method, which is not very reliable – there is no color change on newsprint – is often used in the retail trade for reasons of cost and time.
Carefully created images can be hidden in the background or in a picture on a document. These images cannot be seen without the help of an inexpensive lens of a specific line screening. When placed over the location of the image and rotated the image becomes visible. If the document is photocopied the Halo image is lost. A known implementation isScrambled Indicia.[15]
Halo can be printed on traditional or digital presses. The advantage of traditional presses is that multiple images can be overlaid in the same location and become visible in turn as the lens is rotated.
Halo is used as a technique to authenticate the originality of the document and may be used to verify critical information within the document. For example, the value of a coupon might be encoded as a Halo image that could be verified at the time of redemption or similarly the seat number on a sporting event ticket.
Pressure-sensitive or hot stamped labels characterized with a normal (gray or colored) appearance. When viewed via a special filter (such as a polarizer) an additional, normally latent, image appears. With intaglio printing, a similar effect may be achieved for viewing the banknote from a slanted angle.
False-positive testing derives its name because the testing requires both a false and a positive reaction to authenticate a document. The most common instance is the widely available counterfeit detector marker seen in many banks and stores.
Counterfeit detector markers use a chemical interaction with the substrate, usually paper, of a document turning it a particular color. Usually a marker turns newsprint black and leaves currency or specially treated areas on a document clear or gold. The reaction and coloring varies depending upon the formulation. Banknotes, being a specially manufactured substrate, usually behave differently than standard newsprint or other paper and this difference is how counterfeits are detected by the markers.
False-positive testing can also be done on documents other than currencies as a means to test their authenticity. With the stroke of a marker a symbol, word or value can be revealed that will allow the user to quickly verify the document, such as a coupon. In more advanced applications the marker creates a barcode which can be scanned for verification or reference to other data within the document resulting in a higher degree of assurance of authenticity.
Photocopied documents will lack the special characteristics of the substrate so are easily detectable. False-positive testing generally is a one time test because once done the results remain visible so while useful as part of a coupon this technique is not suitable for ID badges for example.
Fluorescent dyesreact withfluorescenceunderultravioletlight or other unusual lighting. These show up as words, patterns or pictures and may be visible or invisible under normal lighting. This feature is also incorporated into many banknotes and other documents - e.g. Northern Ireland NHS prescriptions show a picture of local '8th wonder' the Giant's Causeway in UV light. Some producers include multi-frequency fluorescence, such that different elements fluoresce under specific frequencies of light.Phosphorescencemay accompany fluorescence and shows an after-glow when the UV light is switched off.
Inks may have identical color characteristics in the visible spectrum but differ in theinfraredspectrum.
Machine-readable features are used inpassportsforborder controland inbanknote processing.
There are the following machine-readable features (extract):
Because of the speed with which they can be read by computer systems, magnetic ink character recognition is used extensively in banking, primarily for personal checks. The ink used inmagnetic ink character recognition(MICR) technology is also used to greatly reduce errors in automated (or computerized) reading. The pigment is dispersed in a binder system (resin, solvent) or a wax compound and applied either by pressing or by hot melt to a carrier film (usually polyethylene).[16]
Some people believe that the magnetic ink was intended as a fraud prevention concept, yet the original intent was to have a non-optical technology so that writing on the cheque, like signatures, would not interfere with reading. The main magnetic fonts (E13-B and CMC7) are downloadable for a small fee and in addition magnetic toner is available for many printers. Some higher resolution toners have sufficient magnetic properties for magnetic reading to be successful without special toner.
Phosphorescencemay accompany fluorescence and shows an after-glow when the UV light is switched off.
In the late twentieth century advances in computer and photocopy technology made it possible for people without sophisticated training to easily copy currency. In an attempt to prevent this, banks have sought to add filtering features to the software and hardware available to the public that senses features of currency, and then locks out the reproduction of any material with these marks. One known example of such a system is theEURion constellation.
With the advent ofRadio Frequency Identification(RFID) which is based onsmart cardtechnology, it is possible to insert extremely small RF-active devices into the printed product to enhance document security. This is most apparent in modern biometric passports, where an RFID chip mirrors the printed information.Biometric passportsadditionally include data for the verification of an individual'sfingerprintorface recognitionat automated border control gates.
Acopy detection patternor adigital watermarkcan be inserted into adigital imagebefore printing the security document. These security features are designed to be copy-sensitive[17]and authenticated with an imaging device.[18]
Most central banks also implement so-calledLevel 3(L3) security features which are kept totally secret for their ingredients as well as their sophisticated measurement. Such covert features may be embedded within the substrate and/or the printing ink and are not commercially available. They are the ultimate safeguard in banknote security and restricted to the use of central banks. The machine-readableM-FeaturefromGiesecke+Devrientis the worldwide leading L3 feature and currently used by more than 70 central banks and more than 100 billion banknotes in circulation.[19]Other products areENIGMAfromDe La Rue[20]andLevel III Authenticationfrom Spectra Systems.[21]
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https://en.wikipedia.org/wiki/Security_printing
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TheClandestine HUMINTpage adheres to the functions within the discipline, includingespionageand active counterintelligence.
The page deals withClandestine HUMINT operational techniques, also known as "tradecraft". It applies to clandestine operations for espionage, and a clandestine phase beforedirect action(DA) orunconventional warfare(UW). Clandestine HUMINT sources at certain times act as local guides forspecial reconnaissance(SR).
Many of the techniques are important incounterintelligence. Defensive counterintelligence personnel needs to recognize espionage, sabotage, and so on, in process. Offensive counterintelligence specialists may use them against foreign intelligence services (FIS).
While DA and UW can be conducted by national military or paramilitary organizations,al-Qaedaand similar non-state militant groups that appear to use considerably differentclandestine cell systemstructure, for command, control and operations, from those used by national forces. Cell systems are evolving to more decentralized models, sometimes because they are enabled by new forms of electronic communications.
This page deals primarily with one's assets. Seedouble agentfor additional information adversary sources that a country has turned to its side.
This description is based around the foreign intelligence service, ofcountry B, operating in and againstcountry A. It may also include operations against non-state organizations operating incountry B, with or withoutcountry Bsupport. It may also involve offensive counterintelligence againstcountry Dassets operating incountry B.
The basic structure here can be pertinent to a domestic service operating against a non-national group within its borders. Depending on the legal structure of the country, there may be significant, or very few, restrictions on domestic HUMINT. The most basic question will be whether criminal prosecution, or stopping operations, is the goal. Typically, criminal prosecution will be the primary goal against drug and slavery groups, with breaking up their operations the secondary goal. These priorities, however, are apt to reverse in dealing with terrorist groups.
If there are separate organizations with diplomatic and non-official cover, there may be two chiefs. Sufficiently large stations may have several independent, compartmented groups.
Nations vary as to how well hidden they choose to have all, part or none of their intelligence personnel under the guise of diplomatic immunity. Frequently, at least one individual is known to the host country, so there can be a deniable channel of communications. If the nations are allies, many of the intelligence personnel may be known and actively cooperating.
Certain diplomatic titles were often assumed to be cover jobs. With the United Kingdom, "passport control officer" was, much of the time, an intelligence position.[1]Today, it may be confusing that some passport control officers actually control passports. With other countries, "cultural attaché" was often a cover job, although, again, it might be legitimate. An intelligence officer covered as a cultural attaché might still do some cultural things.
An intermediate approach has the officers clearly working for their country but without diplomatic immunity and with a cover role that does not immediately suggest intelligence affiliation. For example, the Soviet GRU covered some intelligence officers under theTASSnews agency, or as part of a trade or technical mission, or even as diplomats. The last might seem surprising but this was under a GRU assumption that military attaches would always be assumed to be intelligence officers, but that members of the civilian part of an embassy might actually be diplomats rather than intelligence officers.[2]
It was easier, of course, for the socialist USSR to assign people to state agencies. Western sensitivities tend to be much greater about using, for example, journalistic cover.[citation needed]The US has been emphatic in prohibiting any relationship between intelligence and thePeace Corps[citation needed].
US military intelligence doctrine forbids a HUMINT specialist to pose as:
An example of civilian cover for an American officer involved a German refugee, with the pseudonym "Stephan Haller", who had widely ranging interests and special skills in mathematics and physics, as well as native language skill. His overt role, in 1949, was directing a program that paid subsidies to German scientists, part of a larger program of denying German talent to the Soviets. Initially, he was based inPforzheim, (West)Germany.[4]
During two years in Pforzheim, with a well-established cover, he collected political and scientific intelligence from the scientists and also Germans that he knew in political circles before emigrating. In 1951, he moved toBerlin, directing overall "operations against scientific targets in the East Zone of Germany", while still managing the subsidy program. His new work included encouraging defection of key craftsmen working for the Soviets. He was considered a master craftsman,
He did not grow careless or conceited with success. Here remained a meticulous craftsman. Before he debriefed a source, he mastered the subject to be discussed. His agents were made comfortable not only by his cigars and beer but also by the easy flow of communication. And he did not end until he had every last scrap of useful information. He never failed, moreover, to remain alert for operational leads--potential agents,counterintelligence indicators,propaganda possibilities. When Haller was finished, there were no more questions to be asked. And though he groaned over the chore of putting it on paper, his reporting became thorough-and more than thorough, illuminating-for he rarely failed to make interpretive comments.[quotation?] [citation?]
According toVictor Suvorov, the Soviet reaction to losing networks operated from diplomatic missions – after the countries in which those embassies were located were overrun in the Second World War – was to emphasize "illegal" (i.e., what the US calls non-official cover) stations (i.e., residencies) for HUMINT networks. The illegal residencies were preferred to be in safe locations, perhaps of allies such as the United States, Great Britain and Canada.
Soviet operations were tightly compartmentalized, with strictneed-to-knowan absolute rule. A lesson learned from the loss of espionage networks was to keep them small, subdividing them, with independent reporting to Center, when more agents were recruited.[5]
Suvorov explained that new agents were separated from official Soviet institutions only after the agent has compromised himself by giving Soviet Intelligence a significant quantity of secret material; making it impossible for the agent to go to the police. The separated agent then occupies one of three guises: the separated acting agent, the agent group and the agent residency.
Greatest resources are devoted to these agents; which provide the most important material. Once the central headquarters assesses the materials as sufficiently valuable, the doctrine is to temporarily stop obtaining new material from the agent and improve their security as well as their knowledge of espionage tradecraft. This training is preferably done in a third country, from which the agent might or might not be moved to the Soviet Union. Typical cover for an agent absence would be taking a vacation or holiday.
Less valuable than a separated acting agent but still of importance, was the agent group, which migrated from diplomatic or civilian contact, to the in-country illegalrezidentura(resident and infrastructure), to direct communications with the center. The leader of such a group is called, in Soviet terminology, agropovod, and is conceptually the only member of the group that communicates with Moscow. In reality, clandestine communications personnel may be aware of the direct contact, but newer electronics allow the leader to manage his or her only communications.
Suvorov makes the important point that "A group automatically organises itself. The GRU obviously considers family groups containing the head of the family and his wife and children to be more secure and stable. The members of such a group may work in completely different fields of espionage." The pattern of having groups that are self-organizing and have preexisting ties, making them virtually impossible to infiltrate, has survived the GRU and is common in terrorist networks.
When the GRU attaches one or more illegals (i.e., Soviet officer under an assumed identity), the residency changes from "an agent residency into an illegal residency. This process of increasing the numbers and the gradual self-generation of independent organisations continues endlessly." Suvorov uses a medical metaphor of quarantine designed to contain infection to describe separating agents for improved security.
The GRU kept certain officers immediately ready to go into illegal status, should the host nation intensify security.[5]
Again, Suvorov emphasizes that the process of forming new illegal residencies was the Soviet doctrine for imposing compartmentation. Western countries, especially those in danger of invasion, have a related approach, thestay-behindnetwork. The US military definition, used by most NATO countries, is
Agent or agent organization established in a given country to be activated in the
event of hostile overrun or other circumstances under which normal access would be denied.[6]
In such an approach, both clandestine intelligence and covert operations personnel live normal lives, perhaps carrying out regular military or government functions, but have prepared documentation of assumed identities, safehouses, secure communications, etc.
Vilyam Genrikhovich Fisher, usually better known by his alias, Rudolf Abel, was a Soviet intelligence officer who came to the US under the false identity of a US citizen, Emil Robert Goldfus, who had died in infancy but was used by the USSR to create an elaboratelegendfor Fisher. On coming to the US, entering through Canada, Fisher/Abel took over the control of several existing Soviet HUMINT assets, and also recruited new assets. Key assets for whom he was the case officer includedLona CohenandMorris Cohen, who were not direct intelligence collectors butcouriersfor a number of agents reporting on US nuclear information, includingJulius Rosenberg,Ethel Rosenberg,David Greenglass, andKlaus Fuchs.
His role was that of the "illegal" resident in the US, undernonofficial cover. Soviet practice often was to have tworezidents, one illegal and one a diplomat underofficial cover. He was betrayed to the US by an alcoholic assistant who defected to the FBI.
That Fisher/Abel only had one assistant, with operational responsibilities, is not surprising. Unless a clandestine station has a strong cover identity, the larger the station, the larger the possibility it may be detected by counterintelligence organizations. Beyond the station chief, the most likely person to be associated with the station, not as a case officer, is a communicator, especially if highly specialized secure communication methods are used.
Some clandestine services may have additional capabilities for operations or support. Key operationalagents of influenceare apt to be run as singletons, although political considerations may require communication through cutouts.Useful idiotscan be run by diplomatic case officers, since there is no particular secrecy about their existence or loyalty. Valuable volunteers, depending on the size of the volunteer group, may work either with case officers, or operations officers brought clandestinely into the area of operations.
Proprietaries, which can be large businesses (e.g., the CIA proprietary airlines such asAir America, which, in the interest of cover, often had the latest aircraft and flew commercial as well as secret cargo), often are not controlled from the local area, but by headquarters. Especially when the proprietary is a multinational company, and has some commercial business of its own, central control makes the most sense.
In looking at internal as well as external assets, remember the fundamental rule of clandestine operations: the more secure, the less efficient. Because espionage operations need rigorous security, they are always inefficient — they take a lot of time, energy, and money. Proprietaries can be an exception, but, even though they make money, they can require additional capital to be able to expand in the same way a comparable private business would do so.[7]
Another kind of resource could include foreign offices owned or operated by nationals of the country in question. A step farther is aproprietary, or business, not just individuals, under non-official cover. Both kinds of business can provide information from recruitment, unwitting agents, or support functions. Small and medium aviation-related businesses have been popular US proprietaries, including Air America andSouthern Air Transport.
Once the service has a presence in aviation, it may become aware of persons, in private business, civil service, or the military, who fly to destinations of interest. They may mention it in innocent conversation, such as at the airport's restaurant or bar. They also may be assumed to be going there, based by analysis of flight departure times, aircraft type, duration of trip, and their passengers or cargo.
Having routine access to an airport can reveal: "Who's coming and going, on and off the record? What's in the hangars and warehouses? What are the finances? Political connections and loyalties? Access to planes on the ground? Flight plans?" It must be emphasized that a transportation-related proprietary—truck stops, boat maintenance, and other industry-specific businesses, have to operate as a real business. Occasionally, they may produce a profit, and that can be confusing for headquarters financial managers, provide a local but perhaps traceable source of funds, or both.
Public relations firms have long been useful proprietaries.[8]In a given country of operations, or perhaps adjacent countries that are concerned about the actions of their neighbor, news releases placed by experienced public relations professionals can help mold relevant opinion. Care must be taken that the news release does not "blow back" on the clandestinely sponsoring country.
Another viable industry for proprietaries is natural resources exploration. If, hypothetically, a mining company operated in a country where there are both resources deposits and non-national group sanctuaries, a proprietary company could get information on both, and also provide access and support services. If the proprietary began mining operations, it would naturally have access to explosives, which might be made available to sabotage groups in neighboring areas.
Use ofnongovernmental organizations(NGO) is politically sensitive and may require approval at the highest level of an agency. Sometimes, there is a broader policy need not to have the possibility of drawing suspicion onto an NGO. For example, in World War II, it was occasionally necessary to send supplies to Allied POWs, butRed Cross parcelswere never ever used for this purpose. The decision had been made that Red Cross parcels were important to the survival of the POWs and could never be jeopardized.
"Safehouse" is a term of intelligence tradecraft whose origins may be lost in antiquity. "The Bible is also replete with instances of espionage, including Yahweh's instruction to Moses to send spies into the land of Canaan. The account of the harlotRahabsheltering Israelite spies and betraying the city of Jericho might be the first documented instance of a "safe house.'"[9]
The term is not strictly limited to houses, although many intelligence services use rural houses for extended functions such as debriefing defectors. In a city, a safehouse may be an apartment or house that is not known to be associated with an intelligence service.
Another usage refers to mailing addresses (postal and electronic) and telephone numbers, to which messages can be sent with a reasonable chance of not coming into the awareness ofcounter-intelligence.
Useful idiotis a term attributed toLenin, principally in Soviet use, for a person overtly supporting the interests of one country (e.g., the USSR) in another (e.g., a member of the overt Communist Party of the second country). Soviet intelligence practice was to avoid such people in the actual clandestine operations, regarding them at most useful as distractions to the counterintelligence services.
Agents of influence, who were witting of Communist plans and intended to influence their own country's actions to be consistent with Soviet goals, went to great lengths to conceal any affiliation. "Witting" is a term of intelligence art that indicates that one is not only aware of a fact or piece of information, but also aware of its connection to intelligence activities. theVenona projectcommunications intelligenceexposes thatAlger Hiss and Harry Dexter White, accused of Communist sympathies, were indeed Soviet spies. They were Communist agents, and the Soviets certainly did not treat them as useful idiots. There were communications with them, and the dialogues were clandestine.
Gus Hallalso had overt Communist affiliation, and it is extremely unlikely Soviet clandestine operatives would have had anything to do with him. Still, in situations such as emergency exfiltration, Party members in a Western country might be called upon as a last desperate resort.
Thepropaganda modelof communication explains that people write news favorable to those who pay for their job or that people are hired with favorable viewpoints to the hirer.
This section deals with the recruiting of human resources who do not work for a foreign intelligence service (FIS). For techniques of recruiting FIS personnel, seeCounterintelligence.
In principle and best practice, all country B officers in country A report to an executive function in their home country. In CIA terms, this might be a head of a country desk or a regional desk. Russian practice was to refer to "Center".
Actual recruiting involves a direct approach by a case officer who has some existing access to the potential recruit, an indirect approach through an access agent or proprietary, or has reason to risk a "cold" approach. Before the direct recruitment, there may be a delicate period of development. For details, seeClandestine HUMINT asset recruiting.
This section deals with the general structure of running espionage operations. A subsequent section deals withSpecialized Clandestine Functions, and another withSupport Servicesfor both basic and specialized operations
The agent may join, or even create, a new network. In the latter case, the agent may be called alead agentor aprincipal agent. The latter term is also refers to access agents, who only help in recruiting.
Well-managed agent relationships can run for years and even decades; there are cases where family members, children at the time their parents were recruited, became full members of the network. Not all agents, however, operate in networks. A Western term for agents controlled as individuals issingleton. This term usually is reserved for the first or most sensitive recruitments, although specialized support personnel, such as radio operatives acting alone, are called singletons.[10]In Soviet tradecraft, the equivalent of a singleton is aseparated acting agent. Professional intelligence officers, such asRobert Hanssen, may insist on being singletons, and go even farther, as with Hanssen, refuse in-person meetings. Even as a singleton, the agent will use security measures such assecure communications.
Agents also may operate in networks, for which the classic security structure is thecell system.
The agent may join a proprietary, although that is more likely to be for access or support agents.
Before the agent actually starts to carry out assignment, training intradecraftmay be necessary. For security reasons, this ideally will be done outside the agent's own country, but such may not always be possible. Increasingly less desirable alternatives might be to conduct the training away from the operational area, as in a safe house in a resort, and then a safe house inside the operational area.
Among the first things to be taught are communications tradecraft, beginning with recording the material of interest. Skills here can include the operation of cameras appropriate for espionage, methods of carrying out documents without detection, secret writing.
Once the information is captured, it must be transmitted. The transmission may be impersonal, as with dead drops or car tosses. It may involve carriers. It may be electronic. If there is a need for personal meetings, the agent must know how to request them, and also to alert the network leader or case officer that the agent may be under suspicion.
Teachingcountersurveillancetechniques to agents is a calculated risk.[11]While it may be perfectly valid for an agent to abort a drop or other relatively innocent action, even at the cost of destroying valuable collected material, it is much more dangerous to teach the agent to elude active surveillance. The ability to elude professional counterintelligence personnel following the agent, for example, may confirm the counterintelligence organization's suspicion that they are dealing with a real agent.
Still, the agent may need to have an emergency escape procedure if he confirms he is under surveillance, or even if he is interrogated but released.
Case officers should constantly test their agents for changes in motivation or possible counterintelligence compromise. While "name traces cannot be run on every person mentioned by the agent, do not be stingy with them on persons who have familial, emotional, or business ties with him" to detect any linkages to hostile counterintelligence.[11]Until an agent is well established as reliable, meetings must always be done with care to avoid detection. "The prime emphasis is put on vigilance and checking-has he been planted by the local counterintelligence, are his motives in agreeing to collaborate sincere? The need for personal meetings with such an agent is increased, for they give the opportunity to assess him more completely."[12]
An experienced US operations officer emphasized that field operations personnel should report status and progress often. Only with such reporting can a headquarters staff remain vigilant, looking globally for penetrations, and also aware of political implications. Reporting and headquarters advice is critical for joint operations (i.e., with the intelligence service of another country). Headquarters, aware of all joint operations with a given service, can give advice from a broader viewpoint without compromising the need for local initiative.[11]
Even with the most sensitive agents, occasional personal meetings are important in maintaining psychological control. Nevertheless, some agents, especially trained intelligence officers likeRobert Hanssen, will almost never meet, but provide material good enough to prove their bona fides. A Soviet officer commented, whatever an agent's role in the intelligence net, personal contact should be made with him only when it is impossible to manage without it. The number of meetings should be kept as low as possible, especially with sources of valuable information.
Personal meetings may be held to give an agent his next assignment and instructions for carrying it out, to train him in tradecraft or the use of technical or communications equipment, to transmit documents, reports, technical equipment, money, or other items, or to fulfill several of these purposes. In actual practice several purposes are usually served by a meeting. In addition to its particular objectives more general needs can be filled. A meeting held for training purposes may be a means for clarifying biographic data on the agent or his views on various subjects. At every meeting with an agent one should study him and obtain new data on his potential and talents, thereby providing a better basis for judging his sincerity and deciding how much trust to place in him.[12]
Agents, to varying extents, need reinforcement. Salary is important and also gives a lever of compromise, although pressing it too hard can offend a truly ideologically motivated agent. Some agents benefit from recognition that they can never show, such as a uniform of your service, or decorations from it.
Agents will be more comfortable if they believe that they will have protection, preferably exfiltration, if compromised. Protecting their families may be even more important. When the agent operates in a country with a particularly brutal counterintelligence service, providing them with a "final friend", or means for suicide, can be comforting even if they never use it.[13]
This section deals with skills required of individuals, either agents or support personnel. Most skills are concerned with communications.
A Soviet officer commented, perhaps counterintuitively, that it is harder to have longer meetings with agents when the case officer is under diplomatic cover. The reason is that local counterintelligence is aware of the case officer, where the existence of an illegal (i.e., nonofficial cover in US terms) officer may not be known to them. For the legal officer, "here it is best either to have reliable safehouses or to deliver the agent discreetly to the official residency building. The latter is a serious operational move. If neither is feasible, it is better to have Headquarters dispatch an officer to a third country, either legally or illegally, for the meeting."[12]
It is a case-by-case decision whether the material exchanged should have safeguards against accessing it in other than a precise manner. One straightforward protection method is to have the material on exposed photographic film, in a container that does not suggest that it contains film and might be, innocently, opened in a lighted room. Self-destruct devices also are possibilities, but they confirm that the transfer involved sensitive material.
Under the general term "brush pass" is a wide range of techniques in which one clandestine operative passes a physical item to another operative.[14]"Brush" implies that the two people "brush" past one another, typically in a public place and preferably a crowd, where random people interfere with any visual surveillance. In a properly executed brush pass, the agents do not even stop walking; at most, they may appear to bump into one another.
During the brief contact, a common means of executing the exchange is for both to be carrying otherwise identical objects, such as a newspaper, briefcase, or magazine. The information being exchanged is in one of them. As the two people separate, they still appear to be holding the same object in the same hand.
More challenging versions are reminiscent of passing a baton in arelay race, and would be most commonly done with small objects such as a photographic film cartridge. In this more dangerous method, the transfer is from hand to hand, or from hand into a pocket. While this technique obviously takes better manual dexterity and is more prone to error, it has the countersurveillance advantage that the operatives are not carrying anything after the transfer, and can blend into a crowd even more easily.
A variation of the brush pass is thelive letter drop, in which one agent follows a predefined route, on foot, with a prepared report hidden in a pocket. En route, a second agent unknown to the first agent picks his/her pocket and then passes the report on unread, either to a cut-out or to an intelligence officer. This technique presents opportunities both forplausible deniabilityand for penetration by hostile agents.
Adead dropis a container not easily found, such as a magnetized box attached to a metal rack in an out-of-sight alley. The box could be loosely buried. It should be possible to approach the container to fill or empty it, and not be easily observed from a street or window.
Typically, a clandestine collector will put espionage material, perhaps in encrypted form, into the box, and use some prearranged signal (i.e. signal site) to let a courier know that something needs to be taken out of the box and delivered to the next point on the route to the case officer. Such a route might have several dead drops. In some cases, the dead drop might be equipped with a device to destroy its contents unless it is opened properly.
Signals to tell a courier, or a case officer if there is no intermediate courier, that the dead drop needs service can be as simple as a piece of colored tape on a lamp post or perhaps a set of window shades raised and lowered in a specific pattern. While "wrong number" calls with a predefined apology can be used, they are more vulnerable to surveillance if the phone in question is tapped.
A car toss can take many forms, one of which can be considered a moving dead drop. An agent or courier can put a magnetized box inside a bumper on a parked car.
In some cases, if a car can drive slowly down a street or driveway not easily observed, a courier can toss a message container into an open window, making the transfer method intermediate between a brush pass and a dead drop.
Cars with diplomatic immunity have advantages and disadvantages for tosses. They cannot be searched if the toss is observed, but they also are followed more easily. Diplomatic cars usually have distinctive markings or license plates, and may be equipped with electronic tracking devices. Counterintelligence could wait until the car is out of sight following a toss, then apprehend and interrogate the courier, or simply keep the courier under surveillance to discover another link in the message route.
A message left in a dead drop, or dropped during an improperly executed brush pass, is quite incriminating if counterintelligence personnel can immediately see suspicious information written on it. The ideal material for transfer looks quite innocuous.
At one time,invisible ink, a subset ofsteganography, was popular in espionage communications, because it was not visible to the naked eye without development by heat or chemicals. While computer-based steganographic techniques still are viable, modern counterintelligence laboratories have chemical and photographic techniques that detect the disturbance of paper fibers by the act of writing, so the invisible ink will not resist systematic forensic analysis. Still, if its existence is not suspected, the analysis may not be done.
Another technique, for hiding content that will resist casual examination, is to reduce the message to a photographic transparency or negative, perhaps the size of the dot over the letter "i" in this article. Such a technique needs both a laboratory and considerable technical skill, and is prone to damage and to accidentally falling off the paper. Still, it does have a countersurveillance value.[15]
Encryption, especially using a theoretically secure method, when properly executed, such as theone-time pad,[16]is highly secure, but a counterintelligence agent seeing nonsense characters will immediately become suspicious of the message that has been captured. The very knowledge that a dead drop exists can cause it to be trapped or put under surveillance, and the member of a brush pass that carries it will be hard-pressed to explain it.
One-time pad encryption has the absolute requirement that thecryptographic keyis used only once. Failure to follow this rule caused a serious penetration into Soviet espionage communications, through theVenona projectanalysis.[17]
It is extremely difficult for a nonprofessional to develop acryptosystem, especially without computer support, that is impervious to the attack by a professional cryptanalyst, working for an agency with government resources, such as the USNSAor RussianSpetssvyaz.[16]Still, when the message is very short, the key is random or nearly random, some methods, like the NihilistStraddling checkerboardmay offer some resistance. Improvised methods are most useful when they only have to protect the information for a very short time, such as changing the location or time of an agent meeting scheduled in the same day.
Less suspicious when examined, although very limited in its ability to transfer more than simple content, is plain language code. For example, the final attack order for theBattle of Pearl Harborcame in a radio broadcast of the Japanese phrase, "Climb Mount Niitaka". Subsequent espionage communications referred to ships as different types of dolls at a doll repair shop.
Plain language code is most effective when used to trigger a preplanned operation, rather than transfer any significant amount of information.
Steganography, in the broadest sense of the word, is a technique of hiding information "in plain sight" within a larger message or messaging context. It is hard to detect because the secret message is a very small component of the larger amount, such as a few words hidden in a Web graphic.
Even more sophisticated computer-dependent methods can protect information. The information may or may not be encrypted. Inspread-spectrum communications, the information is sent, in parallel, at very low level through a set of frequencies. Only when the receiver knows the frequencies, the time relationship on when a given frequency or other communications channel will carry content, and how to extract the content, can information be recovered. Basic spread spectrum uses a fixed set of frequencies, but the signal strength in any one frequency is too low to detect without correlation to other frequencies.
Frequency-hopping spread spectrumis a related technique, which can use the parallel transmission of true spread spectrum, not using any one frequency long enough for plausible interception. The pattern of variation among channels may be generated and received using cryptographic methods.
Avoiding detection of radio signals means minimizing the clandestine transmitter's exposure to hostile direction-finding. Modern techniques generally combine several methods:
Exploring agent information often meant a good deal of interaction, in which the home service would clarify what the agent reported, give new orders, etc. One approach used in World War II was theJoan-Eleanor system, which put the case officer into an aircraft at high altitude. From that altitude, there could be fast interaction in voice, so that they get to the key issues faster than with many separately encrypted and transmitted messages.[18]The modern equivalent is a small,low probability of interceptradio transceiver, using a directional antenna aimed at an orbiting satellite communications relay. Avoiding detection of radio communications involves all the principles oftransmission and reception security.
For any number of reasons, a human source operation may need to be suspended for an indefinite time, or definitively terminated. This need rarely eliminates the need for protecting the fact of espionage, the support services, and the tradecraft and tools provided.
One of the most difficult challenges is ending an emotional relationship between the case officer and agent, which can exist in both directions. Sometimes, an agent is unstable, and this is a major complication; perhaps even requiring the evacuation of the agent. More stable agents may be happy with termination bonuses, and perhaps a future emigration opportunity, that do not draw attention to their own side's counterintelligence. In some instances, an intelligence agency may issue a "burn notice", indicating to other such agencies that an individual is an unreliable source of information.
Especially in the case of non-national organizations, termination can be very literal, ranging from having a trusted operative kill the problematic agent, or, when culturally appropriate, sending the agent on a suicide mission.
When the clandestine phase is preparation for a DA mission such as the9/11 attacks, or the assassination attacks, using suicide bombers, by theLiberation Tigers of Tamil Eelam, termination of the operational cells is rather obvious. If there are support cells in the operational area, they may be vulnerable, but it would be good tradecraft to withdraw them shortly before the attack.
Anagent of influence, being witting or unwitting of the goals of a foreign power B, can influence the policy of Country A to be consistent with the goals of Country B.
In Soviet theory, influencing policy was one aspect of what they termedactive measures(aktivnyye meropriyatiya). Active measures have a different connotation than the Western concept ofdirect action(DA), although Soviet active measures could includewet affairs(mokrie dela) conducted by Department V of theKGB, "wet" referring to the spilling of blood.
Intelligence organizations occasionally use live, or even dead, persons to deceive the enemy about their intentions. One of the best-known such operations was the BritishOperation Mincemeat, in which a dead body, bearing carefully misleading documents, was put in British uniform, and floated onto a Spanish beach. In World War II, Spanish security services, while officially neutral, often passed information to the Germans, which, in this case, is exactly what the British wanted done. This operation was under the control of theTwenty Committee, part of the British strategic deception organization, theLondon Controlling Section. A related British operation in World War I was run by a controversial military officer,Richard Meinertzhagen, who prepared a knapsack containing false military plans, which theOttomanallies of theGermanswere allowed to capture. The plans related to false British strategy for theSinai and Palestine Campaign, setting up a successful surprise attack in theBattle of Beershebaand theThird Battle of Gaza.
Active measures, however, reflected a national effort to influence other countries to act in concert with Soviet goals. These measures could involve state organizations up to and including thePolitburo, much as the World War II British organization for strategic deception, theLondon Controlling Section, and its US counterpart, Joint Security Control, could get direct support from the head of government. Much of the Soviet responsibilities for active measures was focused in theKGB. Its "First Main Directorateuses active measures such as agents of influence, propaganda, and disinformation to promote Soviet goals."
In the present political context of Western democracies, the sensitivity, and separation, of clandestine and open contacts do not lend themselves to the process of building agents of influence.
"Active measures is not exclusively an intelligence activity, and in this sense it differs from the similar American concept of covert action. There are many differences between active measures and covert action. One is the Soviet ability to mesh overt and covert influence activities through centralized coordination of party, government, and ostensibly private organizations dealing with foreigners. Despite interagency coordination mechanisms, the United States is too pluralistic to achieve full coordination between all the overt and covert means of exercising influence abroad. Other major differences are in scope, intensity, and importance attributed to active measures and covert action, and in immunity from legal and political constraints."
While deception and influence operations could involve the highest levels of Allied governments in World War II, it is worth noting that while the West generally speaks ofmilitary deception, strategic deception operates at a higher level. A Soviet, and presumably Russian, term of art,maskirovkaor "denial and deception", is much broader than the current Western doctrine of deception being run by lower-level staff groups.
In the military, responsibility for maskirovka easily can be at the level of a deputy chief of the General Staff, who can call upon all levels of government.
Returning to KGB doctrine, presumably still present in theSVR, "Influence operations integrate Soviet views into foreign leadership groups. Propaganda operations take the form of disinformation articles placed in the foreign press. Disinformation operations are false documents designed to incite enmity toward the United States."
TheSecond Main Directorate of the KGB, whose responsibilities are now primarily in the RussianFSB, is responsible for the recruitment of agents among foreigners stationed in the Soviet Union. The KGB influences these people unwittingly, as most regard themselves too sophisticated to be manipulated.
"The second deception program is counterintelligence, which aims to neutralize the efforts of foreign intelligence services. It achieves this through the use of non-Soviet double agents and Soviet double agents. Non-Soviet double agents are foreign nationals who have been 'turned'. A Soviet double agent is a Soviet with access to classified information. These officials may be used as false defectors".[19]
"Influence operations integrate Soviet views into leadership groups. The agent of influence may be a well- placed, 'trusted contact' who consciously serves Soviet interests on some matters while retaining his integrity on others, or an unwitting contact who is manipulated to take actions that advance Soviet interests on specific issues of common concern."
There is no consensus on whether it is, or is not, advisable to intermingle espionage and direct action organizations, even at the headquarters level. SeeClandestine HUMINT and Covert Actionfor more history and detail. A terminology point: current US terminology, ignoring an occasional euphemism, has now consolidated espionage into the National Clandestine Services. These are part of the CIA Directorate of Operations, which has some responsibility fordirect action(DA) andunconventional warfare(UW), although the latter two, when of any appreciable size, are the responsibility of the military.
There is much more argument for doing so at headquarters, possibly not as one unit but with regular consultation. Certain services, such as name checks, communications, cover identities, and technical support may reasonably be combined, although the requirements of a particular field network should be held on a need-to-know basis.
Other countries might have the functions under the same organization, but run them in completely different networks. The only commonality they might have is emergency use of diplomatic facilities.
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https://en.wikipedia.org/wiki/Clandestine_HUMINT_operational_techniques
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TheUnited States Geospatial Intelligence Foundation(USGIF) is a 501(c)(3)non-profiteducational foundation inVirginiadedicated to promoting thegeospatial intelligencetradecraftand developing a strongerGEOINTCommunity with government, industry, academia, professional organizations, and individuals who develop and apply geospatial intelligence to address national security challenges. USGIF achieves its mission through various programs and events and by building the community, advancing the tradecraft, and accelerating innovation. USGIF provides a number of programs and events such as its GEOINT Symposium, an academic accreditation program for college and university geospatial programs, and other live, virtual, and hybrid programs to provide the community with the opportunity to collaborate with senior-level officials across the multiple communities and support the future of the tradecraft.
The United States Geospatial Intelligence Foundation was created in January 2004 by a group of tradecraft professionals recognizing the need for a forum where they could work together——outside their own organizational and corporate interests——toward a mutual goal of improvingnationalandhomeland security. The idea for the Foundation started with an event, Geo-Intel 2003, which drew enough interest to solidify the group’s notion that the tradecraft community needed a forum. This event drew more than 1,000 intelligence professionals. Just months later, USGIF was created, publicly announcing its launch on May 12, 2004.[1]
The United States Geospatial Intelligence Foundation (USGIF) is a Virginia-based nonstock, nonlobbyist, not-for-profit 501(c)(3) corporation. The business and affairs of the Foundation are managed by a Board of Directors which oversees the Foundation through the work of three standing Board committees: Finance and Audit, Management and Compensation, and Nominating and Corporate Governance.
USGIF builds its constituency through memberships at the individual and corporate level. Individuals can join the Foundation as members from academia (faculty and students), law enforcement and first responders, U.S. or foreign government or military members, young professionals, association/not-for-profit/non-governmental organization members, members of the press/media, and members from U.S. or foreign industry. Individuals may also join as lifetime members. Organizations can join USGIF at different tiers of partnership: strategic, premier, associate, sustaining, academic, and small business.
Much of the business of the Foundation is accomplished through its non-Board committees and working groups composed of members of the Foundation. The two non-Board committees are the Planning Committee, which helps plan Foundation events and programs, and the Academic Committee which provides academic outreach to universities and colleges as well as promoting the aims of USGIF in government and industry. USGIF also has nine working groups that serve as topically-oriented communities of interest:
Carrying the torch of Geo-Intel 2003 as an official organization, USGIF rebranded the event as GEOINT Symposium. In November 2004, USGIF held the first GEOINT Symposium in New Orleans and attracted more than 1,500 participants. The annual event is typically held in the spring and has since been hosted in San Antonio, Texas; Orlando, Florida; Nashville, Tennessee; Tampa, Florida; Washington, DC; St. Louis, Missouri; and Denver, Colorado. Typically the event draws more than 3500 attendees, including speakers and exhibitors. The Symposium was not held in 2013 due to a government shutdown and was postponed instead to 2014, and it was cancelled in 2020 due to the global pandemic.
The GEOINT Symposium was described in 2008 byTim Shorrockas "one of the few open windows into the thinking at the highest levels of US intelligence", as it "has become the nation's showcase for intelligence contractors and agencies alike...".[2]In his bookSpies for Hire: The Secret World of Intelligence Outsourcing, Shorrock recounts several notable events at GEOINT Symposiums. Among them, in 2004, the Symposium featured the directors of theCIA, theNSA, and theNGAspeaking at a public session at the same time—the only occasion during the presidency ofGeorge W. Bushwhen such a public collective gathering would occur.[3]He also notes that, in 2005, Deputy Director of National Intelligence for CollectionMary Margaret Grahaminadvertently revealed the amount of money spent by the US government on national intelligence, the first time the budget amount had been revealed since 1998.[4]
In 2006, the GEOINT Symposium featured thenDirector of National IntelligenceJohn D. Negroponteaskeynotespeaker.[5][6]In 2008, the address was delivered by Negroponte's successor,Mike McConnell, whose speech was picked up by multiple media outlets.[7][8][9]As the thenDirector of National Intelligence,James Clapperprovided keynote remarks at every GEOINT Symposium between 2011 and 2016, and also keynoted in 2010 as theUnder Secretary of Defense for Intelligence. Other Undersecretaries of Defense for Intelligence have spoken at the Symposium, includingMichael G. Vickersin 2011 and 2012,Marcel Lettrein 2015 and 2016,Joseph D. Kernanin 2018, andRonald Moultriein 2022. In 2018 and 2019, former Principal Deputy Director of National IntelligenceSusan M. Gordonspoke at the Symposium, in 2021 and 2022 Dr.Stacey Dixonkeynoted, and in 2023 Director of National IntelligenceAvril Haineskeynoted. The GEOINT stage also provides a forum for an annual public address by the Director of theNational Geospatial-Intelligence Agency, and has often included speeches by directors of other intelligence agencies. The GEOINT stage has drawn additional contributors, among themDonald Kerr, GeneralJames Cartwright, Lt. Gen.William G. Boykin, Lt. Gen.Russel L. Honoré, Dr. Christopher K. Tucker, retired Gen.Anthony Zinni,Charles E. Allen, AmbassadorDennis Richardson,Anthony Tether, Al Munson,[10]Bran Ferren, Gen.Michael Hayden,Suzette Kimball, Gen.Stanley A. McChrystal, Gen.Charles Q. Brown Jr.,Robert D. Kaplan, and Dr.Lisa Porter.
For many years, Tech Days was an event at which USGIF Members showcased their technologies without having to compete against speakers or an agenda. This event was held each spring in the D.C. metro area to allow members of US Congress and other government employees convenient access to the latest developments and solutions in geospatial technology. Tech Days was produced in cooperation with theNational Geospatial-Intelligence Agency(NGA), which hosted a classified technology component as part of the event. Tech Days culminated with the GEOGala black-tie dinner.
In addition to the USGIF Speaker Series, USGIF hosts smaller dinner events where USGIF Strategic Partner Members can listen to and speak with leaders in Government, Defense, Intelligence, Academia, and Industry in a more intimate and casual environment. The event, at times, coincides with a classified briefing or other relevant activities. The Chairman’s Events are open only to Strategic Partner Members, USGIF Board of Directors, and select invited guests.
USGIF supports education through several programs. It is the only body accrediting university programs in geospatial intelligence a sub-field ofgeographic information science, under its Geospatial Intelligence Certificate Program.[11]The first four universities accredited wereUniversity of Missouri,Pennsylvania State University,George Mason Universityand theUniversity of Texas at Dallas.[12]The program was launched after several years of planning and community outreach to draft an acceptable set of standards.[13]There are currently 21 colleges and universities with USGIF accredited GEOINT programs.
USGIF also provides scholarships to college and university students in geospatial-related fields as well as to high school students intending higher education in geospatial-related fields, and as of 2024 has awarded more than $1.7 million in scholarships to students. In the past, USGIF maintained theJamesand Susan Clapper Education Initiative Fund to fund earth-science material forprimaryandsecondarystudents.
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https://en.wikipedia.org/wiki/United_States_Geospatial_Intelligence_Foundation
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k-anonymityis a property possessed by certainanonymized data. The termk-anonymity was first introduced byPierangela SamaratiandLatanya Sweeneyin a paper published in 1998,[1]although the concept dates to a 1986 paper by Tore Dalenius.[2]
k-anonymity is an attempt to solve the problem "Given person-specific field-structured data, produce a release of the data with scientific guarantees that the individuals who are the subjects of the data cannot be re-identified while the data remain practically useful."[3][4][5]A release of data is said to have thek-anonymity property if the information for each person contained in the release cannot be distinguished from at leastk−1{\displaystyle k-1}individuals whose information also appear in the release. The guarantees provided byk-anonymity are aspirational, not mathematical.
To usek-anonymity to process a dataset so that it can be released with privacy protection, a data scientist must first examine the dataset and decide whether each attribute (column) is anidentifier(identifying), anon-identifier(not-identifying), or aquasi-identifier(somewhat identifying). Identifiers such as names are suppressed, non-identifying values are allowed to remain, and the quasi-identifiers need to be processed so that every distinct combination of quasi-identifiers designates at leastkrecords.
The example table below presents a fictional, non-anonymized database consisting of the patient records for a fictitious hospital. TheNamecolumn is an identifier,Age,Gender,State of domicile, andReligionare quasi-identifiers, andDiseaseis a non-identifying sensitive value. But what aboutHeightandWeight? Are they also non-identifying sensitive values, or are they quasi-identifiers?
There are 6 attributes and 10 records in this data. There are two common methods for achievingk-anonymity for some value ofk:
The next table shows the anonymized database.
This data has 2-anonymity with respect to the attributesAge,GenderandState of domicile, since for any combination of these attributes found in any row of the table there are always at least 2 rows with those exact attributes. The attributes available to an adversary are calledquasi-identifiers. Each quasi-identifier tuple occurs in at leastkrecords for a dataset withk-anonymity.[6]
The following example demonstrates a failing withk-anonymity: there may exist other data records that can be linked on the variables that are allegedly non-identifying. For instance, suppose an attacker is able to obtain the log from the person who was taking vital signs as part of the study and learns that Kishor was at the hospital on April 30 and is 180 cm tall. This information can be used to link with the "anonymized" database (which may have been published on the Internet) and learn that Kishor has a heart-related disease. An attacker who knows that Kishor visited the hospital on April 30 may be able to infer this simply knowing that Kishor is 180 cm height, roughly 80–82 kg, and comes from Karnataka.
The root of this problem is the core problem withk-anonymity: there is no way to mathematically, unambiguously determine whether an attribute is an identifier, a quasi-identifier, or a non-identifying sensitive value. In fact, all values are potentially identifying, depending on their prevalence in the population and on auxiliary data that the attacker may have. Other privacy mechanisms such asdifferential privacydo not share this problem.
Although k-anonymity safeguards against identity revelations, it does not shield against the disclosure of specific attributes. This becomes problematic when attackers possess background knowledge. Additionally, the absence of diversity in sensitive domains may result in the exposure of personal information. In such scenarios, opting forℓ-Diversitymight offer a more robust privacy safeguard.[1]
Meyerson and Williams (2004) demonstrated that optimalk-anonymity is anNP-hardproblem, however heuristic methods such ask-Optimize as given by Bayardo and Agrawal (2005) often yield effective results.[7][8]A practical approximation algorithm that enables solving thek-anonymization problem with an approximation guarantee ofO(logk){\displaystyle O(\log k)}was presented by Kenig and Tassa.[9]
Whilek-anonymity is a relatively simple-to-implement approach for de-identifying a dataset prior to public release, it is susceptible to many attacks. When background knowledge is available to an attacker, such attacks become even more effective. Such attacks include:
Becausek-anonymization does not include any randomization, attackers can make reliable, unambiguous inferences about data sets that may harm individuals. For example, if the 19-year-old John from Kerala is known to be in the database above, then it can be reliably said that he has either cancer, a heart-related disease, or a viral infection.
K-anonymization is not a good method to anonymize high-dimensional datasets.[11]
It has also been shown thatk-anonymity can skew the results of a data set if it disproportionately suppresses and generalizes data points with unrepresentative characteristics.[12]The suppression and generalization algorithms used tok-anonymize datasets can be altered, however, so that they do not have such a skewing effect.[13]
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https://en.wikipedia.org/wiki/K-anonymity
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Secure multi-party computation(also known assecure computation,multi-party computation(MPC) orprivacy-preserving computation) is a subfield of cryptography with the goal of creating methods for parties to jointly compute a function over their inputs while keeping those inputs private.[1]Unlike traditional cryptographic tasks, where cryptography assures security and integrity of communication or storage and the adversary is outside the system of participants (an eavesdropper on the sender and receiver), the cryptography in this model protects participants'privacyfrom each other.
The foundation for secure multi-party computation started in the late 1970s with the work on mental poker, cryptographic work that simulates game playing/computational tasks over distances without requiring a trusted third party. Traditionally, cryptography was about concealing content, while this new type of computation and protocol is about concealing partial information about data while computing with the data from many sources, and correctly producing outputs. By the late 1980s,Michael Ben-Or, Shafi Goldwasser and Avi Wigderson, and independently David Chaum, Claude Crépeau, andIvan Damgård, had published papers showing "how to securely compute any function in the secure channels setting".
Special purpose protocols for specific tasks started in the late 1970s.[2]Later, secure computation was formally introduced assecure two-party computation(2PC) in 1982 (for the so-calledMillionaires' Problem, a specific problem which is a Boolean predicate), and in generality (for any feasible computation) in 1986 byAndrew Yao.[3][4]The area is also referred to as Secure Function Evaluation (SFE). The two party case was followed by a generalization to the multi-party by Oded Goldreich, Silvio Micali, and Avi Wigderson. The computation is based on secret sharing of all the inputs and zero-knowledge proofs for a potentially malicious case, where the majority of honest players in the malicious adversary case assure that bad behavior is detected and the computation continues with the dishonest person eliminated or his input revealed. This work suggested the very basic general scheme to be followed by essentially all future multi-party protocols for secure computing.[5]This work introduced an approach, known as GMW paradigm, for compiling a multi-party computation protocol which is secure against semi-honest adversaries to a protocol that is secure against malicious adversaries. This work was followed by the first robust secure protocol which tolerates faulty behavior graciously without revealing anyone's output via a work which invented for this purpose the often used `share of shares idea'[6]and a protocol that allows one of the parties to hide its input unconditionally.[7]The GMW paradigm was considered to be inefficient for years because of huge overheads that it brings to the base protocol. However, it is shown that it is possible to achieve efficient protocols,[8]and it makes this line of research even more interesting from a practical perspective. The above results are in a model where the adversary is limited to polynomial time computations, and it observes all communications, and therefore the model is called the `computational model'. Further, the protocol ofoblivious transferwas shown to be complete for these tasks.[9]The above results established that it is possible under the above variations to achieve secure computation when the majority of users are honest.
The next question to solve was the case of secure communication channels where the point-to-point communication is not available to the adversary; in this case it was shown that solutions can be achieved with up to 1/3 of the parties being misbehaving and malicious, and the solutions apply no cryptographic tools (since secure communication is available).[10][11]Adding a broadcast channel allows the system to tolerate up to 1/2 misbehaving minority,[12]whereas connectivity constraints on the communication graph were investigated in the book Perfectly Secure Message Transmission.[13]
Over the years, the notion of general purpose multi-party protocols became a fertile area to investigate basic and general protocol issues properties on, such asuniversal composabilityormobile adversaryas inproactive secret sharing.[14]
Since the late 2000s, and certainly since 2010 and on, the domain of general purpose protocols has moved to deal with efficiency improvements of the protocols with practical applications in mind. Increasingly efficient protocols for MPC have been proposed, and MPC can be now considered as a practical solution to various real-life problems (especially ones that only require linear sharing of the secrets and mainly local operations on the shares with not much interactions among the parties), such as distributed voting, private bidding and auctions, sharing of signature or decryption functions andprivate information retrieval.[15]The first large-scale and practical application of multi-party computation was the execution of an electronic double auction in theDanish Sugar Beet Auction, which took place in January 2008.[16]Obviously, both theoretical notions and investigations, and applied constructions are needed (e.g., conditions for moving MPC into part of day by day business was advocated and presented
in[17]).
In 2020, a number of companies working with secure-multiparty computation founded the MPC alliance with the goal of "accelerate awareness, acceptance, and adoption of MPC technology."
In an MPC, a given number of participants, p1, p2, ..., pN, each haveprivate data, respectively d1, d2, ..., dN. Participants want to compute the value of a public function on that private data: F(d1, d2, ..., dN) while keeping their own inputs secret.
For example, suppose we have three parties Alice, Bob and Charlie, with respective inputs x, y and z denoting their salaries. They want to find out the highest of the three salaries, without revealing to each other how much each of them makes. Mathematically, this translates to them computing:
If there were some trusted outside party (say, they had a mutual friend Tony who they knew could keep a secret), they could each tell their salary to Tony, he could compute the maximum, and tell that number to all of them. The goal of MPC is to design a protocol, where, by exchanging messages only with each other, Alice, Bob, and Charlie can still learnF(x, y, z)without revealing who makes what and without having to rely on Tony. They should learn no more by engaging in their protocol than they would learn by interacting with an incorruptible, perfectly trustworthy Tony.
In particular, all that the parties can learn is what they can learn from the output and their own input. So in the above example, if the output isz, then Charlie learns that hiszis the maximum value, whereas Alice and Bob learn (ifx,yandzare distinct), that their input is not equal to the maximum, and that the maximum held is equal toz. The basic scenario can be easily generalised to where the parties have several inputs and outputs, and the function outputs different values to different parties.
Informally speaking, the most basic properties that a multi-party computation protocol aims to ensure are:
There are a wide range of practical applications, varying from simple tasks such as coin tossing to more complex ones like electronic auctions (e.g. compute the market clearing price), electronic voting, or privacy-preserving data mining. A classical example is the Millionaires' Problem: two millionaires want to know who is richer, in such a way that neither of them learns the net worth of the other. A solution to this situation is essentially to securely evaluate the comparison function.
A multi-party computation protocol must be secure to be effective. In modern cryptography, the security of a protocol is related to a security proof. The security proof is a mathematical proof where the security of a protocol is reduced to that of the security of its underlying primitives. Nevertheless, it is not always possible to formalize thecryptographic protocolsecurity verification based on the party knowledge and the protocol correctness. For MPC protocols, the environment in which the protocol operates is associated with the Real World/Ideal World Paradigm.[18]The parties can't be said to learn nothing, since they need to learn the output of the operation, and the output depends on the inputs. In addition, the output correctness is not guaranteed, since the correctness of the output depends on the parties’ inputs, and the inputs have to be assumed to be correct.
The Real World/Ideal World Paradigm states two worlds: (i) In the ideal-world model, there exists an incorruptible trusted party to whom each protocol participant sends its input. This trusted party computes the function on its own and sends back the appropriate output to each party. (ii) In contrast, in the real-world model, there is no trusted party and all the parties can do is to exchange messages with each other. A protocol is said to be secure if one can learn no more about each party's private inputs in the real world than one could learn in the ideal world. In the ideal world, no messages are exchanged between parties, so real-world exchanged messages cannot reveal any secret information.
The Real World/Ideal World Paradigm provides a simple abstraction of the complexities of MPC to allow the construction of an application under the pretense that the MPC protocol at its core is actually an ideal execution. If the application is secure in the ideal case, then it is also secure when a real protocol is run instead.
The security requirements on an MPC protocol are stringent. Nonetheless, in 1987 it was demonstrated that any function can be securely computed, with security for malicious adversaries[5]and the other initial works mentioned before.
Despite these publications, MPC was not designed to be efficient enough to be used in practice at that time. Unconditionally or information-theoretically secure MPC is closely related and builds on to the problem ofsecret sharing, and more specificallyverifiable secret sharing(VSS), which many secure MPC protocols use against active adversaries.
Unlike traditional cryptographic applications, such as encryption or signature, one must assume that the adversary in an MPC protocol is one of the players engaged in the system (or controlling internal parties). That corrupted party or parties may collude in order to breach the security of the protocol. Letn{\displaystyle n}be the number of parties in the protocol andt{\displaystyle t}the number of parties who can be adversarial. The protocols and solutions for the case oft<n/2{\displaystyle t<n/2}(i.e., when an honest majority is assumed) are different from those where no such assumption is made. This latter case includes the important case of two-party computation where one of the participants may be corrupted, and the general case where an unlimited number of participants are corrupted and collude to attack the honest participants.
Adversaries faced by the different protocols can be categorized according to how willing they are to deviate from the protocol. There are essentially two types of adversaries, each giving rise to different forms of security (and each fits into different real world scenario):
Security against active adversaries typically leads to a reduction in efficiency. Covert security[19]is an alternative that aims to allow greater efficiency in exchange for weakening the security definition; it is applicable to situations where active adversaries are willing to cheat but only if they are not caught. For example, their reputation could be damaged, preventing future collaboration with other honest parties. Thus, protocols that are covertly secure provide mechanisms to ensure that, if some of the parties do not follow the instructions, then it will be noticed with high probability, say 75% or 90%. In a way, covert adversaries are active ones forced to act passively due to external non-cryptographic (e.g. business) concerns. This mechanism sets a bridge between both models in the hope of finding protocols which are efficient and secure enough in practice.
Like manycryptographic protocols, the security of an MPC protocol can rely on different assumptions:
The set of honest parties that can execute a computational task is related to the concept ofaccess structure.Adversary structurescan be static, where the adversary chooses its victims before the start of the multi-party computation, or dynamic, where it chooses its victims during the course of execution of the multi-party computation making the defense harder. An adversary structure can be defined as a threshold structure or as a more complex structure. In a threshold structure the adversary can corrupt or read the memory of a number of participants up to some threshold. Meanwhile, in a complex structure it can affect certain predefined subsets of participants, modeling different possible collusions.
There are major differences between the protocols proposed for two party computation (2PC) and multi-party computation (MPC). Also, often for special purpose protocols of importance a specialized protocol that deviates from the generic ones has to be designed (voting, auctions, payments, etc.)
The two party setting is particularly interesting, not only from an applications perspective but also because special techniques can be applied in the two party setting which do not apply in the multi-party case. Indeed, secure multi-party computation (in fact the restricted case of secure function evaluation, where only a single function is evaluated) was first presented in the two-party setting. The original work is often cited as being from one of the two papers of Yao;[20]although the papers do not actually contain what is now known asYao's garbled circuit protocol.
Yao's basic protocol is secure against semi-honest adversaries and is extremely efficient in terms of number of rounds, which is constant, and independent of the target function being evaluated. The function is viewed as a Boolean circuit, with inputs in binary of fixed length. A Boolean circuit is a collection of gates connected with three different types of wires: circuit-input wires, circuit-output wires and intermediate wires. Each gate receives two input wires and it has a single output wire which might be fan-out (i.e. be passed to multiple gates at the next level). Plain evaluation of the circuit is done by evaluating each gate in turn; assuming the gates have been topologically ordered. The gate is represented as a truth table such that for each possible pair of bits (those coming from the input wires' gate) the table assigns a unique output bit; which is the value of the output wire of the gate. The results of the evaluation are the bits obtained in the circuit-output wires.
Yao explained how to garble a circuit (hide its structure) so that two parties, sender and receiver, can learn the output of the circuit and nothing else. At a high level, the sender prepares the garbled circuit and sends it to the receiver, who obliviously evaluates the circuit, learning the encodings corresponding to both his and the sender's output. He then just sends back the sender's encodings, allowing the sender to compute his part of the output. The sender sends the mapping from the receivers output encodings to bits to the receiver, allowing the receiver to obtain their output.
In more detail, the garbled circuit is computed as follows. The main ingredient is a double-keyed symmetric encryption scheme. Given a gate of the circuit, each possible value of its input wires (either 0 or 1) is encoded with a random number (label). The values resulting from the evaluation of the gate at each of the four possible pair of input bits are also replaced with random labels. The garbled truth table of the gate consists of encryptions of each output label using its inputs labels as keys. The position of these four encryptions in the truth table is randomized so no information on the gate is leaked.
To correctly evaluate each garbled gate the encryption scheme has the following two properties. Firstly, the ranges of the encryption function under any two distinct keys are disjoint (with overwhelming probability). The second property says that it can be checked efficiently whether a given ciphertext has been encrypted under a given key. With these two properties the receiver, after obtaining the labels for all circuit-input wires, can evaluate each gate by first finding out which of the four ciphertexts has been encrypted with his label keys, and then decrypting to obtain the label of the output wire. This is done obliviously as all the receiver learns during the evaluation are encodings of the bits.
The sender's (i.e. circuit creators) input bits can be just sent as encodings to the evaluator; whereas the receiver's (i.e. circuit evaluators) encodings corresponding to his input bits are obtained via a 1-out-of-2 oblivious transfer (OT) protocol. A 1-out-of-2 OT protocol enables the sender possessing of two values C1 and C2 to send the one requested by the receiver (b a value in {1,2}) in such a way that the sender does not know what value has been transferred, and the receiver only learns the queried value.
If one is considering malicious adversaries, further mechanisms to ensure correct behavior of both parties need to be provided. By construction it is easy to show security for the sender if the OT protocol is already secure against malicious adversary, as all the receiver can do is to evaluate a garbled circuit that would fail to reach the circuit-output wires if he deviated from the instructions. The situation is very different on the sender's side. For example, he may send an incorrect garbled circuit that computes a function revealing the receiver's input. This would mean that privacy no longer holds, but since the circuit is garbled the receiver would not be able to detect this. However, it is possible to efficiently apply Zero-Knowledge proofs to make this protocol secure against malicious adversaries with a small overhead comparing to the semi-honest protocol.[8]
Most MPC protocols, as opposed to 2PC protocols and especially under the unconditional setting of private channels, make use of secret sharing. In the secret sharing based methods, the parties do not play special roles (as in Yao, of creator and evaluator). Instead, the data associated with each wire is shared amongst the parties, and a protocol is then used to evaluate each gate. The function is now defined as a "circuit" over a finite field, as opposed to the binary circuits used for Yao. Such a circuit is called an arithmetic circuit in the literature, and it consists of addition and multiplication "gates" where the values operated on are defined over a finite field.
Secret sharing allows one to distribute a secret among a number of parties by distributing shares to each party. Two types of secret sharing schemes are commonly used;Shamir secret sharingand additive secret sharing. In both cases the shares are random elements of a finite field that add up to the secret in the field; intuitively, security is achieved because any non-qualifying set of shares looks randomly distributed.
Secret sharing schemes can tolerate an adversary controlling up totparties out ofntotal parties, wheretvaries based on the scheme, the adversary can be passive or active, and different assumptions are made on the power of the adversary. The Shamir secret sharing scheme is secure against a passive adversary whent<n2{\displaystyle t<{\frac {n}{2}}}and an active adversary whent<n3{\displaystyle t<{\frac {n}{3}}}while achieving information-theoretic security, meaning that even if the adversary has unbounded computational power, they cannot learn any information about the secret underlying a share. The BGW protocol,[21]which defines how to compute addition and multiplication on secret shares, is often used to compute functions with Shamir secret shares. Additive secret sharing schemes can tolerate the adversary controlling all but one party, that ist<n{\displaystyle t<n}, while maintaining security against a passive and active adversary with unbounded computational power. Some protocols require a setup phase, which may only be secure against a computationally bounded adversary.
A number of systems have implemented various forms of MPC with secret sharing schemes. The most popular is SPDZ,[22]which implements MPC with additive secret shares and is secure against active adversaries.
In 2014 a "model of fairness in secure computation in which an adversarial party that aborts on receiving output is forced to pay a mutually predefined monetary penalty" has been described for theBitcoinnetwork or for fair lottery, and has been successfully implemented inEthereum.[23][24]
Many advances have been made on 2PC and MPC systems in recent years.
One of the main issues when working with Yao-based protocols is that the function to be securely evaluated (which could be an arbitrary program) must be represented as a circuit, usually consisting of XOR and AND gates. Since most real-world programs contain loops and complex data structures, this is a highly non-trivial task. The Fairplay system[25]was the first tool designed to tackle this problem. Fairplay comprises two main components. The first of these is a compiler enabling users to write programs in a simple high-level language, and output these programs in a Boolean circuit representation. The second component can then garble the circuit and execute a protocol to securely evaluate the garbled circuit. As well as two-party computation based on Yao's protocol, Fairplay can also carry out multi-party protocols. This is done using the BMR protocol,[25]which extends Yao's passively secure protocol to the active case.
In the years following the introduction of Fairplay, many improvements to Yao's basic protocol have been created, in the form of both efficiency improvements and techniques for active security. These include techniques such as the free XOR method, which allows for much simpler evaluation of XOR gates, and garbled row reduction, reducing the size of garbled tables with two inputs by 25%.[26]
The approach that so far seems to be the most fruitful in obtaining active security comes from a combination of the garbling technique and the "cut-and-choose" paradigm. This combination seems to render more efficient constructions. To avoid the aforementioned problems with respect to dishonest behaviour, many garblings of the same circuit are sent from the constructor to the evaluator. Then around half of them (depending on the specific protocol) are opened to check consistency, and if so a vast majority of the unopened ones are correct with high probability. The output is the majority vote of all the evaluations. Here the majority output is needed. If there is disagreement on the outputs the receiver knows the sender is cheating, but he cannot complain as otherwise this would leak information on his input.
This approach for active security was initiated by Lindell and Pinkas.[27]This technique was implemented by Pinkas et al. in 2009,[26]This provided the first actively secure two-party evaluation of the Advanced Encryption Standard (AES) circuit, regarded as a highly complex (consisting of around 30,000 AND and XOR gates), non-trivial function (also with some potential applications), taking around 20 minutes to compute and requiring 160 circuits to obtain a2−40{\displaystyle 2^{-40}}cheating probability.
As many circuits are evaluated, the parties (including the receiver) need to commit to their inputs to ensure that in all the iterations the same values are used. The experiments of Pinkas et al. reported[26]show that the bottleneck of the protocol lies in the consistency checks. They had to send over the net about 6,553,600 commitments to various values to evaluate the AES circuit. In recent results[28]the efficiency of actively secure Yao-based implementations was improved even further, requiring only 40 circuits, and a much smaller number of commitments, to obtain2−40{\displaystyle 2^{-40}}cheating probability. The improvements come from new methodologies for performingcut-and-chooseon the transmitted circuits.
More recently, there has been a focus on highly parallel implementations based on garbled circuits, designed to be run onCPUswith many cores. Kreuter, et al.[29]describe an implementation running on 512 cores of a powerful cluster computer. Using these resources they could evaluate the 4095-bitedit distancefunction, whose circuit comprises almost 6 billion gates. To accomplish this they developed a custom, better optimized circuit compiler than Fairplay and several new optimizations such as pipelining, whereby transmission of the garbled circuit across the network begins while the rest of the circuit is still being generated. The time to compute AES was reduced to 1.4 seconds per block in the active case, using a 512-node cluster machine, and 115 seconds using one node. Shelat and Shen[30]improve this, using commodity hardware, to 0.52 seconds per block. The same paper reports on a throughput of 21 blocks per second, but with a latency of 48 seconds per block.
Meanwhile, another group of researchers has investigated using consumer-gradeGPUsto achieve similar levels of parallelism.[31]They utilize oblivious transfer extensions and some other novel techniques to design their GPU-specific protocol. This approach seems to achieve comparable efficiency to the cluster computing implementation, using a similar number of cores. However, the authors only report on an implementation of the AES circuit, which has around 50,000 gates. On the other hand, the hardware required here is far more accessible, as similar devices may already be found in many people's desktop computers or games consoles. The authors obtain a timing of 2.7 seconds per AES block on a standard desktop, with a standard GPU. If they allow security to decrease to something akin to covert security, they obtain a run time of 0.30 seconds per AES block. In the passive security case there are reports of processing of circuits with 250 million gates, and at a rate of 75 million gates per second.[32]
One of the primary applications of secure multi-party computation is allowing analysis of data that is held by multiple parties, or blind analysis of data by third parties without allowing the data custodian to understand the kind of data analysis being performed.
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Incryptography, aprivate information retrieval (PIR)protocol is a protocol that allows a user to retrieve an item from a server in possession of adatabasewithout revealing which item is retrieved. PIR is a weaker version of 1-out-of-noblivious transfer, where it is also required that the user should not get information about other database items.
One trivial, but very inefficient way to achieve PIR is for the server to send an entire copy of the database to the user. In fact, this is the only possible protocol (in the classical or thequantumsetting[1]) that gives the userinformation theoretic privacyfor their query in a single-server setting.[2]There are two ways to address this problem: make the server computationally bounded or assume that there are multiple non-cooperating servers, each having a copy of the database.
The problem was introduced in 1995 byChor, Goldreich, Kushilevitz and Sudan[2]in the information-theoretic setting and in 1997 by Kushilevitz and Ostrovsky in the computational setting.[3]Since then, very efficient solutions have been discovered. Single database (computationally private) PIR can be achieved with constant (amortized) communication and k-database (information theoretic) PIR can be done withnO(loglogkklogk){\displaystyle n^{O\left({\frac {\log \log k}{k\log k}}\right)}}communication.
The first single-database computational PIR scheme to achieve communication complexity less thann{\displaystyle n}was created in 1997 by Kushilevitz and Ostrovsky[3]and achieved communication complexity ofnϵ{\displaystyle n^{\epsilon }}for anyϵ{\displaystyle \epsilon }, wheren{\displaystyle n}is the number of bits in the database. The security of their scheme was based on the well-studiedquadratic residuosity problem. In 1999, Christian Cachin,Silvio Micaliand Markus Stadler[4]achieved poly-logarithmic communication complexity. The security of their system is based on thephi-hiding assumption. In 2004,Helger Lipmaa[5]achieved log-squared communication complexityO(ℓlogn+klog2n){\displaystyle O(\ell \log n+k\log ^{2}n)}, whereℓ{\displaystyle \ell }is the length of the strings andk{\displaystyle k}is the security parameter. The security of his system reduces to thesemantic securityof a length-flexible additively homomorphic cryptosystem like theDamgård–Jurik cryptosystem. In 2005 Craig Gentry andZulfikar Ramzan[6]achieved log-squared communication complexity which retrieves log-square (consecutive) bits of the database. The security of their scheme is also based on a variant of the Phi-hiding assumption. The communication rate was finally brought down to1{\displaystyle 1}byAggelos Kiayias,Nikos Leonardos,Helger Lipmaa,Kateryna Pavlyk,Qiang Tang, in 2015.[7]
All previous sublinear-communication computational PIR protocol required linear computational complexity ofΩ(n){\displaystyle \Omega (n)}public-key operations. In 2009,Helger Lipmaa[8]designed a computational PIR protocol with communication complexityO(ℓlogn+klog2n){\displaystyle O(\ell \log n+k\log ^{2}n)}and worst-case computation ofO(n/logn){\displaystyle O(n/\log n)}public-key operations. Amortization techniques that retrieve non-consecutive bits have been considered byYuval Ishai,Eyal Kushilevitz,Rafail OstrovskyandAmit Sahai.[9]
As shown by Ostrovsky and Skeith,[10]the schemes by Kushilevitz and Ostrovsky[3]and Lipmaa[5]use similar ideas based onhomomorphic encryption. The Kushilevitz and Ostrovsky protocol is based on theGoldwasser–Micali cryptosystemwhile the protocol by Lipmaa is based on theDamgård–Jurik cryptosystem.
Achieving information theoretic security requires the assumption that there are multiple non-cooperating servers, each having a copy of the database. Without this assumption, any information-theoretically secure PIR protocol requires an amount of communication that is at least the size of the databasen. Multi-server PIR protocols tolerant of non-responsive or malicious/colluding servers are calledrobustorByzantine robustrespectively. These issues were first considered by Beimel and Stahl (2002). An ℓ-server system that can operate where onlykof the servers respond, ν of the servers respond incorrectly, and which can withstand up totcolluding servers without revealing the client's query is called "t-private ν-Byzantine robustk-out-of-ℓ PIR" [DGH 2012]. In 2012, C. Devet, I. Goldberg, andN. Heninger(DGH 2012) proposed an optimally robust scheme that is Byzantine-robust toν<k−t−1{\displaystyle \nu <k-t-1}which is the theoretical maximum value. It is based on an earlier protocol of Goldberg that usesShamir's Secret Sharingto hide the query. Goldberg has released aC++implementation onSourceForge.[11]
One-way functionsare necessary, but not known to be sufficient, for nontrivial (i.e., with sublinear communication) single database computationally private information retrieval. In fact, such a protocol was proved by Giovanni Di Crescenzo,Tal MalkinandRafail Ostrovskyto imply oblivious transfer (see below).[12]
Oblivious transfer, also called symmetric PIR, is PIR with the additional restriction that the user may not learn any item other than the one she requested. It is termed symmetric because both the user and the database have a privacy requirement.
Collision-resistantcryptographic hash functionsare implied by any one-round computational PIR scheme, as shown by Ishai, Kushilevitz and Ostrovsky.[13]
The basic motivation for private information retrieval is a family of two-party protocols in which one of the parties (thesender) owns a database, and the other part (thereceiver) wants to query it with certain privacy restrictions and warranties. So, as a result of the protocol, if thereceiverwants thei-thvalue in the database he must learn thei-thentry, but thesendermust learn nothing abouti. In a general PIR protocol, a computationally unboundedsendercan learn nothing aboutiso privacy is theoretically preserved. Since the PIR problem was posed, different approaches to its solution have been pursued and some variations were proposed.
A CPIR (computationally private information retrieval) protocol is similar to a PIR protocol: thereceiverretrieves an element chosen by him from the sender's database, so that thesenderobtains no knowledge about which element was transferred.[8]The only difference is that privacy is safeguarded against a polynomially bounded sender.[14]
A CSPIR (computationally symmetric private information retrieval) protocol is used in a similar scenario in which a CPIR protocol is used. If thesenderowns a database, and thereceiverwants to get thei-thvalue in this database, at the end of the execution of a SPIR protocol, thereceivershould have learned nothing about values in the database other than thei-thone.[14]
Numerous Computational PIR and Information theoretic PIR schemes in the literature have been implemented. Here is an incomplete list:
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Incryptography, theZimmermann–Sassaman key-signing protocolis a protocol to speed up thepublic key fingerprintverification part of akey signing party. It requires somework before the event.
The protocol was invented during a key signing party withLen Sassaman,Werner Koch,Phil Zimmermann, and others.
The Sassaman-Efficient method is the first of the 2 types developed. Before the event, all participants email the keysigning coordinator their public keys. The coordinator then makes a text file of all the keys and accompanied fingerprint and then hashes it. They then proceed to make the text file and checksum available to all participants. The participants then download the file and check the validity using the hash. Then the participants print out the list and make sure that their own key is correct.
Everyone brings their own key list so that they know it is correct and not manipulated. Then the coordinator reads aloud or projects the checksums of the keys. Each participant verifies and states that their key is correct and once that is established a check mark can be put by that key. Once all the keys have been checked then the line folds upon itself and the participants then show each other at least 2 government-issued IDs. Once sufficient verification is established with the authenticity of the person, the other participant puts a second check mark by their name.
The participants then fetch the keys from a server or obtain a keyring made for the event. They sign each key on their list with 2 check marks and make sure that the fingerprints match. The signatures are then uploaded to the server or mailed directly to the key owner (if requested).[1]
The Sassaman-Projected method is a modified version of the Sassaman-Efficient, with the purpose for large groups. They both follow the same way with the exception of verifying identity. Instead of doing it individually the 2 forms of ID are projected for everyone to see at once. Once the person has verified that it is their key, the rest of the participants make 2 check marks next to the key.[2]
This cryptography-related article is astub. You can help Wikipedia byexpanding it.
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CryptoParty(Crypto-Party) is agrassrootsglobal endeavour[1]to introduce the basics of practical cryptography such as theTor anonymity network,I2P,Freenet,key signing parties,disk encryptionandvirtual private networksto the general public.[2][3]The project primarily consists of a series of free public workshops.
As a successor to theCypherpunksof the 1990s,[4]CryptoParty was conceived in late August 2012 by the Australian journalist Asher Wolf in aTwitterpost[5]following the passing of the Cybercrime Legislation Amendment Bill 2011 and the proposal of atwo-year data retention law in that country,[6]the Cybercrime Legislation Amendment Bill 2011.[7]TheDIY,self-organizingmovement immediately wentviral,[8]with a dozen autonomous CryptoParties being organized within hours in cities throughout Australia, the US, the UK, and Germany.[9]Many more parties were soon organized or held in Chile, The Netherlands, Hawaii, Asia, etc. Tor usage in Australia itself spiked,[10]and CryptoParty London with 130 attendees—some of whom were veterans of theOccupy Londonmovement—had to be moved fromLondon Hackspaceto the Google campus in east London'sTech City.
As of mid-October 2012 some 30 CryptoParties have been held globally, some on a continuing basis, and CryptoParties were held on the same day in Reykjavik, Brussels, and Manila.[11]
The first draft of the 442-pageCryptoParty Handbook(the hard copy of which is available at cost) was pulled together in three days using thebook sprintapproach, and was released 2012-10-04 under aCC BY-SAlicense.[12]
In May 2014,Wiredreported thatEdward Snowden, while employed byDellas anNSAcontractor, organized a local CryptoParty at a smallhackerspaceinHonolulu,Hawaiion December 11, six months before becoming well known for leaking tens of thousands of secret U.S. government documents. During the CryptoParty, Snowden taught 20 Hawaii residents how to encrypt their hard drives and use the Internet anonymously. The event was filmed by Snowden's then-girlfriend, but the video has never been released online. In a follow-up post to the CryptoParty wiki,[13]Snowden pronounced the event a "huge success."[14]
In 2013, CryptoParty received messages of support from theElectronic Frontier Foundation[15]and (purportedly)AnonyOps, as well as theNSAwhistleblowerThomas Drake,WikiLeakscentral editorHeather Marsh,[16]andWiredreporterQuinn Norton.[17]Eric Hughes, the author ofA Cypherpunk's Manifestonearly two decades before, delivered the keynote address,Putting the Personal Back in Personal Computers, at the Amsterdam CryptoParty on 2012-09-27.[18]
Marcin de Kaminski, founding member ofPiratbyrånwhich in turn foundedThe Pirate Bay, regarded CryptoParty as the most important civic project in cryptography in 2012,[19]andCory Doctorowhas characterized a CryptoParty as being "like aTupperwareparty for learning crypto."[20]Der Spiegelin December 2014 mentioned "crypto parties" in the wake of theEdward Snowdenleaks in an article about theNSA.[21]
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Afriend-to-friend(orF2F) computer network is a type ofpeer-to-peer networkin which users only make direct connections with people they know.Passwordsordigital signaturescan be used forauthentication.
Unlike other kinds ofprivate P2P, users in a friend-to-friend network cannot find out who else is participating beyond their own circle of friends, so F2F networks can grow in size without compromising their users' anonymity.Retroshare,WASTE,GNUnet,FreenetandOneSwarmare examples of software that can be used to build F2F networks, though RetroShare is the only one of these configured for friend-to-friend operation by default.
Many F2F networks support indirectanonymousorpseudonymouscommunication between users who do not know or trust one another. For example, anodein a friend-to-friendoverlaycan automatically forward a file (or a request for a file) anonymously between two friends, without telling either of them the other's name orIP address. These friends can in turn automatically forward the same file (or request) to their own friends, and so on.
Dan Bricklincoined the term "friend-to-friend network" in 2000.[1]
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Self-sovereign identity(SSI) is an approach todigital identitythat gives individuals control over the information they use to prove who they are towebsites, services, andapplicationsacross the web. Without SSI, individuals with persistent accounts (identities) across theinternetmust rely on a number of large identity providers, such asFacebook(Facebook Connect) andGoogle(Google Sign-In), that have control of the information associated with their identity.[2][3][4]If a user chooses not to use a large identity provider, then they have to create new accounts with each service provider, which fragments their web experiences. Self-sovereign identity offers a way to avoid these two undesirable alternatives. In a self-sovereign identity system, the user accesses services in a streamlined and secure manner, while maintaining control over the information associated with their identity.[5][6]
TheTCP/IP protocolprovides identifiers for machines, but not for the people and organisations operating the machines. This makes the network-level identifiers on the internet hard to trust and rely on for information and communication for a number of reasons: 1) hackers can easily change a computer’s hardware or IP address, 2) services provide identifiers for the user, not the network. The absence of reliable identifiers is one of the primary sources of cybercrime, fraud, and threats to privacy on the internet.[7]
With the advent of blockchain technology, a new model for decentralized identity emerged in 2015.[8]TheFIDO Allianceproposed an identity model that was no longer account-based, but identified people through direct, private, peer-to-peer connections secured bypublic/private key cryptography. Self-Sovereign Identity (SSI) summarises all components of the decentralized identity model: digital wallets, digital credentials, and digital connections.[3]
SSI addresses the difficulty of establishing trust in an interaction. In order to be trusted, one party in an interaction will present credentials to the other parties, and those relying on the parties can verify that the credentials came from an issuer that they trust. In this way, the verifier's trust in the issuer is transferred to the credential holder.[9]This basic structure of SSI with three participants is sometimes called "the trust triangle".[3]
It is generally recognized that for an identity system to be self-sovereign, users control theverifiable credentialsthat they hold, and their consent is required to use those credentials.[10]This reduces the unintended sharing of users'personal data. This is contrasted with the centralized identityparadigmwhere identity is provided by some outside entity.[11]
In an SSI system, holders generate and control uniqueidentifierscalleddecentralized identifiers. Most SSI systems aredecentralized, where the credentials are managed usingcrypto walletsand verified usingpublic-key cryptographyanchored on adistributed ledger.[12]The credentials may contain data from an issuer's database, asocial media account, a history of transactions on an e-commerce site, orattestationfrom friends or colleagues.
TheEuropean Unionis exploring decentralized digital identity through a number of initiatives including the International Association for Trusted Blockchain Application (INATBA), the EU Blockchain Observatory & Forum and the European SSI Framework. In 2019, the EU created aneIDAScompatible European Self-Sovereign Identity Framework (ESSIF). The ESSIF makes use ofdecentralized identifiers(DIDs) and the European Blockchain Services Infrastructure (EBSI).[13][14]
The Korean government created a public/private consortium specifically for decentralized identity.[15]
In the German and European legal area, there are two regulations that are of particular importance for the topic. These include the eIDAS Regulation, which forms the most important framework for trust in electronic identification in the EU and is a fundamental building block of the digital single market. The European Blockchain Service Infrastructure (EBSI)[16]has provided the SSI eIDAS Bridge,[17]as a technical implementation that enables a substantial level of trust.[18]The eIDAS SSI legal report also describes several scenarios of how SSI can fulfill the necessary regulatory conditions.
Furthermore, the General Data Protection Regulation (GDPR) forms the legal basis for the handling of personal data. The EBSI GDPR Legal Report provides more information on this.[19]
SSI is a value laden technology whose technical operationalizations differ (see Technical aspects).[20]Therefore, its implementations can vary significantly and embed into the very technology different goals, agenda, and intentions.[21]
The term "self-sovereign identity" can create expectations that individuals have absolute control and ownership over their digital identities, akin to physical possessions. However, in practice, SSI involves complex technical infrastructure, interactions with identity issuers and verifiers, and compliance with legal frameworks. The reality may not align with the perception generated by the term, leading to semantic confusion.[22]
Critics argue that SSI may exacerbate social inequalities and exclude those with limited access to technology or digital literacy.[21][23]SSI assumes reliable internet connectivity, access to compatible devices, and proficiency in navigating digital systems. Consequently, marginalized populations, including the elderly, individuals in developing regions, or those with limited technological resources, may face exclusion and reduced access to the benefits of SSI.[24]
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Avirtual communityis asocial networkof individuals who connect through specificsocial media, potentially crossing geographical and political boundaries in order to pursue mutual interests or goals. Some of the most pervasive virtual communities areonline communitiesoperating undersocial networking services.
Howard Rheingolddiscussed virtual communities in his book,The Virtual Community, published in 1993. The book's discussion ranges from Rheingold's adventures onThe WELL,computer-mediated communication, social groups and information science. Technologies cited includeUsenet,MUDs(Multi-User Dungeon) and their derivativesMUSHesandMOOs,Internet Relay Chat(IRC),chat roomsandelectronic mailing lists. Rheingold also points out the potential benefits for personal psychological well-being, as well as for society at large, of belonging to a virtual community. At the same time, it showed that job engagement positively influences virtual communities of practice engagement.[1]
Virtual communities all encourage interaction, sometimes focusing around a particular interest or just to communicate. Some virtual communities do both. Community members are allowed to interact over a shared passion through various means:message boards,chat rooms,social networkingWorld Wide Web sites, or virtual worlds.[2]Members usually become attached to the community world, logging in and out on sites all day every day, which can certainly become an addiction.[3]
The traditional definition of a community is of geographically circumscribed entity (neighborhoods, villages, etc.). Virtual communities are usually dispersed geographically, and therefore are not communities under the original definition. Some online communities are linked geographically, and are known as community websites. However, if one considers communities to simply possess boundaries of some sort between their members and non-members, then a virtual community is certainly a community.[4]Virtual communities resemble real lifecommunitiesin the sense that they both provide support, information, friendship and acceptance between strangers.[5]While in a virtual community space, users may be expected to feel a sense of belonging and a mutual attachment among the members that are in the space.
One of the most influential part about virtual communities is the opportunity to communicate through several media platforms or networks. Now that virtual communities exists, this had leveraged out the things we once did prior to virtual communities, such as postal services, fax machines, and even speaking on the telephone. Early research into the existence of media-based communities was concerned with the nature ofreality, whether communities actually could exist through the media, which could place virtual community research into the social sciences definition of ontology. In the seventeenth century, scholars associated with theRoyal Societyof London formed a community through the exchange of letters.[4]"Community without propinquity", coined by urban plannerMelvin Webberin 1963 and "community liberated", analyzed byBarry Wellmanin 1979 began the modern era of thinking about non-local community.[6]As well,Benedict Anderson'sImagined Communitiesin 1983, described how different technologies, such as national newspapers, contributed to the development of national and regional consciousness among early nation-states.[7]Some authors that built their theories on Anderson's imagined communities have been critical of the concept, claiming that all communities are based on communication and that virtual/real dichotomy is disintegrating, making use of the word "virtual" problematic or even obsolete.[8]
Virtual communities are used for a variety of social and professional groups; interaction between community members vary from personal to purely formal. For example, an email distribution list could serve as a personal means of communicating with family and friends, and also formally to coordinate with coworkers.
User experienceis the ultimate goal for the program or software used by an internet community, because user experience will determine the software's success.[9]The software for social media pages or virtual communities is structured around the users' experience and designed specifically for online use.
User experience testing is utilized to reveal something about the personal experience of the human being using a product or system.[10]When it comes to testing user experience in a software interface, three main characteristics are needed: a user who is engaged, a user who is interacting with a product or interface, and defining the users' experience in ways that are and observable or measurable.[10]User experience metrics are based on a reliability and repeatability, using a consistent set of measurements to result in comparable outcomes. User experience metrics are based on user retention, using a consistent set of measurements to collect data on user experience.
The widespread use of the Internet and virtual communities by millions of diverse users for socializing is a phenomenon that raises new issues for researchers and developers. The vast number and diversity of individuals participating in virtual communities worldwide makes it a challenge to test usability across platforms to ensure the best overall user experience. Some well-established measures applied to the usability framework for online communities are speed of learning, productivity, user satisfaction, how much people remember using the software, and how many errors they make.[11]The human computer interactions that are measured during a usability experience test focus on the individuals rather than their social interactions in the online community. The success of online communities depend on the integration of usability and social semiotics. Usability testing metrics can be used to determine social codes by evaluating a user's habits when interacting with a program. Social codes are established and reinforced by the regular repetition of behavioral patterns.[12]People communicate their social identities orculture codethrough the work they do, the way they talk, the clothes they wear, their eating habits, domestic environments and possessions, and use of leisure time. Usability testing metrics can be used to determine social codes by evaluating a user's habits when interacting with a program. The information provided during a usability test can determine demographic factors and help define the semiotic social code. Dialogue and social interactions, support information design, navigation support, and accessibility are integral components specific to online communities. As virtual communities grow, so do the diversity of their users. However, the technologies are not made to be any more or less intuitive. Usability tests can ensure users are communicating effectively using social and semiotic codes while maintaining their social identities.[11]Efficient communication requires a common set of signs in the minds of those seeking to communicate.[12]As technologies evolve and mature, they tend to be used by an increasingly diverse set of users. This kind of increasing complexity and evolution of technology does no necessarily mean that the technologies are becoming easier to use.[10]Usability testing in virtual communities can ensure users are communicating effectively through social and semiotic codes and maintenance of social realities and identities.[12]
Recent studies have looked into development of health related communities and their impact on those already suffering health issues. These forms of social networks allow for open conversation between individuals who are going through similar experiences, whether themselves or in their family.[13]Such sites have so grown in popularity that now many health care providers form groups for their patients by providing web areas where one may direct questions to doctors. These sites prove especially useful when related to rare medical conditions. People with rare or debilitating disorders may not be able to access support groups in their physical community, thus online communities act as primary means for such support. Online health communities can serve as supportive outlets as they facilitate connecting with others who truly understand the disease, as well as offer more practical support, such as receiving help in adjusting to life with the disease.[14]Each patient on online health communities are on there for different reasons, as some may need quick answers to questions they have, or someone to talk to.Involvement in social communities of similar health interests has created a means for patients to develop a better understanding and behavior towards treatment and health practices.[15][16]Some of these users could have very serious life-threatening issues which these personal contexts could become very helpful to these users, as the issues are very complex.[17]Patients increasingly use such outlets, as this is providing personalized and emotional support and information, that will help them and have a better experience.[17]The extent to which these practices have effects on health are still being studied.
Studies on health networks have mostly been conducted on groups which typically suffer the most from extreme forms of diseases, for example cancer patients, HIV patients, or patients with other life-threatening diseases. It is general knowledge that one participates in online communities to interact with society and develop relationships.[18]Individuals who suffer from rare or severe illnesses are unable to meet physically because of distance or because it could be a risk to their health to leave a secure environment. Thus, they have turned to the internet.
Some studies have indicated that virtual communities can provide valuable benefits to their users. Online health-focused communities were shown to offer a unique form of emotional support that differed from event-based realities and informational support networks. Growing amounts of presented material show how online communities affect the health of their users. Apparently the creation of health communities has a positive impact on those who are ill or in need of medical information.[19]
It was found that young individuals are more bored with politics and history topics, and instead are more interested in celebrity dramas and topics. Young individuals claim that "voicing what you feel" does not mean "being heard", so they feel the need to not participate in these engagements, as they believe they are not being listened to anyway.[20]Over the years, things have changed, as new forms of civic engagement and citizenship have emerged from the rise of social networking sites. Networking sites act as a medium for expression and discourse about issues in specific user communities. Online content-sharing sites have made it easy for youth as well as others to not only express themselves and their ideas through digital media, but also connect with large networked communities. Within these spaces, young people are pushing the boundaries of traditional forms of engagement such as voting and joining political organizations and creating their own ways to discuss, connect, and act in their communities.[21]
Civic engagement throughonline volunteeringhas shown to have positive effects on personal satisfaction and development. Some 84 percent of online volunteers found that their online volunteering experience had contributed to their personal development and learning.[22]
In his bookThe Wealth of Networksfrom 2006,Yochai Benklersuggests that virtual communities would "come to represent a new form of human communal existence, providing new scope for building a shared experience of human interaction".[23]Although Benkler's prediction has not become entirely true, clearly communications and social relations are extremely complex within a virtual community. The two main effects that can be seen according to Benkler are a "thickening of preexisting relations with friends, family and neighbours" and the beginnings of the "emergence of greater scope for limited-purpose, loose relationships".[23]Despite being acknowledged as "loose" relationships, Benkler argues that they remain meaningful.
Previous concerns about the effects of Internet use on community and family fell into two categories: 1) sustained, intimate human relations "are critical to well-functioning human beings as a matter of psychological need" and 2) people with "social capital" are better off than those who lack it. It leads to better results in terms of political participation.[23]However, Benkler argues that unless Internet connections actually displace direct, unmediated, human contact, there is no basis to think that using the Internet will lead to a decline in those nourishing connections we need psychologically, or in the useful connections we make socially. Benkler continues to suggest that the nature of an individual changes over time, based on social practices and expectations. There is a shift from individuals who depend upon locally embedded, unmediated and stable social relationships to networked individuals who are more dependent upon their own combination of strong and weak ties across boundaries and weave their own fluid relationships. Manuel Castells calls this the "networked society".[23]
In 1997,MCI Communicationsreleased the "Anthem" advertisement, heralding the internet as a utopia without age, race, or gender.Lisa Nakamuraargues in chapter 16 of her 2002 bookAfter/image of identity: Gender, Technology, and Identity Politics, that technology gives us iterations of our age, race and gender in virtual spaces, as opposed to them being fully extinguished. Nakamura uses a metaphor of "after-images" to describe the cultural phenomenon of expressing identity on the internet. The idea is that any performance of identity on the internet is simultaneously present and past-tense, "posthuman and projectionary", due to its immortality.[24]
Sherry Turkle, professor of Social Studies of Science and Technology atMIT, believes the internet is a place where actions of discrimination are less likely to occur. In her 1995 bookLife on the Screen: Identity in the Age of the Internet, she argues that discrimination is easier in reality as it is easier to identify as face value, what is contrary to one's norm. The internet allows for a more fluid expression of identity and thus people become more accepting of inconsistent personae within themselves and others. For these reasons, Turkle argues users existing in online spaces are less compelled to judge or compare themselves to their peers, allowing people in virtual settings an opportunity to gain a greater capacity for acknowledging diversity.[25]
Nakamura argues against this view, coining the termidentity tourismin her 1999 article "Race In/For Cyberspace: Identity Tourism and Racial Passing on the Internet". Identity tourism, in the context of cyberspace, is a term used to the describe the phenomenon of users donning and doffing other-race and other-gender personae. Nakamura finds that performed behavior from these identity tourists often perpetuate stereotypes.[26]
In the 1998 bookCommunities in Cyberspace, authorsMarc A. SmithandPeter Kollock, perceives the interactions with strangers are based upon with whom we are speaking or interacting with. People use everything from clothes, voice,body language,gestures, and power to identify others, which plays a role with how they will speak or interact with them. Smith and Kollock believes that online interactions breaks away of all of the face-to-face gestures and signs that people tend to show in front of one another. Although this is difficult to do online, it also provides space to play with one's identity.[27]
The gaming community is extremely vast and accessible to a wide variety of people, However, there are negative effects on the relationships "gamers" have with the medium when expressing identity of gender.Adrienne Shawnotes in her 2012 article "Do you identify as a gamer? Gender, race, sexuality, and gamer identity", that gender, perhaps subconsciously, plays a large role in identifying oneself as a "gamer".[28]According to Lisa Nakamura, representation in video games has become a problem, as the minority of players from different backgrounds who are not just the stereotyped white teen male gamer are not represented.[29]
The explosive diffusion[30]of the Internet since the mid-1990s fostered the proliferation of virtual communities in the form of social networking services and online communities. Virtual communities may synthesizeWeb 2.0technologies with the community, and therefore have been described as Community 2.0, although strong community bonds have been forged online since the early 1970s on timeshare systems likePLATOand later onUsenet. Online communities depend upon social interaction and exchange between users online. This interaction emphasizes thereciprocityelement of the unwrittensocial contractbetween community members.
An onlinemessage boardis a forum where people can discuss thoughts or ideas on various topics or simply express an idea. Users may choose which thread, or board of discussion, they would like to read or contribute to. A user will start a discussion by making a post.[31]Other users who choose to respond can follow the discussion by adding their own posts to that thread at any time. Unlike in spokenconversations, message boards do not usually have instantaneous responses; users actively go to the website to check for responses.
Anyone can register to participate in an online message board. People can choose to participate in the virtual community, even if or when they choose not to contribute their thoughts and ideas. Unlike chat rooms, at least in practice, message boards can accommodate an almost infinite number of users.
Internet users' urges to talk to and reach out to strangers online is unlike those in real-life encounters where people are hesitant and often unwilling to step in to help strangers. Studies have shown that people are more likely to intervene when they are the only one in a situation. With Internet message boards, users at their computers are alone, which might contribute to their willingness to reach out. Another possible explanation is that people can withdraw from a situation much more easily online than off. They can simply click exit or log off, whereas they would have to find a physical exit and deal with the repercussions of trying to leave a situation in real life. The lack of status that is presented with an online identity also might encourage people, because, if one chooses to keep it private, there is no associated label of gender, age, ethnicity or lifestyle.[32]
Shortly after the rise of interest in message boards and forums, people started to want a way of communicating with their "communities" in real time. The downside to message boards was that people would have to wait until another user replied to their posting, which, with people all around the world in different time frames, could take a while. The development of onlinechat roomsallowed people to talk to whoever was online at the same time they were. This way, messages were sent and online users could immediately respond.
The original development byCompuServe CBhosted forty channels in which users could talk to one another in real time. The idea of forty different channels led to the idea of chat rooms that were specific to different topics. Users could choose to join an already existent chat room they found interesting, or start a new "room" if they found nothing to their liking. Real-time chatting was also brought into virtual games, where people could play against one another and also talk to one another through text. Now, chat rooms can be found on all sorts of topics, so that people can talk with others who share similar interests. Chat rooms are now provided byInternet Relay Chat(IRC) and other individual websites such asYahoo,MSN, andAOL.
Chat room users communicate through text-based messaging. Most chat room providers are similar and include an input box, a message window, and a participant list. The input box is where users can type their text-based message to be sent to the providing server. The server will then transmit the message to the computers of anyone in the chat room so that it can be displayed in the message window. The message window allows the conversation to be tracked and usually places a time stamp once the message is posted. There is usually a list of the users who are currently in the room, so that people can see who is in their virtual community.
Users can communicate as if they are speaking to one another in real life. This "simulated reality" attribute makes it easy for users to form a virtual community, because chat rooms allow users to get to know one another as if they were meeting in real life. The individual "room" feature also makes it more likely that the people within a chat room share a similar interest; an interest that allows them to bond with one another and be willing to form a friendship.[33][34]
Virtual worldsare the most interactive of all virtual community forms. In this type of virtual community, people are connected by living as anavatarin a computer-based world. Users create their own avatar character (from choosing the avatar's outfits to designing the avatar's house) and control their character's life and interactions with other characters in the 3D virtual world. It is similar to a computer game; however, there is no objective for the players. A virtual world simply gives users the opportunity to build and operate a fantasy life in the virtual realm. Characters within the world can talk to one another and have almost the same interactions people would have in reality. For example, characters can socialize with one another and hold intimate relationships online.
This type of virtual community allows for people to not only hold conversations with others in real time, but also to engage and interact with others. The avatars that users create are like humans. Users can choose to make avatars like themselves, or take on an entirely different personality than them. When characters interact with other characters, they can get to know one another through text-based talking and virtual experience (such as having avatars go on a date in the virtual world). A virtual community chat room may give real-time conversations, but people can only talk to one another. In a virtual world, characters can do activities together, just like friends could do in reality. Communities in virtual worlds are most similar to real-life communities because the characters are physically in the same place, even if the users who are operating the characters are not.[35]Second Lifeis one of the most popular virtual worlds on the Internet.Whyvilleoffers an alternative for younger audiences where safety and privacy are a concern. In Whyville, players use the virtual world's simulation aspect to experiment and learn about various phenomena.
Another use for virtual worlds has been in business communications. Benefits from virtual world technology such as photo realistic avatars and positional sound create an atmosphere for participants that provides a less fatiguing sense of presence. Enterprise controls that allow the meeting host to dictate the permissions of the attendees such as who can speak, or who can move about allow the host to control the meeting environment.Zoom, is a popular platform that has grown over theCOVID-19 pandemic. Where those who host meetings on this platform, can dictate who can or cannot speak, by muting or unmuting them, along with who is able to join. Several companies are creating business based virtual worlds includingSecond Life. These business based worlds have stricter controls and allow functionality such as muting individual participants, desktop sharing, or access lists to provide a highly interactive and controlled virtual world to a specific business or group. Business based virtual worlds also may provide various enterprise features such as Single Sign on with third party providers, or Content Encryption.[citation needed]
Social networking servicesare the most prominent type of virtual community. They are either a website or software platform that focuses on creating and maintaining relationships.Facebook,Twitter, andInstagramare all virtual communities. With these sites, one often creates a profile or account, and adds friends or follow friends. This allows people to connect and look for support using the social networking service as a gathering place. These websites often allow for people to keep up to date with their friends and acquaintances' activities without making much of an effort.[36]On several of these sites you may be able to video chat, with several people at once, making the connections feel more like you are together. On Facebook, for example, one can upload photos and videos, chat, make friends, reconnect with old ones, and join groups or causes.[37]
Participatory culture plays a large role in online and virtual communities. In participatory culture, users feel that their contributions are important and that by contributing, they are forming meaningful connections with other users. The differences between being a producer of content on the website and being a consumer on the website become blurred and overlap. According toHenry Jenkins, "Members believe their contributions matter and feel some degree of social connection with one another "(Jenkins, et al. 2005). The exchange and consumption of information requires a degree of "digital literacy", such that users are able to "archive, annotate, appropriate, transform and recirculate media content" (Jenkins). Specialized information communities centralizes a specific group of users who are all interested in the same topic. For example, TasteofHome.com, the website of the magazineTaste of Home, is a specialized information community that focuses on baking and cooking. The users contribute consumer information relating to their hobby and additionally participate in further specialized groups and forums. Specialized Information Communities are a place where people with similar interests can discuss and share their experiences and interests.
Howard Rheingold'sVirtual Communitycould be compared withMark Granovetter's ground-breaking "strength of weak ties" article published twenty years earlier in theAmerican Journal of Sociology. Rheingold translated, practiced and published Granovetter's conjectures about strong and weak ties in the online world. His comment on the first page even illustrates the social networks in the virtual society: "My seven year old daughter knows that her father congregates with a family of invisible friends who seem to gather in his computer. Sometimes he talks to them, even if nobody else can see them. And she knows that these invisible friends sometimes show up in the flesh, materializing from the next block or the other side of the world" (page 1). Indeed, in his revised version ofVirtual Community, Rheingold goes so far to say that had he readBarry Wellman's work earlier, he would have called his book "onlinesocial networks".
Rheingold's definition contains the terms "social aggregation and personal relationships" (page 3). Lipnack and Stamps (1997)[38]and Mowshowitz (1997) point out how virtual communities can work across space, time and organizational boundaries; Lipnack and Stamps (1997)[38]mention a common purpose; and Lee, Eom, Jung and Kim (2004) introduce "desocialization" which means that there is less frequent interaction with humans in traditional settings, e.g. an increase in virtual socialization. Calhoun (1991) presents adystopiaargument, asserting the impersonality of virtual networks. He argues that IT has a negative influence on offline interaction between individuals because virtual life takes over our lives. He believes that it also creates different personalities in people which can cause frictions in offline and online communities and groups and in personal contacts. (Wellman & Haythornthwaite, 2002). Recently, Mitch Parsell (2008) has suggested that virtual communities, particularly those that leverage Web 2.0 resources, can be pernicious by leading to attitude polarization, increased prejudices and enabling sick individuals to deliberately indulge in their diseases.[39]
Internet communities offer the advantage of instant information exchange that is not possible in a real-life community. This interaction allows people to engage in many activities from their home, such as: shopping, paying bills, and searching for specific information. Users of online communities also have access to thousands of specific discussion groups where they can form specialized relationships and access information in such categories as: politics, technical assistance, social activities, health (see above) and recreational pleasures. Virtual communities provide an ideal medium for these types of relationships because information can easily be posted and response times can be very fast. Another benefit is that these types of communities can give users a feeling of membership and belonging. Users can give and receive support, and it is simple and cheap to use.[40]
Economically, virtual communities can be commercially successful, making money through membership fees, subscriptions, usage fees, and advertising commission. Consumers generally feel very comfortable making transactions online provided that the seller has a good reputation throughout the community. Virtual communities also provide the advantage ofdisintermediationin commercial transactions, which eliminates vendors and connects buyers directly to suppliers. Disintermediation eliminates pricey mark-ups and allows for a more direct line of contact between the consumer and the manufacturer.[41]
While instant communication means fast access, it also means that information is posted without being reviewed for correctness. It is difficult to choose reliable sources because there is no editor who reviews each post and makes sure it is up to a certain degree of quality.[42]
In theory, online identities can be kept anonymous which enables people to use the virtual community for fantasy role playing as in the case ofSecond Life's use of avatars. Some professionals urge caution with users who use online communities because predators also frequent these communities looking for victims who are vulnerable to onlineidentity theftoronline predators.[43]
There are also issues surrounding bullying on internet communities. With users not having to show their face, people may use threatening and discriminating acts towards other people because they feel that they would not face any consequences.[44]
There are standing issues with gender and race on the online community as well, where only the majority is represented on the screen, and those of different background and genders are underrepresented.[29]
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In computer security, achain of trustis established by validating each component ofhardwareandsoftwarefrom the end entity up to the root certificate. It is intended to ensure that only trusted software and hardware can be used while still retaining flexibility.
A chain of trust is designed to allow multiple users to create and use the software on the system, which would be more difficult if all the keys were stored directly in hardware. It starts with hardware that will only boot from software that isdigitally signed. The signing authority will only sign boot programs that enforce security, such as only running programs that are themselves signed, or only allowing signed code to have access to certain features of the machine. This process may continue for several layers.
This process results in a chain of trust. The final software can be trusted to have certain properties because if it had been illegally modified its signature would be invalid, and the previous software would not have executed it. The previous software can be trusted, because it, in turn, would not have been loaded if its signature had been invalid. The trustworthiness of each layer is guaranteed by the one before, back to thetrust anchor.
It would be possible to have the hardware check the suitability (signature) for every single piece of software. However, this would not produce the flexibility that a "chain" provides. In a chain, any given link can be replaced with a different version to provide different properties, without having to go all the way back to the trust anchor. This use of multiple layers is an application of a general technique to improve scalability and is analogous to the use of multiple certificates in acertificate chain.
In computer security, digital certificates are verified using a chain of trust.[1]The trust anchor for the digital certificate is the rootcertificate authority(CA).
The certificate hierarchy is a structure of certificates that allows individuals to verify the validity of a certificate's issuer. Certificates are issued and signed by certificates that reside higher in the certificate hierarchy, so the validity and trustworthiness of a given certificate is determined by the corresponding validity of the certificate that signed it.
The chain of trust of a certificate chain is an ordered list of certificates, containing an end-user subscriber certificate andintermediate certificates(that represents the intermediate CA), that enables the receiver to verify that the sender and all intermediate certificates are trustworthy. This process is best described in the pageIntermediate certificate authority. See alsoX.509 certificate chainsfor a description of these concepts in a widely used standard for digital certificates.
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Incryptographicsystems with hierarchical structure, atrust anchoris an authoritative entity for which trust is assumed and not derived.[1]
In theX.509architecture, aroot certificatewould be the trust anchor from which the wholechain of trustis derived. The trust anchor must be in the possession of the trusting party beforehand to make any furthercertificate path validationpossible.
Most operating systems provide a built-in list of self-signedroot certificatesto act as trust anchors for applications. TheFirefoxweb browser also provides its own list of trust anchors. The end-user of an operating system or web browser is implicitly trusting in the correct operation of that software, and the software manufacturer in turn is delegating trust for certain cryptographic operations to thecertificate authoritiesresponsible for the root certificates.
Thiscomputer securityarticle is astub. You can help Wikipedia byexpanding it.
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Anambigramis acalligraphiccomposition ofglyphs(letters, numbers, symbols or other shapes) that can yield different meanings depending on the orientation of observation.[2][3]Most ambigrams are visualpalindromesthat rely on some kind ofsymmetry, and they can often be interpreted asvisual puns.[4]The term was coined byDouglas Hofstadterin 1983–1984.[2][5]
Most often, ambigrams appear as visually symmetrical words. When flipped, they remain unchanged, or they mutate to reveal anothermeaning. "Half-turn" ambigrams undergo apoint reflection(180-degreerotational symmetry) and can be read upside down (for example, the word "swims"), while mirror ambigrams haveaxial symmetryand can be read through areflectivesurface like amirror. Many other types of ambigrams exist.[6]
Ambigrams can be constructed in variouslanguagesandalphabets, and the notion often extends tonumbersand othersymbols. It is a recentinterdisciplinaryconcept, combiningart,literature,mathematics,cognition, andoptical illusions. Drawing symmetrical words constitutes also arecreational activityforamateurs. Numerous ambigramlogosare famous, and ambigramtattooshave become increasingly popular. There are methods to design an ambigram, a field in which someartistshave become specialists.
The wordambigramwas coined in 1983 byDouglas Hofstadter, an American scholar ofcognitive sciencebest known as thePulitzer Prize-winning author of the bookGödel, Escher, Bach.[7][4][5]It is aneologismcomposed of the Latin prefixambi-("both") and the Greek suffix-gram("drawing, writing").[2]
Hofstadter describes ambigrams as "calligraphic designs that manage to squeeze in two different readings."[8]"The essence is imbuing a singlewritten formwithambiguity".[9][10]
Anambigramis avisual punof a special kind: acalligraphic designhaving two or more (clear)interpretationsaswritten words. One can voluntarily jump back and forth between the rivalreadingsusually by shifting one's physicalpoint of view(moving the design in some way) but sometimes by simply altering one'sperceptualbias towards a design (clicking an internal mental switch, so to speak). Sometimes the readings will say identical things, sometimes they will say different things.[11][4]
Hofstadter attributes the origin of the wordambigramto conversations among a small group of friends in 1983.[12]
Prior to Hofstadter'sterminology, other names were used to refer to ambigrams. Among them, the expressions "verticalpalindromes"[13]byDmitri Borgmann[14](1965) andGeorges Perec,[15][16]"designatures" (1979),[17]"inversions" (1980) byScott Kim,[18][19]or simply "upside-down words" byJohn Langdonand Robert Petrick.[20]
Ambigramwas added to theOxford English Dictionaryin March 2011,[6][21]and to theMerriam-Websterdictionary in September 2020.[2][22]Scrabbleincluded the word in its database in November 2022.[23][3][24]
Many ambigrams can be described asgraphicpalindromes.
The firstSator squarepalindrome was found in the ruins ofPompeii, meaning it was created before theEruption of Mount Vesuvius in 79 AD.
A sator square using themirror writingfor the representation of the letters S and N was carved in a stone wall inOppède(France) between theRoman Empireand theMiddle Ages,[26]thus producing a work made up of 25 letters and 8 differentcharacters, 3 naturally symmetrical (A, T, O), 3 others decipherable from left to right (R, P, E), and 2 others from right to left (S, N). This engraving is therefore readable in four directions.[27]
Although the term is recent, the existence ofmirrorambigrams has been attested since at least thefirst millennium. They are generallypalindromesstylizedto be visuallysymmetrical.
Inancient Greek, the phrase"ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ"(wash the sins, not only the face), is apalindromefound in several locations, including the site of the churchHagia Sophiain Turkey.[28][29]It is sometimes turned into a mirror ambigram when written in capital letters with the removal ofspaces, and the stylization of the letter Ν (Ν).
Aboustrophedonis a type ofbi-directional text, mostly seen in ancient manuscripts and other inscriptions. Every other line of writing is flipped or reversed, with reversed letters. Rather than going left-to-right as in modern European languages, or right-to-left as inArabicandHebrew, alternate lines in boustrophedon must be read in opposite directions. Also, the individual characters are reversed, or mirrored. This two-way writing system reveals that modern ambigrams can have quite ancient origins, with an intuitive component in some minds.
Mirror writing inIslamic calligraphyflourished during the early modern period, but its origins may stretch as far back as pre-Islamic mirror-image rock inscriptions in theHejaz.[30]
The earliest known non-naturalrotationalambigram dates to 1893 by artistPeter Newell.[31]Although better known for his children's books and illustrations forMark TwainandLewis Carroll, he published two books ofreversible illustrations, in which the picture turns into a different image entirely when flippedupside down. The last page in his bookTopsys & Turvyscontains the phraseThe end, which, when inverted, readsPuzzle. InTopsys & Turvys Number 2(1902), Newell ended with a variation on the ambigram in whichThe endchanges intoPuzzle 2.[32]
In March 1904 the Dutch-AmericancomicartistGustave Verbeekused ambigrams in three consecutive strips ofThe UpsideDowns of old man Muffaroo and little lady Lovekins.[33]His comics wereambiguous images, made in such a way that one could read the six-panel comic, flip the book and keep reading.
From June to September 1908, the British monthlyThe Strand Magazinepublished a series of ambigrams by different people in its "Curiosities" column.[34]Of particular interest is the fact that all four of the people submitting ambigrams believed them to be a rare property of particular words. Mitchell T. Lavin, whose "chump" was published in June, wrote, "I think it is in the only word in the English language which has this peculiarity," while Clarence Williams wrote, about his "Bet" ambigram, "Possibly B is the only letter of the alphabet that will produce such an interesting anomaly."[34][35]
In theLatin alphabet, many letters are symmetricalglyphs. Thecapital lettersB, C, D, E, H, I, K, O, and X have a horizontal symmetry axis. This means that all words that can be written using only these letters are naturallake reflectionambigrams; examples include BOOK, CHOICE, or DECIDE.
Thelowercaseletters o, s, x, and z arerotationally symmetric, while pairs such as b/q, d/p, n/u, and in sometypefacesa/e, h/y and m/w, are rotations of each other. Among the lowercase letters "l" is unique since its symmetry is broken if it is close to a reference character which establishes a clearx-height. When rotated around the middle of the x-height l/ȷ or lo/oȷ it doesn't appear the same, but it does when rotated around its center like the uppercase-I. Thus, the words "sos", "pod", "suns", "yeah", "swims", "passed", or "dollop", form natural rotational ambigrams.
More generally, a "natural ambigram" is a word that possesses one or moresymmetrieswhen written in its natural state, requiring notypographicstyling. The words "bud", "bid", or "mom", form natural mirror ambigrams when reflected over avertical axis, as does "ليبيا", the name of the countryLibyainArabic. The words "HIM", "TOY, "TOOTH" or "MAXIMUM", in all capitals, form natural mirror ambigrams when their letters are stacked vertically and reflected over a vertical axis. Theuppercaseword "OHIO" can flip a quarter to produce a 90°rotationalambigram when written inserifstyle (with large "feet" above and below the "I").
Like allstrobogrammatic numbers,69is a natural rotational ambigram.
Patterns in natureare regularities found in the natural world.[36]Similarly,patternsin ambigrams are regularities found ingraphemes.
As a consequence to this "natural" property, someshapesappear more or less appropriate to handle for thedesigner. Ambigram candidates can become "almostnatural", when all the letters except maybe one or two are symmetrically cooperative, for example the word "awesome" possesses 5 compatible letters (the central s that flips around itself, and the couples a/e and w/m).
A symmetrical ambigram can be called "homogram" (contraction of "homo-ambigram") when it remains unchanged after reflection, and "heterogram" when it transforms.[11][37]In the most common type of ambigram, the twointerpretationsarise when the image is rotated 180 degrees with respect to each other (in other words, a second reading is obtained from the first by simply rotating the sheet).
Douglas Hofstadtercoined the word "homogram" to define an ambigram with identical letters.[11][37]In this case, the first half of the word turns into the last half.[38]
A symmetrical ambigram may be called a "heterogram"[11][37](contraction of "hetero-ambigram") when it becomes a different word after rotation. Visually, a hetero-ambigram is symmetrical only when both versions of the pairing are shown together. Theaestheticappearance is more difficult to design when a changing ambigram is intended to be shown in one way only, becausesymmetrygenerally enhances the visual appearance of artwork. Technically, there are two times more combinations of letters involved in ahetero-ambigramthan in ahomo-ambigram. For example, the 180° rotational ambigram "yeah" contains only two pairs of letters: y/h and e/a, whereas the heterogram "yeah / good" contains four : y/d, e/o, a/o, and h/g.
There is no limitation to the number of words that can potentially be paired up as hetero-ambigrams, and full ambigramsentenceshave even been published.[15][38]
Ambigrams are exercises ingraphic designthat play withoptical illusions,symmetryandvisual perception.
Some ambigrams feature a relationship between theirformand theircontent. Ambigrams usually fall into one of several categories.
"Half-turn" ambigrams orpoint reflectionambigrams, commonly called "upside-down words", are 180°rotational symmetricalcalligraphies.[7]They can be read right side up or upside down, or both.
Rotation ambigrams are the most common type of ambigrams for good reason. When a word isturned upside down, the top halves of the letters turn into the bottom halves.
And because our eyes pay attention primarily to the top halves of letters when we read, that means that you can essentially chop off the top half of a word, turn it upside down, and glue it to itself to make an ambigram. [...][41]
Amirrororreflectionambigram is a design that can be read when reflected in amirrorvertically, horizontally, or at 45 degrees,[42]giving either the same word or another word or phrase.
When thereflectingsurface is vertical (like amirrorfor example), the calligraphic design is avertical axis mirror ambigram.
The "museum" ambigram is almost natural with mirror symmetry, because the first two letters are easily exchanged with the last two, and the lowercase letter e can be transformed into s by a fairly obvious typographical acrobatics.[43]
Vertical axis mirror ambigrams find clever applications inmirror writing(orspecular writing), that is formed by writing in the direction that is the reverse of the natural way for a given language, such that the result is themirror imageof normal writing: it appears normal when it is reflected in amirror. For example, the word "ambulance" could be read frontward and backward in a vertical axis reflective ambigram. Following this idea, the French artist Patrice Hamel created a mirror ambigram saying "entrée" (entrance, in French) one way, and "sortie" (exit) the other way, displayed in the giant glass façade of theGare du NordinParis, so that the travelers coming in readentrance, and those leaving readway out.[44]
When the reflecting surface is horizontal (like amirroring lakefor example), the calligraphic design is ahorizontal axis mirror ambigram.
The bookAmbigrams Revealedfeatures several creations of this type, like the word "Failure" mirroring in the water of a pond to give "Success", or "Love" changing into "Lust".[45]
In afigure / groundambigram, letters fit together so thenegative spacearound and between one word spells another word.[42]
InGestalt psychology,figure–ground perceptionis known as identifying afigurefrom the background. For example, black words on a printed paper are seen as the "figure", and the white sheet as the "background".
In ambigrams, thetypographic spaceof the background is used asnegative spaceto form new letters and new words. For example, inside acapitalH, one can easily insert a lowercasei.
The oil paintingYou & Me(US) byJohn Langdon(1996) belongs to this category. The word "me" fills the space between the letters of "you".[46]
WithEscher-liketessellationsassociated to wordpatterns, ambigrams can be oriented in three, four, and up to six directions via rotational symmetries of 120°, 90° and 60° respectively,[47]such as those created by French artist Alain Nicolas.[48]Some words can also transform in thenegative space, but the multiplication of constraints often has the effect of reducing either the readability or thecomplexityof thedesignedwords.
Ambigram tessellations are wordpuzzles, in whichgeometrysets the rules.[48]
Media related toAmbigram tessellationsat Wikimedia Commons.
A chain ambigram is a design where a word (or sometimes words) are interlinked, forming a repeating chain.[42]Letters are usually overlapped: a word will start partway through another word.
Sometimes chain ambigrams are presented in the form of a circle.
For example, the chain "...sunsunsunsun..." can flip upside down, but not the word "sun" alone, written horizontally.
A chain ambigram can be constituted of one to several elements. A single element ambigram chain is like asnake eating its own tail. A two-elements ambigram chain is like a snake eating the neighbor's tail with the neighbor eating the first snake's, and so on.
Scott Kim's "Infinity" works, and that ofJohn Langdon"Chain reaction", are alsoself-referential, since the first is infinite in the literal sense of the word, and the second, both reversible at 180° and interfering around the letter O, evokes a chain reaction.[49]
Aspinonym[de]is a type of ambigram in which a word is written using the sameglyphrepeated in differentorientations.[38]WEB is an example of a word that can easily be made into a spinonym.
Perceptualshiftambigrams, also called "oscillation" ambigrams, are designs with nosymmetrybut can be read astwo different wordsdepending on how the curves of the letters are interpreted.[42]These ambigrams work on the principle ofrabbit-duck-style ambiguous images.
For exampleDouglas Hofstadterexpresses the dual nature of light as revealed by physics with his perceptual shift ambigramWave / Particle.
"Quarter-turn" ambigrams or 90°rotationalambigrams turnclockwiseorcounterclockwiseto express different meanings.[4]For example, the letter U can turn into a C and reciprocally, or the letters M or W into an E.[38]
A totem ambigram is an ambigram whose letters are stacked like atotem, most often offering a vertical axismirror symmetry.
This type helps when several letters fit together, but hardly the whole word.
For example, in theMariamonogram[hu], the letters M, A and I are individually symmetrical, and the pairing R/A is almost naturally mirroring.
When adequately stacked, the 5 letters produce a nice totem ambigram, whereas the whole name "Maria" would not offer the same cooperativeness.
The ambigrammist artistJohn Langdondesigned several totemic assemblages, such as the word "METRO" composed of the symmetrical letter M, then section ETR, and below O; or the sentence "THANK YOU", vertical assembly of T, H, A, then of the symmetric NK couple, then finally Y, O, U.[50]
In mathematics, afractalis a geometricalshapethat exhibitsinvarianceunder scaling.
A piece of the whole, if enlarged, has the same geometrical features as the entire object itself.
A fractal ambigram is a sort of space-filling ambigrams where thetiledword branches from itself and then shrinks in aself-similarmanner, forming afractal.[51]In general, only a few letters are constrained in a fractal ambigram. The other letters don't need to look like any other, and thus can be shaped freely.
A3Dambigram is a design where an object is presented that will appear to read several letters or words when viewed from different angles.
Such designs can be generated usingconstructive solid geometry, a technique used insolid modeling, and then physically constructed with therapid prototypingmethod.
3-dimensional ambigramsculpturescan also be achieved inplastic arts. They arevolumeambigrams.
The original 1979 edition ofHofstadter'sGödel, Escher, Bachfeatured two 3-D ambigrams on the cover.[52]
Complex ambigrams are ambigrams involving more than one symmetry, or satisfying the criteria for several types. For example, a complex ambigram can be both rotational and mirror with a 4-folddihedralsymmetry. Or a spinonym that reads upside down is also a complex ambigram.
Ambigrams exist in many languages. With theLatin alphabet, they generally mixlowercaseanduppercaseletters. But words can also be symmetrical in other alphabets, likeArabic,Bengali,Cyrillic,Greek, and even inChinese charactersand Japanesekanji.
InKorean,곰(bear) and문(door),공(ball) and운(luck), or물(water) and롬(ROM) form a natural rotational ambigram. Some syllables like응(yes),표(ticket/signage) or를(object particle), and words like "허리피라우" (straighten your back) also make full ambigrams.
Thehan charactermeaning "hundred" is written百, that makes a natural 90° rotational ambigram when theglyphmakes a quarter turn counterclockwise, one sees "100".[53]
Media related toAmbigrams by languageat Wikimedia Commons.
An ambigram of numbers, ornumeral ambigram, containsnumerical digits, like1,2,3...[38]
Inmathematics, apalindromic number(also known as anumeral palindrome) is a number that remains the same when its digits arereversedthrough a vertical axis (but not necessarily visually). The palindromic numbers containing only 1, 8, and 0, constitute natural numeric ambigrams (visuallysymmetricalthrough amirror). Also, because theglyph2is graphically themirror imageof5, it means numbers like 205 or 85128 are natural numeral mirror ambigrams. Though not palindromic in the mathematical sense, they read frontward and backward like real ambigrams.
Astrobogrammatic numberis a number whose numeral isrotationally symmetric, so that it appears the same when rotated 180 degrees. The numeral looks the same right-side up and upside down (e.g., 69, 96, 1001).[54][55][56]
Somedatesare natural numeral ambigrams.[57]In March 1961, artistNorman Mingocreated an upside-down cover forMad magazinefeaturing an ambigram of the current year. The title says "No matter how you look at it... it's gonna be aMadyear. 1961, the first upside-down year since 1881."[58]Tuesday, 22 February 2022, was a palindrome and ambigram date called "Twosday" because it contained reversible 2 (two).[59][60][61]
Ambigrams of numbers receive most attention in the realm ofrecreational mathematics.[4][62]
Ambigrams with numbers sometimes combine letters and numerical digits. Because the number 5 is approximately shaped like the letter S, the number 6 like a lowercase b, the number 9 like the letter g, it is possible to play on these similarities to design ambigrams. A good example is theSochi 2014 (Olympic games)logo where the fourglyphscontained in 2014 are exact symmetries of the four letters S, o, i and h, individually.[63]
Asalphabet lettersareglyphsused in thewriting systemsto express thelanguagesvisually, othersymbolsare also used in the world to code other fields, like theprosignsin theMorse codeor themusical notesinmusic.
Similarly to the ambigrams of letters, the ambigrams with other symbols are generally visually symmetrical, eitherpoint reflectiveorreflective through an axis.
The internationalMorse codedistress signalSOS▄ ▄ ▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄ ▄ ▄is a natural ambigram constituted of dots and dashes. It flips upside down or through a mirror.
In morse code, the letter P coded▄ ▄▄▄ ▄▄▄ ▄and the letter R coded▄ ▄▄▄ ▄are individually symmetrical, like many other letters and numbers. Also, the letter G coded▄▄▄ ▄▄▄ ▄is the exact reverse of the letter W coded▄ ▄▄▄ ▄▄▄. Thus, the combination▄▄▄ ▄▄▄ ▄/▄ ▄▄▄ ▄▄▄coding the pairing G/W constitutes a natural ambigram. Consequently, meaningful natural ambigrams written in morse code certainly exist, like for example the words "gnaw"▄▄▄ ▄▄▄ ▄▄▄▄ ▄▄ ▄▄▄▄ ▄▄▄ ▄▄▄, "Dou"▄▄▄ ▄ ▄▄▄▄ ▄▄▄ ▄▄▄▄ ▄ ▄▄▄or "mom"▄▄▄ ▄▄▄▄▄▄ ▄▄▄ ▄▄▄▄▄▄ ▄▄▄.[7][64][65]
Inmusic, the interlude fromAlban Berg's operaLuluis apalindrome, thus thescoremade up ofmusical notesis almost symmetrical through a vertical axis.[66]
Inbiology, researchers study the ambigrammatic property ofnarnavirusesby using visual representations of the symmetrical sequences.[36][1][67]
Instead of simply writing them, ambigramletteringcovers theartofdrawingletters. In ambigram calligraphy, each letter acts as anillustration, each letter is created with attention to detail and has a unique role within acomposition. Lettering ambigrams do not translate into combinations of alphabet letters that can be used like atypeface, since they are created with a specific candidate in mind.
Thecalligrapher,graffitiwriter andgraphic designerNiels Shoe Meulmancreated several rotational ambigrams like the number "fifty",[69]the names "Shoe / Patta",[70]and the opposition "Love / Fear".[71]
Thecoverof the 7th volume of thetypographybookTypismis an ambigram drawn byNikita Prokhorov.[72]
The AmericantypedesignerMark Simonsondesigned poetic andhumorousambigrams, such as the words "Revelation", "Typophile", and the symbiosis "Drink / Drunk".[73]The last one makes avisual punwhen printed on ashot glass, sold commercially.[74]
Since they are visually striking, and sometimes surprising, ambigram words find large application incorporate logosandwordmarks, setting the visualidentityof many organizations, trademarks and brands.[75]
In 1968[76]or 1969,Raymond Loewydesigned the rotationalNew Man[fr]ambigram logo.[77][78][79]
The mirror ambigramDeLorean Motor Companylogo, designed by Phil Gibbon, was first used in 1975.[80][81][82]
Robert Petrick designed the invertibleAngellogo[83]in 1976.
The logoSun(Microsystems) designed by professorVaughan Pratt[84]in 1982 fulfills the criteria of several types: chain ambigram, spinonym, 90° and 180° rotational symmetries.
The Swedish pop groupABBAowns a mirror ambigram logo stylizedAᗺBAwith a reversed B, designed byRune Söderqvist[sv][85]in 1976.[86]
TheVenturalogo of the Visitors & Convention Bureau's board, in California, costUS$25,000 and was created in 2014 by the DuPuis group. It uses a 180° rotational symmetry.[87][88]
Other famous ambigram logos include:
the insurance companyAviva;[89]theacronymCRD(Capital Regional District) in the Canadian province of British Columbia;[90]the American multinational corporationDXCTechnology;
the two-sided marketplace for residential cleaningHandy;[91][92]the brand name of French premium high-speed train servicesInOui;[93]the French company specializing in ticketing and passenger information systemsIXXI;
the century-old brandMaoamof the confectionery manufacturer Haribo;[94]the American industrial rock bandNIͶ;
the Japanese food companyNissin;
the biotechnology companyNoxxonPharma, founded in 1997;
the online travel agencyOpodoin 2001;[95]the brand of food productsOXO[96]born in 1899;
the video gamePod;
the American developer and manufacturer of audio productsSonos;[97]the American professional basketball team PhoenixSuns;[98][99]the German manufacturer of adhesive productsUHU;
the quadruple symmetrical logoUAfrom the American clothing brandUnder Armour;
the Canadian corporation mandated to operate intercity passenger rail serviceVIAin 1978;[100]the American international broadcasterVOA, born in 1942;
and the Malaysian mobile virtual network operatorXOX. The student edition of theTesco Clubcardused 180° rotational symmetry.[101]
Because they arevisual puns,[4]ambigrams generally attract attention, and thus can be used invisual communicationto broadcast amarketingorpoliticalmessage.
In France, a mirror ambigram "Penelope/benevole" legible through a horizontal axis became amemeon the web after its diffusion onWikimedia Commons.[102]Penelope Fillon, wife of French politician and former Prime Minister of FranceFrançois Fillon, is suspected of having received wages for a fictitious job. Ironically, her name through the mirror becomesbenevole(voluntaryin French), suggesting dedication for a free service. Shared tens of thousands of times on thesocial networks, thishumorousambigram made thebuzzvia several French,[103]Belgian[104][105]and Swiss[102]medias.
Ambigrams are regularly used bycommunication agenciessuch asPublicisto engage the reader or the consumer through two-way messages.[106]Thus, in 2021, male first names transformed into female first names are included in a Swissadvertising campaignaimed at raising awareness aboutgender equality. An intriguingcatchphrasetypography upside down invites the reader to rotate the magazine, in which the first names "Michael" or "Peter" are transformed into "Nathalie" or "Alice".[107][108]
In 2015 iSmart's logo on one of its travelchargerswentviralbecause the brand's name turned out to be a natural ambigram that read "+Jews!" upside down. The company noted that "...we learned a powerful lesson of what not to do when creating alogo."[109]
Cinema posterssometimes seduce observers with ambigram titles, such as that ofTenetbyChristopher Nolan, by central symmetry.[27]orAnnabyLuc Bessonaround a vertical axis,[110][111]
The American artist and writerPeter Newellpublished arotationalambigram in 1893 saying "Puzzle / The end" in the book containingreversible illustrationsTopsys & Turvys.[31]
In March 1904 the Dutch-AmericancomicartistGustave Verbeekused ambigrams in three consecutive strips ofThe UpsideDowns of old man Muffaroo and little lady Lovekins.[33]His comics wereambiguous images, made in such a way that one could read the six-panel comic, flip the book and keep reading. InThe Wonderful Cure of the Waterfall(13 March 1904) an Indian medicine man says 'Big waters would make her very sound', while when flipped the medicine man turns into an Indian woman who says 'punos dery, ery apew poom, serlem big'. Which is explained as, 'poor deary' several foreign words that meant that she would call the 'Serlem Big'. The next comic calledAt the House of the Writing Pig(20 March 1904), where two ambigram wordballoonsare featured. The first features an angry pig trying to make the main protagonist leave by showing a sign that says; 'big boy go away, dis am home of mr h hog', up side down it reads 'Boy yew go away. We sip. Home of hog pig.' The protagonist asks the pig if it wants a big bun, upon which it replies 'Why big buns? Am mad u!', which flips into 'In pew we sang big hym'. Finally inThe Bad Snake and the Good Wizard(1904 Mar 27) there are two more ambigrams. The first turns 'How do you do' into the name of a wizard called 'Opnohop Moy', the second features a squirrel telling the protagonist 'Yes further on' only to inform it that there are 'No serpents here' on his way back. In a 2012 Swedish remake of the book,[112]the artist Marcus Ivarsson redrawsThe Bad Snake and the Good Wizardin his own style. He removes the squirrel, but keeps the other ambigram. 'How do you do' is replaced by 'Nejnej' (Swedish for no) and the wizard is now called 'Laulau'.
Media related toAmbigrams by Gustave Verbeekat Wikimedia Commons.
Oubapo,workshop of potentialcomic book art, is acomicsmovement which believes in the use offormalconstraintsto push the boundaries of the medium.Étienne Lécroart,cartoonist, is a founder and key member of Oubapo association, and has composed cartoons that could be read either horizontally, vertically, or in diagonal, and vice versa, sometimes including appropriate ambigrams.[113]
The Britishpainter, designer and illustratorRex Whistler, published in 1946 a rotational ambigram "¡OHO!" for the cover of a book gatheringreversible drawings.[114]
The artistJohn Langdon, specialist of ambigrams,[75]designed many colorpaintingsfeaturing ambigrams of all kinds, figure-ground, rotational, mirror or totem. Among other influences, he particularly admiresM. C. Escher'sdrawings.[115]
The Canadian artist Kelly Klages painted severalacrylicsoncanvaswith ambigram words and sentences referring to famous writers' novels written byWilliam ShakespeareorAgatha Christie, such asThird Girl,The Tempest,After the Funeral,The Hollow, Reformation,Sherlock Holmes, andElephants Can Remember.[116]
The GermanconceptualartistMia Florentine Weissbuilt a sculptural ambigramLove Hate[de],[117]that has traveled Europe as a symbol of peace and change of perspective.[118]Depending on which side the viewer looks at it, the sculpture says "Love" or "Hate". A similar concept was installed in front of theReichstag buildinginBerlinwith the words "Now / Won". Both sculptures are mirror type ambigrams, symmetrical around a vertical axis.[119]
The Swiss sculptorMarkus Raetzmade several three-dimensional ambigram works, featuring words generally with related meanings, such as
YES-NO (2003),[120]ME-WE (2004, 2010),[121]OUI-NON (2000–2002) in French,[122][123]SI–NO (1996)[124]and TODO-NADA (1998) in Spanish[125][126]These areanamorphicworks, which change in appearance depending on the angle of view of the observer.
The OUI–NON ambigram is installed on the Place du Rhône, inGeneva,Switzerland, at the top of a metal pole. Physically, the letters have the appearance of iron twists. With the perspective, this work demonstrates that reality can beambiguous.[123]
Some ambigram sculptures by the French conjurerFrancis Tabary[fr]are reversible by a half-turn rotation, and can therefore be exhibited on a support in two different ways.[127][128]
One of the most dynamic sectors that harbors ambigrams istattooing. Because they possess two ways of reading, ambigram tattoos inked on the skin benefit from a "mind-blowing" effect. On the arm,sleeve tattoosflip upside-down, on the back or jointly on two wrists they are more striking with amirror symmetry. A large range ofscriptsandfontsis available. Experienced ambigram artists can create anoptical illusionwith a complexvisual design.[129]
In 2015, an ambigramtattoowentviralfollowing anadvertising campaigndeveloped by thePublicisgroup two years earlier. TheSamaritans of Singaporeorganization, active in suicide prevention, has a 180° reversible "SOS" ambigram logo,acronymof its name andhomonymof the famousSOSdistress signal.
In 2013, this center orders advertisements that could be inserted in magazines to make readers aware of the problem ofdepressionamong young people, and the communication agency notices the symmetrical aspect of the logo. As a result, it begins to produce several ambigrammatic visuals, staged in photographic contexts, where sentences such as "I'm fine", "I feel fantastic" or "Life is great" turn into "Save me", "I'm falling apart", and "I hate myself". Readers noticing this logo placed at the upper left corner of the page with an upside-down typographicalcatchphraserotate the newspaper and visualize the double calligraphed messages, which call out with theSOS.[106][130]These ads are so influential that Bekah Miles, an American student herself coming out of a severe depression, chooses to use the "I'm fine / Save me" ambigram to get a tattoo on her thigh. Posted on Facebook, the two-sided photography immediately appeals to many young people, impressed or sensitive to this difficulty.[131][132]To educate its students,George Fox Universityin the United States then relays the optical illusion in its official journal, through a video totaling more than three million views[133]and the information is also reproduced in several local media and international organizations, thus helping to popularize this famous two-way tattoo.[134][135]Less fortunate, another teenage girl, aged 16, committed suicide, with her also this ambigram found on a note in her room, "I'm fine / Save me", reversible calligraphy today printed on badges and bracelets, for educational purposes.[136]
Ambigrams are sorts ofvisualpalindromes.[137]Some words turn upside down, others are symmetrical through a mirror. Natural ambigram palindromes exist, like the words "wow", "malayalam"[138](Dravidian language), or the biotechnology companyNoxxonthat possesses apalindromicname associated to a rotational ambigram logo. But some words are natural ambigrams, though not palindromes in the literary acception, like "bud" for example, because b and d are different letters. As a result, some words and sentences are good candidates for ambigrammists, but not for palindromists, and reciprocally, since theconstraintsdiffer slightly. Authors of ambigrams also benefit from a certain flexibility by playing on thetypefaceandgraphicaladjustments to influence the reading of their visual palindromes.
Oulipo,workshop of potential literature, seeks to create works usingconstrained writingtechniques.[13]Georges Perec, French novelist and member of the Oulipo group, designed a rotational ambigram, that he called "vertical palindrome".[15]Sibylline, the sentence "Andin Basnoda a une épouse qui pue" in French means "Andin Basnoda has a smelly wife". Perec did not care about punctuation spaces, but hiscreationflips easily with a classical font likeArial.
Visual palindromes sometimes perfectly illustrate literary contents. The American authorDan BrownincorporatedJohn Langdon's designs into the plot of his bestsellerAngels & Demons, and his fictional characterRobert Langdon's surname was a homage to theambigram artist.[139]
The fantasy novelAbarat, written and illustrated byClive Barker, features an ambigram of the title on its cover.[140]
Acalligramis text arranged in such a way that it forms a thematically related image. It can be a poem, a phrase, a portion ofscripture, or a single word. The visual arrangement can rely on certain use of thetypeface,calligraphyorhandwriting. The image created by the words illustrates the text by expressing visually what it says, or something closely associated.
InIslamic calligraphy, symmetrical calligrams appear in ancient and modern periods, forming mirror ambigrams inArabiclanguage.[30]
The word "OK" turned 90°counterclockwiseevokes a human icon, with the letter O forming the head and the letter K the arms and the legs. The Norwegian Climbing ClubOslo Klatreklubb[no](acronym"OK") borrowed the concept of this naturalcalligramfor their official logo.[141]
As described byDouglas Hofstadter, ambigrams arevisual punshaving two or more (clear)interpretationsaswritten words.[4]
Multilingualambigrams can be read one way in alanguage, and another way in a different language oralphabet.[42]Multi-lingual ambigrams can occur in all of the various types of ambigrams, with multi-lingual perceptual shift ambigrams being particularly striking.
Like certainanagramswith providential meanings such as "Listen / Silent" or "The eyes / They see", ambigrams also sometimes take on a timely sense, for example "up" becomes the abbreviation "dn", very naturally by rotation of 180°.[142]But on the other hand, it happens that the luck of the letters makes things bad. This is the case with the weird anagram "Santa/ Satan", as it is with a rotational ambigram that has goneviralbecause of theparadoxicaland unintentional message it expresses. Spotted in 2015 on a metal medal marketed without bad intention, the text "hope" displays upside down with a fairly obvious reading "Adolf". This coincidence photographed by an Internet user was relayed by several media and constitutes anambiguous image.[143][144]
Recreational mathematicsis carried out forentertainmentrather than as a strictly research and application-based professional activity.[62]An ambigrammagic squareexists, with the sums of the numbers in each row, each column, and both main diagonals the same right side up and upside down (180° rotational design). Numeral ambigrams also associate with alphabet letters. A "dissection" ambigram of "squaring the circle" was achieved in a puzzle where each piece of the word "circle" fits inside a perfect square.[4]
Burkard Polster, professor of mathematics inMelbourne[145]conducted researches on ambigrams and published several books dealing with the topic, includingEye Twisters, Ambigrams & Other Visual Puzzles to Amaze and Entertain.[146]In the abstractMathemagical Ambigrams, Polster performs several ambigrams closely related to his realm, like the words "algebra", "geometry", "math", "maths", or "mathematics".[4]
Calculator spellingis anunintended characteristicof theseven-segment displaytraditionally used bycalculators, in which, when read upside-down, the digits resemble letters of theLatin alphabet. Also,palindromic numbersandstrobogrammatic numberssometimes attract attention of mathematician ambigrammists.[55][54]
Ambigramtessellationsand3Dambigrams are two types particularly fun for the mathematician ingeometry. Wordpatternsin tessellations can start from 35 different fundamentalpolygons, such as therhombus, theisoscelesright triangle, or theparallelogram.[47]
Word puzzlesare used as a source ofentertainment, but can additionally serve aneducationalpurpose. The AmericanpuzzledesignerScott Kimpublished several ambigrams inScientific AmericaninMartin Gardner's
"Mathematical Games" column, among them long sentences like"Martin Gardner's celebration ofmind"turning into "Physics, patterns andprestidigitation".[147]
Legibilityis an important aspect in successful ambigrams. It concerns the ease with which a reader decodes symbols. If the message is lost or difficult to perceive, an ambigram does not work.[8]Readability is related toperception, or how our braininterpretsthe forms we see through our eyes.[148]
Symmetryin ambigrams generally improves the visual appearance of thecalligraphicwords.[38]Hermann Rorschach, inventor of theRorschach Testnotices that asymmetric figures are rejected by many subjects. Symmetry supplies part of the necessary artistic composition.[149]
For manyamateurs, designing ambigrams represents arecreational activity, whereserendipitycan play a fertile role, when the author makes an unplanned fortunate discovery.[4][34]
In the word "ambigram", the rootambi-means "both" and is a popular prefix in aworld of dualities, such as day/night, left/right, birth/death, good/evil.[150]InWordplay: The Philosophy, Art, and Science of Ambigrams,[151]John Langdonmentions theyin and yangsymbol as one of his major influences to create upside down words.
Ambigrams are mentioned inMetamagical Themas, an eclectic collection of articles thatDouglas Hofstadterwrote for thepopular sciencemagazineScientific Americanduring the early 1980s.[9]
Seeking the balance point ofanalogiesis anaestheticexercise closely related to the aesthetically pleasing activity of doing ambigrams, where shapes must be concocted that are poised exactly at the midpoint between twointerpretations. But seeking the balance point is far more than just aesthetic play; it probes the very core of how people perceiveabstractions, and it does so without their even knowing it. It is a crucial aspect ofCopycatresearch.[9]
Inmagic, ambigrams work likevisual illusions, revealing an unexpected new message from a particular written word.[153]
In the first series of the British showTrick or Treat, the show's host and creatorDerren Brownuses cards with rotational ambigrams.[154][155]These cards can read either 'Trick' or 'Treat'.
Ambiguous images, of which ambigrams are a part, cause ambiguity in different ways. For example, by rotational symmetry, as in the Illusion ofThe CookbyGiuseppe Arcimboldo(1570);[156]sometimes by afigure-groundambivalence as inRubin vase; by perceptual shift as in therabbit–duck illusion, or throughpareidolias; or again, by the representation ofimpossible objects, such asNecker cubeorPenrose triangle. For all these types of images, certain ambigrams exist, and can be combined withvisualsof the same type.
John Langdondesigned afigure-groundambigram "optical illusion" with the two words "optical" and "illusion", one forming the figure and the other the background. "Optical" is easier to see initially but "illusion" emerges with longer observation.[157]
Adidasmarketed a line ofsneakerscalled "Bounce", with an ambigramtypographyprinted inside the shoe.
Several clothing brands, such asHelly Hansen(HH),Under Armour(UA), orNew Man[fr], raise an ambigram logo as their visualidentity.[79]
Mirror ambigrams are also sometimes placed onT-shirts,towelsandhats, whilesocksare more adapted to rotational ambigrams. TheconceptualartistMia Florentine Weissmarketed T-shirts and other products with her mirror ambigramLove Hate[de].[158][118]Likewise, the city ofVenturain California sells sweatshirts, caps, jackets, and other fashion accessories printed with its rotational ambigram logo.[159]
TheCD coverof the thirteenth studio albumFuneralby American rapperLil Waynefeatures a 180° rotational ambigram reading "Funeral / Lil Wayne".[160]
Thespecial editionpaper sleeve (CD with DVD) of the solo albumChaos and Creation in the BackyardbyPaul McCartneyfeatures an ambigram of the singer's name.[161]
TheGrateful Deadhave used ambigrams several times, including on their albumsAoxomoxoa[162]andAmerican Beauty.[163]
Although the words spelled by most ambigrams are relatively short in length, oneDVDcover forThe Princess Bridemovie creates a rotational ambigram out of two words "Princess Bride", whether viewed right side up or upside down.[164]
The cover of the studio albumCreate/Destroy/Createby rock bandGoodnight, Sunriseis an ambigram composition constituted of two invariant words, "create" and "destroy", designed by Polish artist Daniel Dostal.[165]
The reversibleshot glasscontaining a changing message "Drink / Drunk", created by thetypographerMark Simonsonwas manufactured and sold in the market.[74]
The concept of reversible sign that some merchants use through their windows to indicate that the store is sometimes "open", sometimes "closed", was inaugurated at the beginning of the 2000s, by a rotational ambigram "Open / Closed" developed by David Holst.[43]
Different ambigramartists, sometimes calledambigrammists,[9][166]may create distinctive ambigrams from the same words, differing in bothstyleandform.
There are no universal guidelines for creating ambigrams, and differentwaysof approaching problems coexist.
A number of books suggestmethodsforcreation, includingWordPlay,[75]Eye Twisters,[146]andAmbigrams Revealed,[38]in English.
Computerizedmethods toautomaticallycreate ambigrams have been developed.[167][168]
John LangdonandScott Kimeach believed that they had invented ambigrams in the 1970s.[169]
Douglas Hofstadtercoined the term.[4]
To explain visually the numerous types of possible ambigrams, Hofstadter created many pieces with different constraints and symmetries.[170]Hofstadter has had several exhibitions of his artwork in various university galleries.[171][172]
According toScott Kim, Hofstadter once created a series of 50 ambigrams on the name of all the states in the US.[173]
In 1987 a book of 200 of his ambigrams, together with a long dialogue with his alter ego Egbert G. Gebstadter on ambigrams andcreativity, was published in Italy.[5][12]
John Langdonis aself-taughtartist,graphic designerandpainter, who started designing ambigrams in the late 1960s and early 70s.Letteringspecialist, Langdon is a professor oftypographyandcorporate identityatDrexel UniversityinPhiladelphia.[174]
John Langdon produced a mirror image logo "Starship" in 1972-1973,[175][176]that was sold to the rock bandJefferson Starship.
Langdon's ambigram bookWordplaywas published in 1992. It contains about 60 ambigrams. Each design is accompanied by a brief essay that explores the word's definition, its etymology, its relationship to philosophy and science, and its use in everyday life.[75]
Ambigrams became more popular as a result ofDan Brownincorporating John Langdon's designs into the plot of his bestseller,Angels & Demons, and the DVD release of theAngels & Demonsmovie contains a bonus chapter called "This is an Ambigram". Langdon also produced the ambigram that was used for some versions of the book's cover.[169]Brown used the nameRobert Langdonfor the hero in his novels as an homage to John Langdon.[139][177]
Blacksmith Records, the music management company andrecord label, possesses a rotational ambigram logo[178]designed by John Langdon.[179]
Scott Kimis one of the best-known masters of the art of ambigrams.[78]He is an Americanpuzzledesigner andartistwho published in 1981 a book calledInversionswith ambigrams of many types.[18][177]
Nikita Prokhorovis agraphic designer,lettering artistand ambigram designer. His bookAmbigrams Revealedshowcases ambigram designs of all types, from all around the world.[38][180]
Born in 1946,Alain Nicolasis a specialist of figurative and ambigramtessellations. In his book, he performed many tilings with various words like "infinity", "Einstein" or "inversion" legible in many orientations.[47]According toThe Guardian, Nicolas has been called "the world's finest artist ofEscher-styletilings".[181]
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Anagrammatic poetryispoetrywith the constrained form that either each line or each verse is ananagramof all other lines or verses in the poem.
A poet that specializes in anagrams is an anagrammarian.[1]
Writing anagrammatic poetry is a form of aconstrained writingsimilar to writingpangramsor longalliterations.
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Anagrams(also published under names includingAnagram,SnatchandWord Making and Taking) is atile-basedword gamethat involvesrearrangingletter tiles to form words.
The game pieces are a set of tiles with letters on one side. Tiles are shuffled face-down then turned over one by one, players forming words by combining them with existing words, their own or others'. The game has never been standardized and there are many varieties of sets and rules. Anagrams is often played with tiles from another word game, such asScrabbleorBananagrams.
Reputed to have originated as aVictorianword game, Anagrams has appeared in many versions since then.
An early modern version is Charles Hammett'sWord Making and Taking, released in 1877.[1]The first version to include the wordAnagramsin its name may have beenThe Game of Letters and Anagrams on Wooden Blocks, published byParker Brothersaround 1890. Another game calledAnagramswas published in 1934 bySelchow and Righter, which publishedScrabblein 1953.Spelling and Anagrams(a set incorporating two distinct games,SpellingandAnagrams) was also published in the 1930s.[2]In 1975, Selchow publishedScrabble Scoring Anagrams, which featured tiles with point values like those inScrabble. Another version was published in the 1960s by the now defunctTransogram. The Embossing Company, formerly Halsam Products Company, also produced a yellow-on-blackEye-Restset.Leslie Scott(the creator ofJenga) published a variation calledSwipein the early 1980s, and since 1990, Scott's company,Oxford GamesLtd, has publishedAnagram.TycopublishedUp For Grabsin 1995. Prodijeux has been marketing a variant,WordXchange, since 2000, and Portobello Games produced a version,Snatch-It, in 2001.One Up!is a version that adds a "wild" tile that can be any letter, like a blank tile inScrabble.
Some players use several sets of tiles from games such asScrabbleorUpwordsto play Anagrams, and a version of the game is popular among tournamentScrabbleplayers. WritersJohn Ciardi,James Merrill,John Malcolm Brinnin, andRichard Wilburreputedly played together regularly inKey West, Florida, sometimes also with novelistJohn Hersey.[3]
Different editions of the game use different rules, and players now often play by house rules, but most[citation needed]are variants of the rules given here, taken from Snatch-It.[4]
To begin, all tiles are placed face down in a pool in the middle of the table.Playersthen take turns flipping over tiles until somebody notices a word of three or more letters. A word can be formed by either:
When a player sees a word, they call it immediately (irrespective of who flipped the last tile) and place the word in front of them. The game then continues with further tiles being flipped.
All words must be at least three letters long. When a word is expanded with tiles from the pool, the added tiles may not simply be a suffix (like -S or -ING).
The game ends when all tiles are face up and no further words can be formed. Players then score according to the words they have in front of them: a 3-letter word is worth 1 point, a 4-letter word 2 points, and so on.[4]
A host of variations come from both different versions and players'house rules.
Other scoring systems include:
The minimum acceptable word length can be adjusted to a player's skill level (for example, in a game with adults and children playing together, the children may be permitted to form four-letter words while the adults are restricted to words of at least five or six letters). TournamentScrabbleplayers often play with a minimum length of six or seven.
In some editions of the game, such as the Milton Bradley[6]and Selchow & Righter versions, only the player whose turn it is may form words. In the Selchow & Righter edition, a word may be stolen by any playerimmediatelyafter it was made if they form a longer word with tiles from the pool.[5]
TheNational Scrabble Associationhas published a set of rules for competitive Anagrams play in tournament setting. On a player's turn, after revealing a tile, they have a ten-second window during which only they can call a word. If a player calls a word on their own turn they take an extra turn. After 100 turns, the order of play reverses. Minimum word length is six letters.[7]
One variation is to have each player have a "bank" of tiles in front of themselves, which affords players a clearer view of the "pool" of face-up letter tiles in the middle of the table.
A faster-paced version—sometimes known as "Alaskan rules"—has each of the players (or several, if there are too many) simultaneously put a tile into the pool. This results in many more possibilities being available at a time.
Players may not create a word by creating a word that is already on the table or steal one resulting in such a word.
Some versions of the game name the winner as the person who, after the round of turns has finished, first acquires eight words. If more than one player has done so, then the winner is the player is the one with the most tiles. There may be a tie. A very similar rule found in The Embossing Company set simply says the "first player to complete ten words, wins."
Players are permitted to combine two or more existing words with zero or more letters from the pool to create a single new word. This is often difficult in practice.
A game of Anagrams is played in theAlfred Hitchcock1941 thriller filmSuspicion.
A game of Anagrams is played inIra Levin's debut novel,A Kiss Before Dying.
Though there are many variants, one standard letter distribution of 188 letters (given in the Rust Hills article) is as follows:
A variant with 220 letters:
The distribution of 180 letters forScrabble Scoring Anagrams(according to a review on funagain.com):
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Ananadrome[1][2][3][4][a]is a word or phrase whose letters can be reversed to spell a different word or phrase. For example,dessertsis an anadrome ofstressed. An anadrome is therefore a special type ofanagram. The English language is replete with such words.
The wordanadromecomes from Greekanádromos(ἀνάδρομος), "running backward", and can be compared topalíndromos(παλίνδρομος), "running back again" (whencepalindrome).
There is a long history (dating at least to the fourteenth century, as withTreborandS. Uciredor) of alternate and invented names being created out of anadromes of real names; such a contrivedproper nounis sometimes called anananym, especially if it is used as personalpseudonym. Unlike typical anadromes, these anadromic formations often do not conform to any real names or words. Similarly cacographic anadromes are also characteristic of Victorianback slang, where for exampleyobstands forboy.
The English language has a very large number of single-word anadromes, by some counts more than 900.[3]Some examples:
An anadrome can also be a phrase, as inno tops↔spot on. The wordredrum(i.e., "red rum") is used this way formurderin theStephen KingnovelThe Shining(1977) andits film adaptation(1980).[11]
Anadromes exist in other written languages as well, as can be seen, for example, inSpanishorar↔raroorFrenchl'ami naturel("the natural friend") ↔le rut animal("the animal rut").
Many jazz titles were written by reversing names or nouns:Ecarohinverts the spelling of its composerHorace Silver's Christian name.Sonny Rollinsdedicated toNigeriaa tune called "Airegin".
A number ofPokémonspecies, such as the snake PokémonEkansandArbok(cobrabackwards with a K), have anadromic names.
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Ablanagram(portmanteauof blank andanagram) is a word which is ananagramof another but for the substitution of a single letter. The term has its origin in competitiveScrabble, where a blank tile on a player's rack may be used to form any of several possible words in conjunction with the player's other tiles.
Many seven- and eight-letter words, such as KILOVOLT andQUIXOTIC, have no acceptable blanagrams; such words typically contain a subset of the letters JKQVWXZ.
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Constrained writingis aliterary techniquein which the writer is bound by some condition that forbids certain things or imposes a pattern.[1]
Constraints are very common inpoetry, which often requires the writer to use a particular verse form.
Constraints on writing are common and can serve a variety of purposes. For example, a text may place restrictions on itsvocabulary, e.g.Basic English,copula-free text,defining vocabularyfor dictionaries, and other limited vocabularies for teachingEnglish as a second languageor to children.
In poetry, formal constraints abound in both mainstream and experimental work. Familiar elements of poetry likerhymeandmeterare often applied as constraints. Well-established verse forms like thesonnet,sestina,villanelle,limerick, andhaikuare variously constrained by meter, rhyme, repetition, length, and other characteristics.
Outside of established traditions, particularly in theavant-garde, writers have produced a variety of work under more severe constraints; this is often what the term "constrained writing" is specifically applied to. For example:
TheOulipogroup is a gathering of writers who use such techniques. TheOutrapogroup usestheatrical constraints.[3]
There are a number of constrained writing forms that are restricted by length, including:
Notable examples ofconstrained comics:
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Aheterogram(fromhetero-, meaning 'different', +-gram, meaning 'written') is a word, phrase, or sentence in which noletterof the alphabet occurs more than once. The termsisogramandnonpattern wordhave also been used to mean the same thing.[1][2][3]
It is not clear who coined or popularized the term "heterogram". The concept appears inDmitri Borgmann's 1965 bookLanguage on Vacation: An Olio of Orthographical Odditiesbut he uses the termisogram.[4]In a 1985 article, Borgmann claims to have "launched" the termisogramthen.[5]He also suggests an alternative term,asogram, to avoid confusion with lines of constant value such ascontour lines, but usesisogramin the article itself.
Isogramhas also been used to mean a string where each letter present is used the same number of times.[6][2][7]Multiple terms have been used to describe words where each letter used appears a certain number of times. For example, a word where every featured letter appears twice, like "noon", might be called apair isogram,[8]asecond-order isogram,[2]or a2-isogram.[3]
A perfectpangramis an example of a heterogram, with the added restriction that it uses all the letters of the alphabet.
A ten-letter heterogram can be used as the key to asubstitution cipherfor numbers, with the heterogram encoding the string 1234567890 or 0123456789. This is used in businesses where salespeople and customers traditionally haggle over sales prices, such as used-car lots and pawn shops. The nominal value or minimum sale price for an item can be listed on a tag for the salesperson's reference while being visible but meaningless to the customer.[9][10]
A twelve-letter cipher could be used to indicate months of the year.
In the bookLanguage on Vacation: An Olio of Orthographical Oddities,Dmitri Borgmanntries to find the longest such word. The longest one he found was "dermatoglyphics" at 15 letters. He coins several longer hypothetical words, such as "thumbscrew-japingly" (18 letters, defined as "as if mocking athumbscrew") and, with the "uttermost limit in the way of verbal creativeness", "pubvexingfjord-schmaltzy" (23 letters, defined as "as if in the manner of the extremesentimentalismgenerated in some individuals by the sight of a majesticfjord, which sentimentalism is annoying to the clientele of an English inn").[4]
The word "subdermatoglyphic" was constructed by Edward R. Wolpow.[11]Later, in the bookMaking the Alphabet Dance,[12]Ross Ecklerreports the word "subdermatoglyphic" (17 letters) can be found in an article by Lowell Goldsmith calledChaos: To See a World in a Grain of Sand and a Heaven in a Wild Flower.[13]He also found the name "Melvin Schwarzkopf" (17 letters), a man living inAlton, Illinois, and proposed the name "Emily Jung Schwartzkopf" (21 letters). In an elaborate story, Eckler talked about a group of scientists who name the unavoidable urge to speak inpangramsthe "Hjelmqvist-Gryb-Zock-Pfund-Wax syndrome".
The longest German heterogram is "Heizölrückstoßabdämpfung" (heating oil recoil dampening) which uses 24 of the 30 letters in theGerman alphabet, asä,ö,ü, andßare considered distinct letters froma,o,u, andsin German.[citation needed]It is closely followed by "Boxkampfjuryschützlinge" (boxing-match jury protégés) and "Zwölftonmusikbücherjagd" (twelve-tone music book chase) with 23 letters.[citation needed]
There are hundreds of eleven-letter isograms, over one thousand ten-letter isograms and thousands of such nine-letter words.[14]
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Aletter bankis a relative of theanagramwhere all the letters of one word (the "bank") can be used as many times as desired (minimum of once each) to make a new word or phrase. For example, IMPS is a bank of MISSISSIPPI and SPROUT is a bank ofSUPPORT OUR TROOPS. As a convention, the bank should have no repeat letters within itself.
The term was coined byWill Shortz, whose first letter bank (BLUME -> BUMBLEBEE) appeared in his 1979 book, "Brain Games". In 1980, Shortz introduced letter banks to theNational Puzzlers' League(of which he is the historian), in the form of a contest puzzle. In 1981, the letter bank was announced an official puzzle type in the NPL’s magazine "The Enigma".[1]
Letter banks are the basis for the word gameAlpha Blitz.[citation needed]
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These aregeographic anagrams and anadromes.Anagramsare rearrangements of the letters of another name or word.Anadromes(also called reversals or ananyms) are other names or words spelled backwards. Technically, a reversal is also an anagram, but the two are derived by different methods, so they are listed separately.
Place names created by anagramming fall into three distinct groups:
[19]
A few places names were constructed by arranging a preselected set of letters in an order that made a pronounceable name.
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In the biologicalnomenclature codes, ananagramcan be used to name a newtaxon.
Wordplaysare one source of inspiration allowing organisms to receivescientific names.[1]In thebinomial nomenclature, as scientists have latitude in naminggeneraandspecies, a taxon name can therefore be an anagram, provided it remains pronounceable.[2]For example, in theInternational Code of Nomenclature for algae, fungi, and plants, a new generic name can be taken from the name of a person by using an anagram or abbreviation of it.[3]
William Elford Leachwas among the firstnaturaliststo use taxonomic anagrams, and, in 1818, he described severalisopodgenera that were
each other's anagrams of 'Caroline' :Conilera,Lironeca,Nerocila,Olencira, andRocinela.[1]
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London Underground anagram mapis aparodymap of theLondon Undergroundwith the station and line names replaced withanagrams. The anagram map was circulated on the web in February 2006.
The map was featured on thousands ofblogsbefore aTransport for Londonlawyer requested that the map be removed. It inspired some people to create anagram versions of their hometown'smetrosystem with similar legal repercussions. The fact that it was appreciated internationally, despite some not knowing the stations behind the anagrams, is a recognition ofHarry Beck'siconicTube mapdesign.[citation needed]
The map was created by 'Barry Heck' using aphotoshoppedTube mapand an online anagram generator, on 7 February 2006. It was originally shown in a thread on the Thingbox chat forum and, after being submitted by one of the site owners, appeared onBoingBoinga couple of days later receiving 31,000 hits within the next six days. (The name Barry Heck is apseudonymchosen because it is an anagram andspoonerismofHarry Beck.)
The idea came fromThe Great Bear, a 1992 artwork by UK artistSimon Pattersonon display atTate Modernin London, but it was not until Dorian Lynskey's music genre tube map appeared in a newspaper in 2006 that Barry Heck decided to make it.
In 2000, Scottish artist Mark Campbell created the Glasgow Anagram Tour[1]which involved a large light box anagram map of Glasgow and Pop-out Maps with anagrams of Glasgow's cultural establishments which were for sale to commemorate Scotland's Year of the Artist 2000.
Transport for London claimed the image was a copyright infringement and had one of their lawyers ask for the map to be removed from the web. The site hosting it complied and it was removed on 22 February 2006 with the action being reported on BoingBoing again.
Transport for London also censored other websites that hosted the image such as the www.geofftech.co.uk site. But an online backlash against TFL's lawyers meant that many other websites made mirrors of Geoff's page, thus resulting in more copies of tube map "mash-ups" on the internet.
The owner of the site –Geoff Marshall, was interviewed onBBC Radio 5 Liveby Chris Vallance about "map-mashing" (making parody maps) in which the London Underground anagram map was discussed. This was broadcast on 14 March 2006.
BoingBoing has reported thatWashington,Toronto,Amsterdam,Chicago,Oslo,Boston,New York City,Atlanta,Sydney, andViennahad anagram maps created for their metro systems, inspired by the London map.
The anagram map was featured in thousands of blogs and its progress can be tracked atTechnorati.com. Because of similarities withNeverwhereit was mentioned in the letters page of authorNeil Gaiman's blog, with his fanbase ensuring over 1,700 others linked to it. But nearly 21,000 other blogs linked to BoingBoing's article alone.
BlackwallandHornchurchstations could not be properly anagrammatized and instead were split into their component words and reversed to produce "Wall Black" and "Church Horn" respectively. Burch Chow/Chow Burch (from the gynaecological Burch procedure) was rejected as an anagram forBow Church, because of a dislike for uncommon proper nouns, leaving it reversed as "Church Bow".Bankwas anagrammatized into 'nabk', the edible berry of theZiziphus lotustree.
Not all derivatives for other cities followed this pattern. For Toronto, the impossible stations were named after streets, so the namesake's designation as "Avenue" or "Street" was appended before anagramming (Queen became Queen Street became Queerest Ten).[2]
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Apalindrome(/ˈpæl.ɪn.droʊm/) is a word,number, phrase, or other sequence of symbols that reads the same backwards as forwards, such asmadamorracecar, the date "22/02/2022" and the sentence: "A man, a plan, a canal –Panama". The 19-letterFinnishwordsaippuakivikauppias(asoapstonevendor) is the longest single-word palindrome in everyday use, while the 12-letter termtattarrattat(fromJames JoyceinUlysses) is the longest in English.
The wordpalindromewas introduced by English poet and writerHenry Peachamin 1638.[1]The concept of a palindrome can be dated to the 3rd-century BCE, although no examples survive. The earliest known examples are the 1st-century CE Latinacrosticword square, theSator Square(which contains both word and sentence palindromes), and the 4th-century Greek Byzantine sentence palindromenipson anomemata me monan opsin.[2][3]
Palindromes are also found in music (thetable canonandcrab canon) and biological structures (mostgenomesincludepalindromic gene sequences). Inautomata theory, the set of all palindromes over analphabetis acontext-freelanguage, but it is notregular.
The wordpalindromewas introduced by English poet and writerHenry Peachamin 1638.[1]It is derived from the Greek rootsπάλιν'again' andδρóμος'way, direction'; a different word is used in Greek, καρκινικός 'carcinic' (lit.'crab-like') to refer to letter-by-letter reversible writing.[2][3]
The ancient Greek poetSotades(3rd-century BC) invented a form ofIonic metercalled Sotadic orSotadeanverse, which is sometimes said to have been palindromic,[4]since it is sometimes possible to make a sotadean line by reversing a dactylic hexameter.[5][6][7]
A 1st-century Latin palindrome was found as a graffito atPompeii. This palindrome, known as theSator Square, consists of a sentence written in Latin:sator arepo tenet opera rotas'The sower Arepo holds with effort the wheels'. It is also anacrosticwhere the first letters of each word form the first word, the second letters form the second word, and so forth. Hence, it can be arranged into aword squarethat reads in four different ways: horizontally or vertically from either top left to bottom right or bottom right to top left. Other palindromes found at Pompeii include "Roma-Olim-Milo-Amor", which is also written as an acrostic square.[8][9]Indeed, composing palindromes was "a pastime of Roman landed gentry".[10]
Byzantinebaptismal fontswere often inscribed with the 4th-century Greek palindrome,ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ(orΑΝΟΜΗΜΑ)ΜΗ ΜΟΝΑΝ ΟΨΙΝ("Nipson anomēmata mē monan opsin") 'Wash [your] sin(s), not only [your] face', attributed toGregory of Nazianzus;[11]most notably in the basilica ofHagia SophiainConstantinople. The inscription is found on fonts in many churches in Western Europe:Orléans(St. Menin's Abbey);Dulwich College;Nottingham(St. Mary's);Worlingworth;Harlow;Knapton;London(St Martin, Ludgate); andHadleigh (Suffolk).[12]
A 12th-century palindrome with the same square property is theHebrewpalindrome,פרשנו רעבתן שבדבש נתבער ונשרףperashnu: ra`avtan shebad'vash nitba`er venisraf'We explained the glutton who is in the honey was burned and incinerated', credited in 1924 to the medieval Jewish philosopherAbraham ibn Ezra,[13][unreliable fringe source?]and referring to thehalachicquestion as to whether a fly landing in honey makes the honeytreif(non-kosher).
The palindromic Latin riddle "In girum imus nocte et consumimur igni" 'we go in a circle at night and are consumed by fire' describes the behavior of moths. It is likely that this palindrome is from medieval rather than ancient times. The second word, borrowed from Greek, should properly be spelledgyrum.
In English, there are many palindromewordssuch aseye,madam, anddeified, but English writers generally cited Latin and Greek palindromic sentences in the early 19th century;[14]thoughJohn Taylorhad coined one in 1614: "Lewd did I live, & evil I did dwel" (with theampersandbeing something of a "fudge"[15]). This is generally considered the first English-language palindrome sentence and was long reputed, notably by the grammarianJames "Hermes" Harris, to be theonlyone, despite many efforts to find others.[16][17](Taylor had also composed two other, "rather indifferent", palindromic lines of poetry: "Deer Madam, Reed", "Deem if I meed".[4]) Then in 1848, a certain "J.T.R." coined "Able was I ere I saw Elba", which became famous after it was (implausibly) attributed toNapoleon(alluding to his exile on Elba).[18][17][19]Otherwell-known English palindromesare: "A man, a plan, a canal – Panama" (1948),[20]"Madam, I'm Adam" (1861),[21]and "Never odd or even" (1930).[22]
The most familiar palindromes in English are character-unit palindromes, where the characters read the same backward as forward. Examples arecivic,radar,level,rotor,kayak,madam, andrefer. The longest common ones arerotator, deified, racecar, andreviver; longer examples such asredivider,kinnikinnik, andtattarrattatare orders of magnitude rarer.[23]
There are also word-unit palindromes in which the unit of reversal is the word ("Is it crazy how saying sentences backwards creates backwards sentences saying how crazy it is?"). Word-unit palindromes were made popular in therecreational linguisticscommunity byJ. A. Lindonin the 1960s. Occasional examples in English were created in the 19th century. Several in French and Latin date to theMiddle Ages.[24]
There are also line-unit palindromes, most oftenpoems. These possess an initial set of lines which, precisely halfway through, is repeated in reverse order, without alteration to word order within each line, and in a way that the second half continues the "story" related in the first half in a way that makes sense, this last being key.[25]
Palindromes often consist of a sentence or phrase, e.g., "A man, a plan, a canal, Panama", "Mr. Owl ate my metal worm",
"Do geese see God?", or "Was it a car or a cat I saw?". Punctuation, capitalization, and spaces are usually ignored. Some, such as "Rats live on no evil star", "Live on time, emit no evil", and "Step on no pets", include the spaces.
Some names are palindromes, such as thegiven namesHannah, Ava, Aviva, Anna, Eve, Bob, and Otto, or thesurnamesHarrah, Renner, Salas, and Nenonen.Lon Nol(1913–1985) was Prime Minister of Cambodia.Nisio Isinis a Japanese novelist andmangawriter, whose pseudonym (西尾 維新,Nishio Ishin) is a palindrome when romanized using theKunrei-shikior theNihon-shikisystems, and is often written as NisiOisiN to emphasize this. Some people have changed their name in order to make it palindromic (including as the actorRobert Treborand rock-vocalistOla Salo), while others were given a palindromic name at birth (such as the philologistRevilo P. Oliver, the flamenco dancerSara Baras, the runnerAnuța Cătună, the creator of theEden ProjectTim Smit, and the Mexican racing driverNoel León).
Savas (Emil M. Savas) Painter fromDenmarkchristened with the palindrome Savas. The family name is ofKurdishorigin and derived from Savaş written with Ş /ʃ/ (Hawar alphabet,Bedirxan 1932). The spelling was changed in accordance with theDano-Norwegian alphabetwhen Savas’ father was granted danish citizenship.
There are also palindromic names in fictional media. "Stanley Yelnats" is the name of the main character inHoles, a 1998 novel and2003 film. Five of the fictionalPokémonspecieshave palindromic names in English (Eevee, Girafarig, Farigiraf, Ho-Oh, and Alomomola), as does the region Alola.
The 1970s pop bandABBAis a palindrome using the starting letter of the first name of each of the four band members.
The digits of a palindromic number are the same read backwards as forwards, for example, 91019;decimalrepresentation is usually assumed. Inrecreational mathematics, palindromic numbers with special properties are sought. For example, 191 and 313 arepalindromic primes.
WhetherLychrel numbersexist is an unsolved problem in mathematics about whether all numbers become palindromes when they are continuously reversed and added. For example, 56 is not a Lychrel number as 56 + 65 = 121, and 121 is a palindrome. The number 59 becomes a palindrome after three iterations: 59 + 95 = 154; 154 + 451 = 605; 605 + 506 = 1111, so 59 is not a Lychrel number either. Numbers such as 196 are thought to never become palindromes when this reversal process is carried out and are therefore suspected of being Lychrel numbers. If a number is not a Lychrel number, it is called a "delayed palindrome" (56 has a delay of 1 and 59 has a delay of 3). In January 2017 the number 1,999,291,987,030,606,810 was published in OEIS asA281509, and described as "The Largest Known Most Delayed Palindrome", with a delay of 261. Several smaller 261-delay palindromes were published separately asA281508.
Every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater.[26]
A day or timestamp is a palindrome when its digits are the same when reversed. Only the digits are considered in this determination and the component separators (hyphens, slashes, and dots) are ignored. Short digits may be used as in11/11/1111:11or long digits as in2 February 2020.
A notable palindrome day is this century's 2 February 2020 because this date is a palindrome regardless of thedate format by country(yyyy-mm-dd, dd-mm-yyyy, or mm-dd-yyyy) used in various countries. For this reason, this date has also been termed as a "Universal Palindrome Day".[27][28]Other universal palindrome days include, almost a millennium previously,11/11/1111, the future12/12/2121, and in a millennium03/03/3030.[29]
A phonetic palindrome is a portion ofspeechthat is identical or roughly identical when reversed. It can arise in context where language is played with, for example in slang dialects likeverlan.[30]In theFrench language, there is the phraseune Slave valse nue("a Slavic woman waltzes naked"), phonemically/ynslavvalsny/.[31]John Oswalddiscussed his experience of phonetic palindromes while working on audio tape versions of thecut-up techniqueusing recorded readings byWilliam S. Burroughs.[32][33]A list of phonetic palindromes discussed byword puzzlecolumnist O.V. Michaelsen (Ove Ofteness) include "crew work"/"work crew", "dry yard", "easy", "Funny enough", "Let Bob tell", "new moon", "selfless", "Sorry, Ross", "Talk, Scott", "to boot", "top spot" (also an orthographic palindrome), "Y'all lie", "You're caught. Talk, Roy", and "You're damn mad, Roy".[34]
The longest single-word palindrome in theOxford English Dictionaryis the 12-letteronomatopoeicwordtattarrattat, coined byJames JoyceinUlysses(1922) for a knock on the door.[35][36][37]TheGuinness Book of Recordsgives the title to the 11-letterdetartrated, thepreteriteand past participle ofdetartrate, a chemical term meaning to removetartrates. The 9-letter wordRotavator, a trademarked name for an agricultural machine, is listed in dictionaries as being the longest single-word palindrome. The 9-letter termredivideris used by some writers, but appears to be an invented or derived term; onlyredivideandredivisionappear in the Shorter Oxford English Dictionary; the 9-letter wordMalayalam, a language of southern India, is also of equal length.
According toGuinness World Records, theFinnish19-letter wordsaippuakivikauppias(asoapstonevendor), is the world's longest palindromic word in everyday use.[12]
English palindrome sentences of notable length include mathematicianPeter Hilton's "Doc, note: I dissent. A fast never prevents a fatness. I diet on cod",[38]and Scottish poetAlastair Reid's "T. Eliot, top bard, notes putrid tang emanating, is sad; I'd assign it a name: gnat dirt upset on drab pot toilet."[39]
In English, two palindromic novels have been published:Satire: Veritasby David Stephens (1980, 58,795 letters), andDr Awkward & Olson in Osloby Lawrence Levine (1986, 31,954 words).[40]Another palindromic English work is a 224-word long poem, "Dammit I'm Mad", written byDemetri Martin.[41]"Weird Al" Yankovic's song "Bob" is composed entirely of palindromes.[42]
Joseph Haydn'sSymphony No. 47in G is nicknamed "the Palindrome". In the third movement, aminuetandtrio, the second half of the minuet is the same as the first but backwards, the second half of the ensuing trio similarly reflects the first half, and then the minuet is repeated.
The interlude fromAlban Berg's operaLuluis a palindrome,[43]as are sections and pieces, inarch form, by many other composers, includingJames Tenney, and most famouslyBéla Bartók.George Crumbalso used musical palindrome to text paint theFederico García Lorcapoem "¿Por qué nací?", the first movement of three in his fourth book ofMadrigals.Igor Stravinsky's final composition,The Owl and the Pussy Cat, is a palindrome.[44][unreliable source?]
The first movement fromConstant Lambert'sballetHoroscope(1938) is entitled "Palindromic Prelude". Lambert claimed that the theme was dictated to him by the ghost ofBernard van Dieren, who had died in 1936.[45]
British composerRobert Simpsonalso composed music in the palindrome or based on palindromic themes; the slow movement of hisSymphony No. 2is a palindrome, as is the slow movement of hisString Quartet No. 1. His hour-longString Quartet No. 9consists of thirty-two variations and a fugue on a palindromic theme of Haydn (from the minuet of his Symphony No. 47). All of Simpson's thirty-two variations are themselves palindromic.
Hin und Zurück("There and Back": 1927) is an operatic 'sketch' (Op. 45a) in one scene by Paul Hindemith, with a German libretto by Marcellus Schiffer. It is essentially a dramatic palindrome. Through the first half, a tragedy unfolds between two lovers, involving jealousy, murder and suicide. Then, in the reversing second half, this is replayed with the lines sung in reverse order to produce a happy ending.
The music ofAnton Webernis often palindromic. Webern, who had studied the music of the Renaissance composerHeinrich Isaac, was extremely interested in symmetries in music, be they horizontal or vertical. An example of horizontal or linear symmetry in Webern's music is the first phrase in the second movement of thesymphony, Op. 21. A striking example of vertical symmetry is the second movement of thePiano Variations, Op. 27, in which Webern arranges every pitch of thisdodecaphonicwork around the central pitch axis of A4. From this, each downward reaching interval is replicated exactly in the opposite direction. For example, a G♯3—13 half-steps down from A4 is replicated as a B♭5—13 half-steps above.
Just as the letters of a verbal palindrome are not reversed, so are the elements of a musical palindrome usually presented in the same form in both halves. Although these elements are usually single notes, palindromes may be made using more complex elements. For example,Karlheinz Stockhausen's compositionMixtur, originally written in 1964, consists of twenty sections, called "moments", which may bepermutedin several different ways, including retrograde presentation, and two versions may be made in a single program. When the composer revised the work in 2003, he prescribed such a palindromic performance, with the twenty moments first played in a "forwards" version, and then "backwards". Each moment is a complex musical unit and is played in the same direction in each half of the program.[46]By contrast,Karel Goeyvaerts's 1953 electronic composition,Nummer 5(met zuivere tonen)is anexactpalindrome: not only does each event in the second half of the piece occur according to an axis of symmetry at the centre of the work, but each event itself is reversed, so that the note attacks in the first half become note decays in the second, and vice versa. It is a perfect example of Goeyvaerts's aesthetics, the perfect example of the imperfection of perfection.[47]
Inclassical music, acrab canonis acanonin which one line of the melody is reversed in time and pitch from the other.
A large-scale musical palindrome covering more than one movement is called "chiastic", referring to the cross-shaped Greek letter "χ" (pronounced /ˈkaɪ/.) This is usually a form of reference to the crucifixion; for example, theCrucifixusmovement of Bach'sMass in B minor. The purpose of such palindromic balancing is to focus the listener on the central movement, much as one would focus on the centre of the cross in the crucifixion. Other examples are found in Bach's cantata BWV 4,Christ lag in Todes Banden, Handel'sMessiahand Fauré'sRequiem.[48]
Atable canonis a rectangular piece of sheet music intended to be played by two musicians facing each other across a table with the music between them, with one musician viewing the music upside down compared to the other. The result is somewhat like two speakers simultaneously reading theSator Squarefrom opposite sides, except that it is typically in two-part polyphony rather than in unison.[49]
Palindromic motifs are found in mostgenomesor sets ofgeneticinstructions. The meaning of palindrome in the context of genetics is slightly different, from the definition used for words and sentences. Since theDNAis formed by two paired strands ofnucleotides, and the nucleotides always pair in the same way (Adenine(A) withThymine(T),Cytosine(C) withGuanine(G)), a (single-stranded) sequence of DNA is said to be a palindrome if it is equal to its complementary sequence read backward. For example, the sequenceACCTAGGTis palindromic because its complement isTGGATCCA, which is equal to the original sequence in reverse complement.
A palindromicDNAsequence may form ahairpin. Palindromic motifs are made by the order of thenucleotidesthat specify the complex chemicals (proteins) that, as a result of thosegeneticinstructions, thecellis to produce. They have been specially researched inbacterialchromosomes and in the so-called Bacterial Interspersed Mosaic Elements (BIMEs) scattered over them. In 2003, a research genome sequencing project discovered that many of the bases on theY-chromosomeare arranged as palindromes.[50]A palindrome structure allows the Y-chromosome to repair itself by bending over at the middle if one side is damaged.
It is believed that palindromes are also found in proteins,[51][52]but their role in the protein function is not clearly known. It has been suggested in 2008[53]that the prevalent existence of palindromes in peptides might be related to the prevalence of low-complexity regions in proteins, as palindromes frequently are associated with low-complexity sequences. Their prevalence might also be related to analpha helicalformation propensity of these sequences,[53]or in formation of protein/protein complexes.[54]
Inautomata theory, asetof all palindromes in a givenalphabetis a typical example of alanguagethat iscontext-free, but notregular. This means that it is impossible for afinite automatonto reliably test for palindromes.
In addition, the set of palindromes may not be reliably tested by adeterministic pushdown automatonwhich also means that they are notLR(k)-parsableorLL(k)-parsable. When reading a palindrome from left to right, it is, in essence, impossible to locate the "middle" until the entire word has been read completely.
It is possible to find thelongest palindromic substringof a given input string inlinear time.[55][56]
Thepalindromic densityof an infinite wordwover an alphabetAis defined to be zero if only finitely many prefixes are palindromes; otherwise, letting the palindromic prefixes be of lengthsnkfork=1,2,... we define the density to be
Among aperiodic words, the largest possible palindromic density is achieved by theFibonacci word, which has density 1/φ, where φ is theGolden ratio.[57]
Apalstaris aconcatenationof palindromic strings, excluding the trivial one-letter palindromes – otherwise all strings would be palstars.[55]
February 2, 2020, was the most recent palindromic date which was can perfectly fit to any date formats in 8-digit FIGURES. And it happens very rare in any Millennium. The next of it will occur on December 12, 2121, which will be the last in this 3rd millennium.
3rd Millennium: February 2, 2020, and December 12, 2121.
4th Millennium: March 3, 3030
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Apangramorholoalphabetic sentenceis asentenceusing every letter of a givenalphabetat least once. Pangrams have been used to displaytypefaces, test equipment, and develop skills inhandwriting,calligraphy, andtyping.
The best-known English pangram is "The quick brown fox jumps over the lazy dog".[1]It has been used since at least the late 19th century[1]and was used byWestern Unionto testTelex/TWXdata communication equipment for accuracy and reliability.[2]Pangrams like this are now used by a number of computer programs to display computer typefaces.
Short pangrams in English are more difficult to devise and tend to use uncommon words and unnatural sentences. Longer pangrams afford more opportunity for humor, cleverness, or thoughtfulness.
The following are examples of pangrams that are shorter than "The quick brown fox jumps over a lazy dog" (which has 35 letters) and use standard written English without abbreviations or proper nouns:
A perfect pangram contains every letter of the alphabet only once and can be considered ananagramof the alphabet. The only known perfect pangrams of the English alphabet use abbreviations or other non-dictionary words, such as "Blowzy night-frumps vex'd Jack Q." or "Mr. Jock, TV quiz PhD, bags few lynx."[3]or they include words so obscure that the phrase is challenging to understand, such as "Cwm fjord-bank glyphs vext quiz",[3]in whichcwmis aloan wordfrom theWelsh languagemeaning an amphitheatre-like glaciated depression,vextis an uncommon way to spellvexed, andquizis used in anarchaicsense to mean a puzzling or eccentric person. It means that symbols in the bowl-like depression on the edge of a long steep sea inlet confused an eccentric person.
Other writing systems may present more options: TheIrohais a well-known perfect pangram of the Japanesesyllabary, while theHanacarakais a perfect pangram for theJavanesescript and is commonly used to order its letters in sequence.
Whereas the English language uses all 26 letters of the Latin alphabet in native and naturalized words, many other languages using the same alphabet do not. Pangram writers in these languages are forced to choose between only using those letters found in native words or incorporating exotic loanwords into their pangrams. Some words, such as theGaelic-derivedwhisk(e)y, which has been borrowed by many languages and uses the lettersk,wandy, are a frequent fixture of many foreign pangrams.
There are also languages that use other Latin characters thatdo not appearin the traditional 26 letters of the Latin alphabet. This differs further from English pangrams, with letters such asç,ä, andš.
Non-Latin alphabetic or phonetic scripts such as Greek, Armenian, and others can also have pangrams.[12]In some writing systems, exactly what counts as a distinct symbol can be debated. For example, many languages have accents or other diacritics, but one might count "é" and "e" as the same for pangrams. A similar problem arises for older English orthography that includes thelong s("ſ").
("Mr.Sangkhaphant Hengpithakfang - an elderly man who earns a living by selling bottles - was arrested for prosecution by police because he stole Lady Chatchada Chansamat's watch.") contains all the letters in theThai alphabet, both obsolete and non-obsolete.
Logographic scripts, or writing systems such as Chinese that do not use an alphabet but are composed principally oflogograms, cannot produce pangrams in a literal sense (or at least, not pangrams of reasonable size). The total number of signs is large and imprecisely defined, so producing a text with every possible sign is practically impossible. However, various analogies to pangrams are feasible, including traditional pangrams in aromanization.
InJapanese, although typical orthography useskanji(logograms), pangrams can be made using everykana, orsyllabiccharacter. TheIrohais a classic example of a perfect pangram in non-Latin script.
In Chinese, theThousand Character Classicis a 1000-character poem in which each character is used exactly once; however, it does not include allChinese characters. The single character永(permanence) incorporates all the basic strokes used to write Chinese characters, using each stroke exactly once, as described in theEight Principles of Yong.
Amongabugidascripts, an example of a perfect pangram is theHanacaraka (hana caraka; data sawala; padha jayanya; maga bathanga)of theJavanese script, which is used to write theJavanese languageinIndonesia.
A self-enumerating pangram is a pangrammaticautogram, or a sentence that inventories its own letters, each of which occurs at least once. The first example was produced byRudy Kousbroek, a Dutch journalist and essayist, who publicly challengedLee Sallows, a Britishrecreational mathematicianresident in the Netherlands, to produce an English translation of his Dutch pangram. In the sequel, Sallows built an electronic "pangram machine", that performed a systematic search among millions of candidate solutions. The machine was successful in identifying the following 'magic' translation:[13][14][15]
Chris Patuzzo was able to reduce the problem of finding a self-enumerating pangram to theboolean satisfiability problem. He did this by using a made-to-orderhardware description languageas a stepping stone and then applied theTseytin transformationto the resulting chip.[16][17]
The pangram "The quick brown fox jumps over the lazy dog", and the search for a shorter pangram, are the cornerstone of the plot of the novelElla Minnow PeabyMark Dunn.[18]The search successfully comes to an end when the phrase "Pack my box with five dozen liquor jugs" is discovered (which has only 6 duplicated vowels).
The scientific paperCneoridium dumosum(Nuttall) Hooker F. Collected March 26, 1960, at an Elevation of about 1450 Meters on Cerro Quemazón, 15 Miles South of Bahía de Los Angeles, Baja California, México, Apparently for a Southeastward Range Extension of Some 140 Mileshas a pangrammatic title, seemingly by pure chance.
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Arebus(/ˈriːbəs/REE-bəss) is apuzzledevice that combines the use of illustrated pictures with individual letters to depict words or phrases. For example: the word "been" might be depicted by a rebus showing an illustrated bumblebee next to a plus sign (+) and the letter "n".
It was a favourite form ofheraldicexpression used in theMiddle Agesto denote surnames. For example, in its basic form, threesalmon(fish) are used to denote the surname "Salmon". A more sophisticated example was the rebus of BishopWalter Lyhart(d. 1472) of Norwich, consisting of astag(orhart) lying down in a conventional representation of water. The composition alludes to the name, profession or personal characteristics of the bearer, and speaks to the beholderNon verbis, sed rebus, whichLatinexpression signifies "not by words but by things"[1](res, rei(f), a thing, object, matter;rebusbeingablativeplural).[2]
Rebuses are used extensively as a form of heraldic expression as a hint to the name of the bearer; they are not synonymous withcanting arms. A man might have a rebus as a personal identification device entirely separate from his armorials, canting or otherwise. For example,Sir Richard Weston(d. 1541) bore as arms:Ermine, on a chief azure fivebezants, whilst his rebus, displayed many times in terracotta plaques on the walls of his mansionSutton Place, Surrey, was a "tun" or barrel, used to designate the last syllable of his surname.
An example of canting arms proper are those of theBorough of Congletonin Cheshire consisting of acongereel, a lion (in Latin,leo) and a tun (barrel). This word sequence "conger-leo-tun" enunciates the town's name. Similarly, the coat of arms ofSt. Ignatius Loyolacontains wolves (in Spanish,lobo) and a kettle (olla), said by some (probably incorrectly) to be a rebus for "Loyola". The arms ofElizabeth Bowes-Lyonfeaturebowsand lions.
A modern example of the rebus used as a form ofword playis:
By extension, it also uses the positioning of words or parts of words in relation to each other to convey a hidden meaning, for example:
A rebus made up solely of letters (such as "CU" for "See you") is known as agramogram, grammagram, or letteral word. This concept is sometimes extended to include numbers (as in "Q8" for "Kuwait", or "8" for "ate").[3]Rebuses are sometimes used incrosswordpuzzles, with multiple letters or a symbol fitting into a single square.[4]
The termrebusalso refers to the use of apictogramto represent a syllabic sound. This adapts pictograms intophonograms. A precursor to the development of the alphabet, this process represents one of the most important developments of writing. Fully developed hieroglyphs read in rebus fashion were in use atAbydosin Egypt as early as 3400 BCE.[5]In Mesopotamia, the principle was first employed onProto-Cuneiformtablets, beginning in theJemdet Nasr period(c. 3100–2900 BC).[6][7]
The writing of correspondence in rebus form became popular in the eighteenth century and continued into the nineteenth century.Lewis Carrollwrote the children he befriended picture-puzzle rebus letters, nonsense letters, andlooking-glassletters, which had to be held in front of a mirror to be read.[8]Rebus letters served either as a sort ofcodeor simply as apastime.
Inlinguistics, therebus principleis the use of existing symbols, such as pictograms, purely for their sounds regardless of their meaning, to represent new words. Many ancient writing systems used what we now term 'the rebus principle' to represent abstract words, which otherwise would be hard to represent with pictograms. An example that illustrates the Rebus principle is the representation of the sentence "I can see you" by using the pictographs of "eye—can—sea—ewe".
Some linguists believe that the Chinese developed their writing system according to the rebus principle,[9]and Egyptian hieroglyphs sometimes used a similar system. A famous rebus statue ofRamses IIuses three hieroglyphs to compose his name:Horus(asRa), forRa; the child,mes; and the sedge plant (stalk held in left hand),su; the name Ra-mes-su is then formed.[10]
Canada
United Kingdom
United States
India
The word "rebus" has also come to denote unconventional crossword answers requiring numerals, multiple letters in a single square, or other variations from the customary one-letter-one-square format.[11]The answers do not necessarily involve true rebuses in the traditional sense.
In Japan, the rebus known ashanjimono(判じ物)[18]was immensely popular during theEdo period.[19]A piece byukiyo-eartistKunisadawas "Actor Puzzles" (Yakusha hanjimono) that featured rebuses.[20]
Today the most often seen of these symbols is a picture of a sickle, a circle, and the letternu(ぬ), read askama-wa-nu(鎌輪ぬ, sickle circlenu), interpreted askamawanu(構わぬ), the old-fashioned form ofkamawanai(構わない, don't worry, doesn't matter). This is known as thekamawanu-mon(鎌輪奴文, kamawanu sign), and dates to circa 1700,[21]being used in kabuki since circa 1815.[22][23]
Kabukiactors would wearyukataand other clothing whose pictorial design, in rebus, represented theirYagō"guild names", and would distributetenuguicloth with their rebused names as well. The practice was not restricted to the acting profession and was undertaken by townsfolk of various walks of life. There were also pictorial calendars calledegoyomithat represented theJapanese calendarin rebus so it could be "read" by the illiterate.
Today a number of abstract examples following certain conventions are occasionally used for names, primarily for corporatelogosor product logos and incorporating some characters of the name, as in amonogram; seeJapanese rebus monogram. The most familiar example globally is the logo forYamasasoy sauce, which is a ∧ with a サ under it. This is read asYama, foryama(山, mountain)(symbolized by the ∧) +sa(サ, katakana character forsa).
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TheSator Square(orRotas-Sator SquareorTemplar Magic Square) is a two-dimensionalacrosticclass ofword squarecontaining a five-wordLatinpalindrome.[1]The earliest squares were found at Roman-era sites, all in ROTAS-form (where the top line is "ROTAS", not "SATOR"), with the earliest discovery atPompeii(and also likely pre-AD 62).[a]The earliest square with Christian-associated imagery dates from the sixth century.[b]By theMiddle Ages, Sator squares had been found acrossEurope,Asia Minor, andNorth Africa.[1][2]In 2022, theEncyclopedia Britannicacalled it "the most familiar lettered square in the Western world".[3]
A significant volume of academic research has been published on the square, but after more than a century, there is no consensus on its origin and meaning.[1][4][5]The discovery of the "Paternoster theory" in 1926 led to a brief consensus among academics that the square was created by early Christians, but the subsequent discoveries at Pompeii led many academics to believe that the square was more likely created as a Roman word puzzle (as per theRoma-Amor puzzle), which was later adopted by Christians. This origin theory, however, fails to explain how a Roman word puzzle then became such a powerful religious and magical medieval symbol. It has instead been argued that the square was created in its ROTAS-form as a Jewish symbol, embedded with cryptic religious symbolism, which was later adopted in its SATOR-form by Christians.[1][2][6]There are many other less-supported academic origin theories, such as aPythagoreanorStoicpuzzle, aGnosticorOrphicor Italian paganamulet, a crypticMithraicorSemiticnumerology charm, or that it was simply a device for working out wind directions.[1]
The square has long associations with magical powers throughout its history (and even up to the 19th century in North and South America), including a perceived ability to extinguish fires, particularly in Germany. The square appears in early and late medieval medical textbooks such as theTrotula, and was employed as a medieval cure for many ailments, particularly for dog bites andrabies, as well as for insanity, and relief during childbirth.[1][2]
It has featured in a diverse range of contemporary artworks including fiction books, paintings, musical scores, and films,[5]and most notably inChristopher Nolan's 2020 filmTenet.[7]In 2020,The Daily Telegraphcalled it "one of the closest things the classical world had to ameme".[8]
The Sator square is arranged as a 5 × 5 grid consisting of five 5-letter words, thus totaling 25 characters. It uses 8 different Latin letters: 5 consonants (S, T, R, P, N) and 3 vowels (A, E, O). In some versions, the vertical and horizontal lines of the grid are also drawn, but in many cases, there are no such lines. The square is described as a two-dimensionalpalindrome, orword square, which is a particular class of adouble acrostic.[3][9]
The square comes in two forms: ROTAS (left, below), and SATOR (right, below):[2][6]
R O T A SO P E R AT E N E TA R E P OS A T O R
S A T O RA R E P OT E N E TO P E R AR O T A S
The earliest Roman-era versions of the square have the word ROTAS as the top line (called a ROTAS-form square, left above), but the inverted version with SATOR in the top line became more dominant from early medieval times (called a SATOR-form square, right above).[1]Some academics call it a Rotas-Sator Square,[2][6]and some of them refer to the object as arebus,[1][10]or amagic square.[2]Since medieval times, it has also been known as a Templar Magic Square.[1][11]
The existence of the square was long recognized from early medieval times, and various examples have been found in Europe,Asia Minor,North Africa(in mainlyCopticsettlements), and the Americas.[1][10]Medieval examples of the square in SATOR-form abound, including the earliest French example in aCarolingianBiblefrom AD 822 at the monastery ofSaint-Germain-des-Prés. Many medieval European churches and castles have Sator square inscriptions.[1][10]
The first recognized serious academic study of the square was the 1881 publication ofReinhold Köhler's[de]historical survey inZeitschrift für Ethnologie[de], titled "Sator-Arepo-Formel", and a considerable body of academic research has been subsequently published on the meaning of the square.[1][10]
Up until the 1930s, a Coptic papyrus with the square in the ROTAS-form dating from the fourth or fifth century AD was considered the earliest version.[b][10][13]In 1889, Britishancient historianFrancis Haverfieldidentified the 1868 discovery of a Sator square found in ROTAS-form scratched on a plaster wall in the Roman settlement ofCoriniumatCirencesterto be of Roman origin; however, his assertion was discounted at the time by most academics who considered the square to be an "early medieval charm".[1][14]
Haverfield was ultimately proved right by the 1931-32 excavations atDura-EuroposinSyriathat uncovered three separate Sator square inscriptions, all in ROTAS-form, on the interior walls of a Roman military office (and a fourth a year later) that were dated from circa AD 200.[1][15]
Five years later in 1936, Italian archaeologistMatteo Della Corte[it]discovered a Sator square, also in ROTAS-form, inscribed on a column in thePalestra Grande[it](the gymnasium) near theAmphitheatre of Pompeii(CIL IV8623).[16]This discovery led Della Corte to reexamine a fragment of a square, again also in ROTAS-form, that he had made in 1925 at the house of Publius Paquius Proculus, also at Pompeii (CIL IV8123). The square at the house of Publius Paquius Proculus was dated between AD 50 and AD 79 (based on the decorative style of the interior), and the palestra square find was dated pre-AD 62 (and therefore theearthquake of AD 62),[a]making it the oldest known Sator square discovery to date.[1][10]
The words are in Latin, and the following translations are known by scholars:[2][6]
The most direct sentence translation is: "The sower (or, farmer) Arepo holds the wheels with care (or, with care the wheels)".[1][10][14][4][17]Similar translations include: "The farmer Arepo works his wheels",[18]or "Arepo the sower (sator) guides (tenet) the wheel (rotas) with skill (opera)".[19]
Some academics, such as French historianJules Quicherat,[10]believe the square should be read in aboustrophedonstyle (i.e. in alternating directions).[20]The boustrophedon style, which in Greek means "as the ox plows", emphasizes the agricultural aspect of the text of the square.[1]Such a reading when applied to the SATOR-form square, and repeating the central word TENET, gives SATOR OPERA TENET – TENET OPERA SATOR, which has been very loosely interpreted as: "as ye sow, so shall ye reap",[10]while some believe the square should be read as just three words – SATOR OPERA TENET, which they loosely translate as: "The Creator (the author of all things) maintains his works"; both of which could imply Graeco-RomanStoicand/orPythagoreanorigins.[1][5]
British academic Duncan Fishwick observes that the translation from the boustrophedon approach fails when applied to a ROTAS-form square;[10]however, Belgian scholarPaul Grosjeanreversed the boustrophedon rule on the ROTAS-form (i.e. starting on the right-hand side instead of the left) to get SAT ORARE POTEN, which loosely translates into the Jewish call to prayer, "are you able to pray enough?".[1][10]
The word AREPO is ahapax legomenon, appearing nowhere else in attested Latin literature. Some academics believe it is likely a proper name, or possibly atheophoric name, that was adapted from a non-Latin word or was invented specifically for the Sator square.[10]French historianJerome Carcopinobelieved that it came from theGaulishword for a 'plough'; however, this has been discounted by other academics.[c][10]American ancient legal historianDavid Daubebelieved that AREPO represented aHebreworAramaicrendition of the ancient Greek foralpha(Ἄλφα) andomega(ω), bespeaking the "Alpha-Omega" concept (cf.Isiah 44.6, andRevelation 1:8) from early Judeo-Christianity.[1]J. Gwyn Griffithscontended that the term AREPO came, viaAlexandria, from the attested Egyptian name "Hr-Hp" (ḥrḥp), which he took to mean "the face ofApis".[1][21]In 1983, Serbian-American scholarMiroslav Marcovichproposed the term AREPO as a Latinized abbreviation ofHarpocrates(or "Horus-the-child"), god of the rising sun, also calledΓεωργός `Aρπον, which Marcovich suggests corresponds to SATOR AREPO. This would translate the square as: "The sower Horus/Harpocrates keeps in check toils and tortures".[1][22][5]
Duncan Fishwick, among other academics, believed that AREPO was simply a residual word that was required to complete what is a complex and sophisticated palindrome (which Fishwick believed was embedded with hidden Jewish symbolism, per the "Jewish Symbol" origin theory below), and to expect more from the word was unreasonable from its likely Jewish creators.[2]
Attempts have been made to discover "hidden meanings" by theanagrammatic methodof rearranging the letters of which the square is composed.[1]
The origin and meaning of the square has eluded a definitive academic consensus even after more than a century of study.[6][4][5]In 1938, British classical historian Donald Atkinson said the square occupied the "mysterious region where religion, superstition, and magic meet, where words, numbers, and letters are believed, if properly combined, to exert power over the processes of nature ...".[13]Even by 2003, American academic Rose Mary Sheldon called it "one of the oldest unsolved word puzzles in the world".[1]In 2018, American ancient classical historian Megan O'Donald still noted that "most interpretations of the ROTAS square have failed to gain consensus due to failings", and, in particular, reconciling the archeological evidence with the square's later adoption as a religious and magical object.[23]
Irrespective of the theory of its origin, the evidence that the Sator square, particularly in its SATOR-form, became adopted into Christian imagery is not disputed by academics.[1][2]Academics note the repeated association of Christ with the "sower" (or SATOR),[1]and the words of the Sator square have been discovered in Christian settings even in very early medieval times, including:
The Sator square appears in diverse Christian communities, such as inAbyssiniawhere in theEthiopian Book of the Dead, the individual nails in Christ's cross were called: Sador, Alador, Danet, Adera, Rodas.[1]These are likely derived from even earlierCoptic Christianworks that also ascribe the wounds of Christ and the nails of the cross with names that resemble the five words from the square.[1]
While there is little doubt among academics that Christians adopted the square, it was not clear that they had originated the symbol.[1][14]
During 1924 to 1926, three people separately discovered,[d]or rediscovered, that the square could be used to write the name of theLord's Prayer, the "Paternoster", twice and intersecting in a cross-form (see image opposite). The remaining residual letters (twoAs and twoOs) could be placed in the four quadrants of the cross and would represent theAlpha and Omegathat are established inChristian symbolism.[2][18]The positioning of theAs andOs was further supported by the fact that the position of theTs in the Sator square formed the points of a cross – there are obscure references in theEpistle of BarnabastoT being a symbol of the cross– and that theAs andOs also lay in the four quadrants of this cross.[10]At the time of this discovery, the earliest known Sator square was from the fourth century,[b][1]further supporting the dating of the Christian symbolism inherent in the Paternoster theory.[2]Academics considered the Christian origins of the square to be largely resolved.[1][14][2][6][15]
With the subsequent discovery of Sator squares at Pompeii, dating pre-79 AD,[a]the Paternoster theory began to lose support, even among notable supporters such as French historianGuillaume de Jerphanion.[10][15]Jerphanion noted: that (1) it was improbable that many Christians were present at Pompeii, that (2)first-century Christianswould have written the square in Greek and not Latin, that (3) the Christian concepts of Alpha and Omega only appear after the first century, that (4) thesymbol of the crossonly appears from about AD 130–131, and that (5) cryptic Christian symbols only appeared during thepersecutions of the third century.[1][10][15]
Jérôme Carcopinoclaimed the Pompeii squares were added at a later date by looters. The lack of any disturbance to the volcanic deposits at the palestra, however, meant that this was unlikely,[10][14][15]and the Paternoster theory as a proof of Christian origination lost much of its academic support.[1][10][6][15][24]
Regardless of its Christian origins, many academics considered the Paternoster discovery as being a random occurrence to be mathematically impossible.[13]Several examined this mathematical probability including German historianFriedrich Focke[de]and British historianHugh Last, but without reaching a conclusion.[1]A 1987 computer analysis by William Baines derived a number of "pseudo-Christian formulae" from the square but Baines concluded it proved nothing.[6]
There is considerable contemporary academic support for the theory that the square originated as a Roman-era word puzzle.[1][6][23]Italian historianArsenio Frugonifound it written in the margin of theCarme delle scolte modenesibeside the Roma-Amor palindrome,[1]and Italian classicistMargherita Guarduccinoted it was similar to the ROMA OLIM MILO AMOR two-dimensional acrostic word puzzle that was also found at Pompeii (seeWiktionaryfor details on the Pompeiian graffito), and at Ostia and Bolonia.[1]Similarly, another ROTAS-form square scratched into a Roman-era wall in the basement of theBasilica di Santa Maria Maggiore, was found alongside the Roma-Amor, and the Roma-Summus-Amor, palindromes.[26]Duncan Fishwick noted the "composition of palindromes was, in fact, a pastime of Roman landed gentry".[10]American classicalepigraphistRebecca Benefiel, noted that by 2012, Pompeii had yielded more than 13,000 separate inscriptions and that the house of Publius Paquius Proculus (where a square was found) had more than 70 pieces of graffiti alone.[4]
A 1969 computer study by Charles Douglas Gunn started with a Roma-Amor square and found 2,264 better versions, of which he considered the Sator square to be the best.[1]The square's origin as a word puzzle solved the problem of AREPO (a word that appears nowhere else in classical writing), as being a necessary component to complete the palindrome.[23]
Fishwick still considered this interpretation as unproven and clarified that the apparent discovery of the Roma-Amor palindrome written beside the 1954 discovery of a square on a tile at Aquincum, was incorrectly translated (if anything it supported the square as a charm).[10]Fishwick, and others, consider the key failing of the Roman puzzle theory of origin is the lack of any explanation as to why the square would later become so strongly associated with Christianity, and with being a medieval charm.[10][23][15]Some argue that this can be bridged if considered as aPythagorean-Stoicpuzzle creation.[1][5]
In 2018, Megan O'Donnell argued that the square is less of a pure word puzzle but more a piece of Latin Romangraffitothat should be readfigurativelyas a wheel (i.e. the ROTAS), and that the textual-visual interplay had parallels with other forms of graffito found in Pompeii, some of which later became adopted as charms.[23]
Some prominent academics, including British-Canadian ancient Roman scholar Duncan Fishwick,[2]American ancient legal historianDavid Daube,[1]and British ancient historianMary Beard,[27]consider the square as being likely of Jewish origin.[1]
Fishwick notes that the failings of the Paternoster theory (above) are resolved when looked at from a Jewish perspective.[2]Large numbers of Latin-speaking Jews had been settled in Pompeii, and their affinity for cryptic and mystical word symbols was well known.[2][10]The Alpha and Omega concept appears much earlier in Judaism (Ex. 3.14; Is. 41.4, and44.6), and the letters "aleph" and "tau" are used in theTalmudas symbols of totality.[2][10]TheTs of TENET may be explained not as Christian crosses, but as a Latin form of the Jewish "tau" salvation symbol (from Ezekiel), and its archaic form (+ or X) appears regularly onossuariesof bothHellenisticand early Roman times.[2][10]Fishwick highlights the central position of the letterN, as Jews attached significance to the utterance of the "Name" (or nomen).[2][10]
In addition, Fishwick believes a Jewish origin provides a satisfactory explanation for the Paternoster cross (or X) as the configuration is an archaic Jewish "tau" (+ or X).[2][10]Fishwick draws attention to some liturgical prayers in Judaism, where several prayers refer to "Our Father".[2][10]None of these liturgical prayers, however, can be dated to beforeJesus.[28][29]Fishwick concludes that the translations of the words ROTAS OPERA TENET AREPO SATOR are irrelevant, except to the extent that they make some sense and thereby hide a Jewish cryptic charm, and to require them to mean more is "to expect the impossible".[2][10]The motivation for the creation square might have been the Jewishpogromsof AD 19 or AD 49; however, it fell into disuse only to be revived later by Christians facing their own persecution, and who appreciated its hidden Paternoster and Alpha and Omega symbolism, but who focused on the SATOR-form (which gave an emphasis on the "sower", which was associated with Christ).[2]
Research in 2006 by French classical scholar Nicolas Vinel drew on recent discoveries on the mathematics of ancient magic squares to propose that the square was a "Jewish cryptogram using Pythagorean arithmetic".[25]Vinel decoded several Jewish concepts in the square, including the reason for AREPO, and was able to explain the word SAUTRAN that appears beside the square that was discovered on the palestra column in Pompeii.[25]Vinel addressed a criticism of the Jewish origin theory – why would the Jews have then abandoned the symbol? – by noting the Greek texts that they also abandoned (e.g. theSeptuagint) in favor of Hebrew versions.[25]
The amount of academic research published on the Rotas-Sator square is regarded as being considerable (and even described by one source as "immense");[4]American academicRose Mary Sheldonattempted to catalog and review the most prominent works in a 2003 paper published inCryptologia.[1]Among the more diverse but less supported theories Sheldon recorded were:
In 2003,Rose Mary Sheldonnoted: "Long after the fall of Rome, and long after the general public had forgotten about classical word games, the square survived among people who might not even read Latin. They continued to use it as a charm against illness, evil and bad luck. By the end of the Middle Ages, the "prophylactic magic" of the square was firmly established in the superstition of Italy, Serbia, Germany, and Iceland, and eventually even crossed to North America".[1]The square appears in versions of several popular magical manuscripts from the early and late Middle Ages magical text such as theTabula Smaragdinaand theClavicula Salomonis.[32]
In Germany in the Middle Ages, the square was inscribed on disks that were then thrown into fires to extinguish them.[1]An edict in 1743 byDuke Ernest Auguste of Saxe-Weimar-Eisenachrequired all settlements to make Sator square disks to combat fires.[1]By the fifteenth century the square was being used as a touchstone against fire at theChâteau de ChinonandChâteau de Jarnac[fr]in France.[10]
The square appears as a remedy during labour in the twelfth-century Latin medical text, theTrotula,[33]and was widely cited as a cure for dog bites andrabiesin medieval Europe;[1]in both cases, the remedy/cure is administered by eating bread inscribed with the words of the square.[1][33]By the sixteenth century, the use of the square to cure insanity and fever was being documented in books such asDe Varia Quercus Historia(1555) by Jean du Choul, andDe Rerum Varietate(1557) byGerolamo Cardano. Jean du Choul describes a case where a person fromLyonrecovered from insanity after eating three crusts of bread inscribed with the square.[10]After the meal, the person then recited five paternosters for the five wounds of Christ, linking to the Christian imagery believed encoded into the square.[10]
Scholars have found medieval Sator-based charms, remedies, and cures, for a diverse range of applications from childbirth, to toothaches, to love potions, to ways of warding off evil spells, and even to determine whether someone was a witch.[1]Richard Cavendishnotes a medieval manuscript in theBodleiansays: "Write these [five sator] words on in parchment with the blood of a Culver [pigeon] and bear it in thy left hand and ask what thou wilt and thou shalt have it. fiat."[35]Other examples include Bosnia, where the square was used as a remedy foraquaphobia, and in Iceland, it was etched into the fingernails to curejaundice.[1]
There are examples from the nineteenth century in South America, where the Sator square was used as a cure for dog bites and snake-bites in Brazil,[1]and in enclaves of German settlers (ormountain whites) in theAllegheny Mountainswho used the square to prevent fire, stop fits, and prevent miscarriages.[1]The Sator square features in eighteenth-century books onPow-wow folk medicineof thePennsylvania Dutch, such asThe Long Lost Friend(see image).[34]
The Sator square has inspired many works in the arts, including some classical and contemporary composers such as works by Austrian composerAnton Webernand Italian composerFabio Mengozzi,[39]writers such as Brazilian writerOsman Lins(whose novelAvalovara(1973) follows the structure of the square), and painters such as American artistDick HigginswithLa Melancolia(1983),[5]and American artistGary StephanwithSator Arepo Tenet Opera Rotas(1982).[40]
British-American directorChristopher Nolan's 2020 filmTenethas a story structure that mimics the square's concept of interlinked multiple directions of meaning, and incorporates all five of the names from the Sator square:[7]
American authorLawrence Watt-Evansnotes thatSir Terry Pratchettnamed the main square in the fictional city ofAnkh-Morporkin hisDiscworldbook series, "Sator Square", in a deliberate reference to the symbol. Watt-Evans notes that the Discworld series is full of other incidental references to unusual symbols and concepts.[41]
The songTenetby the Nordic neo-folk bandHeilungis based on the Sator square. All its individual musical parts, melodies and instruments (and even at times the lyrics) play the same both forward and backwards.[42]
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Aspoonerismis an occurrence of speech in which correspondingconsonants,vowels, ormorphemesare switched (seemetathesis) between two words of a phrase.[1][a]These are named after the Oxford don and priestWilliam Archibald Spooner, who reportedly commonly spoke in this way.[2]
Examples include saying "blushing crow" instead of "crushing blow", or "runny babbit" instead of "bunny rabbit". While spoonerisms are commonly heard as slips of the tongue, they can also be used intentionally as aword play.
The first known spoonerisms were published by the 16th-century authorFrançois Rabelaisand termedcontrepèteries.[3]In his novelPantagruel, he wrote"femme folle à la messe et femme molle à la fesse"("insane woman at Mass, woman with flabby buttocks").[4]
Spoonerisms are named for the ReverendWilliam Archibald Spooner(1844–1930), Warden from 1903 to 1924 ofNew College, Oxford, who was allegedly susceptible to this mistake.[5][6][7]TheOxford English Dictionaryrecords the wordspoonerismas early as 1900.[8]The term was well-established by 1921. An article inThe Timesfrom that year reports that:
The boys of Aldro School,Eastbourne, ... have been set the following task for the holidays: Discover and write down something about: The Old Lady of Threadneedle-street, a Spoonerism, a Busman's Holiday...[9]
An article in theDaily Heraldin 1928 reported spoonerisms to be a "legend". In that piece Robert Seton, once a student of Spooner's, admitted that Spooner:
...made, to my knowledge, only one "Spoonerism" in his life, in 1879, when he stood in the pulpit and announced the hymn: 'Kinkering Kongs their Titles Take' ["Conquering Kings their Titles Take"]...Later, a friend and myself brought out a book of "spoonerisms".[10]
In 1937,The Timesquoted a detective describing a man as "a bricklabourer's layer" and used "Police Court Spoonerism" as the headline.[11]
A spoonerism is also known as amarrowskyormorowski, purportedly after an 18th-centuryPolishcount who suffered from the same impediment.[12][8]
Most of the quotations attributed to Spooner are apocryphal;The Oxford Dictionary of Quotations(3rd edition, 1979) lists only one substantiated spoonerism: "The weight of rages will press hard upon the employer" (instead of "rate of wages"). Spooner himself claimed[5]that "The Kinquering Congs Their Titles Take" (in reference to a hymn)[13]was his sole spoonerism. Most spoonerisms were probably never uttered by William Spooner himself but rather invented by colleagues and students as a pastime.[14]Richard Lederer, calling "Kinkering Kongs their Titles Take" (with an alternative spelling) one of the "few" authenticated Spoonerisms, dates it to 1879, and he gives nine examples "attributed to Spooner, most of them spuriously".[15]They are as follows:
In modern terms,spoonerismgenerally refers to any changing of sounds in this manner.
Writing in tribute for the inauguralRonnie Barker Talk,Ben Eltonwrote:
What an honour. I grew up loving Ronnie Barker and can only hope the news that I am to give a talk in his name doesn't leave him spitting spiritedly splenetic spoonerisms in comedy heaven.[16]
He once proclaimed, "Hey,belly jeans"When he found a stash of jelly beans.But when he says hepepped in stewWe'll tell him he should wipe his shoe.
Spoonerisms are used incryptic crossword cluesand use aplay on words, in which the initial sounds or syllables of two words are switched to provide a solution. The clue type is generally indicated by a direct reference to 'Spooner', although more tricky examples might refer to him only as 'Rev', or use such phrases as 'in a manner of speaking', or 'slip of the tongue'. Uniquely, in cryptic crosswords the words used to create the Spoonerism might only be hinted at, not explicitly stated.[24]
Example:"Spooner's criminal with nurse finding hiding places."(4,3,6)
Solution: NOOK AND CRANNY (Spoonerism of CROOK AND NANNY).
On the 3 December 1950 episode ofThe Jack Benny Program,Jack mentions that he ran into his butler Rochester while in his car that was on a grease rack. Mary Livingston was supposed to say "How could you run into him on a grease rack?" but flubbed her line with "How could you run into him on a grass reek?" The audience laughed so much that Jack was unable to reply as the show ran out of time.[36]
Spoonerisms are used sometimes infalse etymologies. For example, according to linguistGhil'ad Zuckermann, some wrongly believe that the English wordbutterflyderives fromflutterby.[37]: p.78
As complements tospoonerism,Douglas Hofstadterused thenonce wordskniferismandforkerismto refer to changing, respectively, the vowels or the final consonants of two syllables, giving them a new meaning.[38]Examples of so-called kniferisms include a British television newsreader once referring to the police at a crime scene removing a 'hypodeemic nerdle'; a television announcer once saying that "All the world was thrilled by the marriage of the Duck and Doochess of Windsor";[39]and during a live radio broadcast in 1931, radio presenterHarry von Zellaccidentally mispronouncing U.S. PresidentHerbert Hoover's name as "Hoobert Heever".[39][40]
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Atautonymis ascientific nameof a species in which both parts of the name have the same spelling, such asRattus rattus. The first part of the name is the name of the genus and the second part is referred to as thespecific epithetin theInternational Code of Nomenclature for algae, fungi, and plantsand thespecific namein theInternational Code of Zoological Nomenclature.
Tautonymy (i.e., the usage of tautonymous names) is permissible in zoological nomenclature (seeList of tautonymsfor examples). In past editions of the zoological code, the term tautonym was used, but it has now been replaced by the more inclusive "tautonymous names"; these includetrinomial namesfor subspecies such asGorilla gorilla gorillaandBison bison bison.
Tautonyms can be formed when animals are given scientific names for the first time, or when they are reclassified and given new scientific names.[1]An example of the former is the hidden mirror skipper of Brazil with the scientific nameSpeculum speculum, which comes from a Latin word for "mirror" in reference to the shiny, mirror-like coloring on its wings.[2][3]An example of the latter isNombe nombe, an extinct kangaroo from the late Pleistocene epoch found in Papua New Guinea's Nombe Rockshelter that was classified asProtemnodon nombeuntil 2022 when it was reclassified in light of a more recent review of the animal's dental attributes.[4]Animals with tautonymous names can also be reclassified so that they no longer have tautonymous names, as was the case withPolyspila polyspila(nowCalligrapha polyspila).[5]
For animals, a tautonym implicitly (though not always) indicates that the species is thetype speciesof its genus.[6]This can also be indicated by a species name with the specific epithettypusortypicus,[7]although more commonly the type species is designated another way.
Regarding other living organisms, tautonyms were prohibited in bacteriological nomenclature from 1947 until 1975, but they are now permitted for all bacteria andprokaryotes.[8]Tautonyms are prohibited by the codes of nomenclature for botany and for cultivated plants, but they are not prohibited by the code of nomenclature for viruses.[9]
In the current rules forbotanical nomenclature(which apply retroactively), tautonyms are explicitly prohibited.[10]The reason for prohibiting tautonyms is not explained in current or historical botanical nomenclatural codes, but it appears to have resulted from concerns over a century ago that identical taxon names could result in confusion where those names share identical spelling and identical capitalization.[11]
One example of a former botanical tautonym is 'Larix larix'. The earliest name for theEuropean larchisPinus larixL. (1753) butGustav Karl Wilhelm Hermann Karstendid not agree with the placement of the species inPinusand decided to move it toLarixin 1880. His proposed name created a tautonym. Under rules first established in 1906, which are applied retroactively,Larix larixcannot exist as a formal name. In such a case either the next earliest validly published name must be found, in this caseLarix deciduaMill. (1768), or (in its absence) a new epithet must be published.
However, it is allowed for both parts of the name of a species to mean the same (pleonasm), without being identical in spelling. For instance,Arctostaphylos uva-ursimeansbearberrytwice, in Greek and Latin respectively;Picea omorikauses the Latin and Serbian terms for aspruce.
Instances that repeat the genus name with a slight modification, such asLycopersicon lycopersicum(Greek and Latinized Greek, a rejected name for thetomato) andZiziphus zizyphus, have been contentious, but are in accord with the Code of Nomenclature.[12]
In April 2023, a proposal was made to permit tautonyms in botanical nomenclature on a non-retroactive basis, noting that tautonyms have been allowed in zoological and bacteriological codes for decades without incident, and that allowing tautonyms would simplify botany's nomenclatural code while eliminating certain naming problems and preserving the epithets originally assigned to species.[13]
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Word playorwordplay[1](also:play-on-words) is aliterary techniqueand a form ofwitin which words used become the main subject of the work, primarily for the purpose of intended effect oramusement. Examples of word play includepuns, phonetic mix-ups such asspoonerisms, obscure words and meanings, cleverrhetoricalexcursions, oddly formed sentences,double entendres, and telling character names (such as in the playThe Importance of Being Earnest,Ernestbeing agiven namethat sounds exactly like the adjectiveearnest).
Word play is quite common inoral culturesas a method of reinforcing meaning. Examples of text-based (orthographic) word play are found in languages with or without alphabet-based scripts, such ashomophonic puns in Mandarin Chinese.
Most writers engage in word play to some extent, but certain writers are particularly committed to, or adept at, word play as a major feature of their work .Shakespeare's "quibbles" have made him a noted punster. Similarly,P.G. Wodehousewas hailed byThe Timesas a "comic genius recognized in his lifetime as a classic and an old master of farce" for his own acclaimed wordplay.[6]James Joyce, author ofUlysses, is another noted word-player. For example, in hisFinnegans WakeJoyce's phrase "they were yung and easily freudened" clearly implies the more conventional "they were young and easily frightened"; however, the former also makes an apt pun on the names of two famouspsychoanalysts,JungandFreud.
Anepitaph, probably unassigned to anygrave, demonstrates use in rhyme.
Crossword puzzlesoften employ wordplay to challenge solvers.Cryptic crosswordsespecially are based on elaborate systems of wordplay.
An example of modern word play can be found on line 103 ofChildish Gambino's "III. Life: The Biggest Troll".
H2O plus my D, that's my hood, I'm living in it
RapperMilouses a play on words in his verse on "True Nen".[7]
A farmer says, "I got soaked for nothing, stood out there in the rain bang in the middle of my land, a complete waste of time. I'll like to kill the swine who said you can win theNobel Prizefor being out standing in your field!".
TheMario Partyseries is known for its mini-game titles that usually are puns and various plays on words; for example: "Shock, Drop, and Roll", "Gimme a Brake", and "Right Oar Left". These mini-game titles are also different depending onregional differencesand take into account that specific region's culture.
Many of the books the characterGromitin theWallace & Gromit seriesreads or the music Grommit listens to are plays on words, such as "Pup Fiction" (Pulp Fiction), "Where Beagles Dare" (Where Eagles Dare), "Red Hot Chili Puppies" (Red Hot Chili Peppers) and "The Hound of Music" (The Sound of Music).
Word play can enter common usage asneologisms.
Word play is closely related toword games; that is, games in which the point is manipulating words. See alsolanguage gamefor a linguist's variation.
Word play can cause problems for translators: e.g., in the bookWinnie-the-Pooha character mistakes the word "issue" for the noise of asneeze, a resemblance which disappears when the word "issue" is translated into another language.
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In mathematics,arithmetic geometryis roughly the application of techniques fromalgebraic geometryto problems innumber theory.[1]Arithmetic geometry is centered aroundDiophantine geometry, the study ofrational pointsofalgebraic varieties.[2][3]
In more abstract terms, arithmetic geometry can be defined as the study ofschemesoffinite typeover thespectrumof thering of integers.[4]
The classical objects of interest in arithmetic geometry are rational points:sets of solutionsof asystem of polynomial equationsovernumber fields,finite fields,p-adic fields, orfunction fields, i.e.fieldsthat are notalgebraically closedexcluding thereal numbers. Rational points can be directly characterized byheight functionswhich measure their arithmetic complexity.[5]
The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields,étale cohomologyprovidestopological invariantsassociated to algebraic varieties.[6]p-adic Hodge theorygives tools to examine when cohomological properties of varieties over thecomplex numbersextend to those overp-adic fields.[7]
In the early 19th century,Carl Friedrich Gaussobserved that non-zerointegersolutions tohomogeneous polynomialequations withrationalcoefficients exist if non-zero rational solutions exist.[8]
In the 1850s,Leopold Kroneckerformulated theKronecker–Weber theorem, introduced the theory ofdivisors, and made numerous other connections between number theory andalgebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as histwelfth problem, which outlines a goal to have number theory operate only with rings that are quotients ofpolynomial ringsover the integers.[9]
In the late 1920s,André Weildemonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to theMordell–Weil theoremwhich demonstrates that the set of rational points of anabelian varietyis afinitely generated abelian group.[10]
Modern foundations of algebraic geometry were developed based on contemporarycommutative algebra, includingvaluation theoryand the theory ofidealsbyOscar Zariskiand others in the 1930s and 1940s.[11]
In 1949,André Weilposed the landmarkWeil conjecturesabout thelocal zeta-functionsof algebraic varieties over finite fields.[12]These conjectures offered a framework between algebraic geometry and number theory that propelledAlexander Grothendieckto recast the foundations making use ofsheaf theory(together withJean-Pierre Serre), and later scheme theory, in the 1950s and 1960s.[13]Bernard Dworkproved one of the four Weil conjectures (rationality of the local zeta function) in 1960.[14]Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together withMichael ArtinandJean-Louis Verdier) by 1965.[6][15]The last of the Weil conjectures (an analogue of theRiemann hypothesis) would be finally proven in 1974 byPierre Deligne.[16]
Between 1956 and 1957,Yutaka TaniyamaandGoro Shimuraposed theTaniyama–Shimura conjecture(now known as the modularity theorem) relatingelliptic curvestomodular forms.[17][18]This connection would ultimately lead tothe first proofofFermat's Last Theoremin number theory through algebraic geometry techniques ofmodularity liftingdeveloped byAndrew Wilesin 1995.[19]
In the 1960s, Goro Shimura introducedShimura varietiesas generalizations ofmodular curves.[20]Since the 1979, Shimura varieties have played a crucial role in theLanglands programas a natural realm of examples for testing conjectures.[21]
In papers in 1977 and 1978,Barry Mazurproved thetorsion conjecturegiving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certainmodular curves.[22][23]In 1996, the proof of the torsion conjecture was extended to all number fields byLoïc Merel.[24]
In 1983,Gerd Faltingsproved theMordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstratesfinite generationof the set of rational points as opposed to finiteness).[25][26]
In 2001, the proof of thelocal Langlands conjectures for GLnwas based on the geometry of certain Shimura varieties.[27]
In the 2010s,Peter Scholzedevelopedperfectoid spacesand new cohomology theories in arithmetic geometry over p-adic fields with application toGalois representationsand certain cases of theweight-monodromy conjecture.[28][29]
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Arithmetic topologyis an area ofmathematicsthat is a combination ofalgebraic number theoryandtopology. It establishes an analogy betweennumber fieldsand closed, orientable3-manifolds.
The following are some of the analogies used by mathematicians between number fields and 3-manifolds:[1]
Expanding on the last two examples, there is an analogy betweenknotsandprime numbersin which one considers "links" between primes. The triple of primes(13, 61, 937)are "linked" modulo 2 (theRédei symbolis −1) but are "pairwise unlinked" modulo 2 (theLegendre symbolsare all 1). Therefore these primes have been called a "proper Borromean triple modulo 2"[2]or "mod 2 Borromean primes".[3]
In the 1960s topological interpretations ofclass field theorywere given byJohn Tate[4]based onGalois cohomology, and also byMichael ArtinandJean-Louis Verdier[5]based onÉtale cohomology. ThenDavid Mumford(and independentlyYuri Manin) came up with an analogy betweenprime idealsandknots[6]which was further explored byBarry Mazur.[7][8]In the 1990s Reznikov[9]and Kapranov[10]began studying these analogies, coining the termarithmetic topologyfor this area of study.
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Themathematicaldisciplines ofcombinatoricsanddynamical systemsinteract in a number of ways. Theergodic theoryof dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field ofarithmetic combinatorics. Alsodynamical systems theoryis heavily involved in the relatively recent field ofcombinatorics on words. Also combinatorial aspects of dynamical systems are studied. Dynamical systems can be defined on combinatorial objects; see for examplegraph dynamical system.
Thiscombinatorics-related article is astub. You can help Wikipedia byexpanding it.
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Inarithmetic dynamics, anarboreal Galois representationis acontinuousgroup homomorphismbetween theabsolute Galois groupof a field and theautomorphism groupof an infinite, regular, rootedtree.
The study of arboreal Galois representations of goes back to the works of Odoni in 1980s.
LetK{\displaystyle K}be afieldandKsep{\displaystyle K^{sep}}be itsseparable closure. TheGalois groupGK{\displaystyle G_{K}}of the extensionKsep/K{\displaystyle K^{sep}/K}is called theabsolute Galois groupofK{\displaystyle K}. This is aprofinite groupand it is therefore endowed with its natural Krull topology.
For a positive integerd{\displaystyle d}, letTd{\displaystyle T^{d}}be the infinite regular rootedtreeof degreed{\displaystyle d}. This is an infinite tree where one node is labeled as the root of the tree and every node has exactlyd{\displaystyle d}descendants. AnautomorphismofTd{\displaystyle T^{d}}is a bijection of the set of nodes that preserves vertex-edge connectivity. The groupAut(Td){\displaystyle Aut(T^{d})}of all automorphisms ofTd{\displaystyle T^{d}}is a profinite group as well, as it can be seen as theinverse limitof the automorphism groups of the finite sub-treesTnd{\displaystyle T_{n}^{d}}formed by all nodes at distance at mostn{\displaystyle n}from the root. The group of automorphisms ofTnd{\displaystyle T_{n}^{d}}is isomorphic toSd≀Sd≀…≀Sd{\displaystyle S_{d}\wr S_{d}\wr \ldots \wr S_{d}}, the iteratedwreath productofn{\displaystyle n}copies of thesymmetric groupof degreed{\displaystyle d}.
An arboreal Galois representation is acontinuousgroup homomorphismGK→Aut(Td){\displaystyle G_{K}\to Aut(T^{d})}.
The most natural source of arboreal Galois representations is the theory of iterations of self-rational functionson theprojective line. LetK{\displaystyle K}be afieldandf:PK1→PK1{\displaystyle f\colon \mathbb {P} _{K}^{1}\to \mathbb {P} _{K}^{1}}a rational function of degreed{\displaystyle d}. For everyn≥1{\displaystyle n\geq 1}letfn=f∘f∘…∘f{\displaystyle f^{n}=f\circ f\circ \ldots \circ f}be then{\displaystyle n}-fold composition of the mapf{\displaystyle f}with itself. Letα∈K{\displaystyle \alpha \in K}and suppose that for everyn≥1{\displaystyle n\geq 1}the set(fn)−1(α){\displaystyle (f^{n})^{-1}(\alpha )}containsdn{\displaystyle d^{n}}elements of thealgebraic closureK¯{\displaystyle {\overline {K}}}. Then one can construct an infinite, regular, rootedd{\displaystyle d}-ary treeT(f){\displaystyle T(f)}in the following way: the root of the tree isα{\displaystyle \alpha }, and thenodesat distancen{\displaystyle n}fromα{\displaystyle \alpha }are the elements of(fn)−1(α){\displaystyle (f^{n})^{-1}(\alpha )}. A nodeβ{\displaystyle \beta }at distancen{\displaystyle n}fromα{\displaystyle \alpha }is connected with an edge to a nodeγ{\displaystyle \gamma }at distancen+1{\displaystyle n+1}fromα{\displaystyle \alpha }if and only iff(β)=γ{\displaystyle f(\beta )=\gamma }.
The absolute Galois groupGK{\displaystyle G_{K}}actsonT(f){\displaystyle T(f)}via automorphisms, and the induced homomorphismρf,α:GK→Aut(T(f)){\displaystyle \rho _{f,\alpha }\colon G_{K}\to Aut(T(f))}is continuous, and therefore is called the arboreal Galois representation attached tof{\displaystyle f}with basepointα{\displaystyle \alpha }.
Arboreal representations attached to rational functions can be seen as a wide generalization ofGalois representationsonTate modulesofabelian varieties.
The simplest non-trivial case is that of monic quadratic polynomials. LetK{\displaystyle K}be a field ofcharacteristicnot 2, letf=(x−a)2+b∈K[x]{\displaystyle f=(x-a)^{2}+b\in K[x]}and set the basepointα=0{\displaystyle \alpha =0}. Theadjusted post-critical orbitoff{\displaystyle f}is the sequence defined byc1=−f(a){\displaystyle c_{1}=-f(a)}andcn=fn(a){\displaystyle c_{n}=f^{n}(a)}for everyn≥2{\displaystyle n\geq 2}. A resultant argument[1]shows that(fn)−1(0){\displaystyle (f^{n})^{-1}(0)}hasdn{\displaystyle d^{n}}elements for evern{\displaystyle n}if and only ifcn≠0{\displaystyle c_{n}\neq 0}for everyn{\displaystyle n}. In 1992, Stoll proved the following theorem:[2]
The following are examples of polynomials that satisfy the conditions of Stoll's Theorem, and that therefore have surjective arboreal representations.
In 1985 Odoni formulated the following conjecture.[4]
Although in this very general form the conjecture has been shown to be false by Dittmann and Kadets,[5]there are several results whenK{\displaystyle K}is anumber field. Benedetto and Juul proved Odoni's conjecture forK{\displaystyle K}a number field andn{\displaystyle n}even, and also when both[K:Q]{\displaystyle [K:\mathbb {Q} ]}andn{\displaystyle n}are odd,[6]Looper independently proved Odoni's conjecture forn{\displaystyle n}prime andK=Q{\displaystyle K=\mathbb {Q} }.[7]
WhenK{\displaystyle K}is aglobal fieldandf∈K(x){\displaystyle f\in K(x)}is a rational function of degree 2, the image ofρf,0{\displaystyle \rho _{f,0}}is expected to be "large" in most cases. The following conjecture quantifies the previous statement, and it was formulated by Jones in 2013.[8]
Jones' conjecture is considered to be a dynamical analogue of Serre's open image theorem.
One direction of Jones' conjecture is known to be true: iff{\displaystyle f}satisfies one of the above conditions, then[Aut(T(f)):Im(ρf,0)]=∞{\displaystyle [Aut(T(f)):Im(\rho _{f,0})]=\infty }. In particular, whenf{\displaystyle f}is post-critically finite thenIm(ρf,α){\displaystyle Im(\rho _{f,\alpha })}is a topologically finitely generated closed subgroup ofAut(T(f)){\displaystyle Aut(T(f))}for everyα∈K{\displaystyle \alpha \in K}.
In the other direction, Juul et al. proved that if theabc conjectureholds for number fields,K{\displaystyle K}is anumber fieldandf∈K[x]{\displaystyle f\in K[x]}is a quadratic polynomial, then[Aut(T(f)):Im(ρf,0)]=∞{\displaystyle [Aut(T(f)):Im(\rho _{f,0})]=\infty }if and only iff{\displaystyle f}ispost-critically finiteor noteventually stable. Whenf∈K[x]{\displaystyle f\in K[x]}is a quadratic polynomial, conditions (2) and (4) in Jones' conjecture are never satisfied. Moreover, Jones and Levy conjectured thatf{\displaystyle f}iseventually stableif and only if0{\displaystyle 0}is not periodic forf{\displaystyle f}.[9]
In 2020, Andrews and Petsche formulated the following conjecture.[10]
Two pairs(f,α),(g,β){\displaystyle (f,\alpha ),(g,\beta )}, wheref,g∈K(x){\displaystyle f,g\in K(x)}andα,β∈K{\displaystyle \alpha ,\beta \in K}areconjugateover a field extensionL/K{\displaystyle L/K}if there exists aMöbius transformationm=ax+bcx+d∈PGL2(L){\displaystyle m={\frac {ax+b}{cx+d}}\in PGL_{2}(L)}such thatm∘f∘m−1=g{\displaystyle m\circ f\circ m^{-1}=g}andm(α)=β{\displaystyle m(\alpha )=\beta }. Conjugacy is anequivalence relation. The Chebyshev polynomials the conjecture refers to are a normalized version, conjugate by theMöbius transformation2x{\displaystyle 2x}to make them monic.
It has been proven that Andrews and Petsche's conjecture holds true whenK=Q{\displaystyle K=\mathbb {Q} }.[11]
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Inmathematics, theEnriques–Kodaira classificationgroupscompactcomplex surfacesinto ten classes, each parametrized by amoduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.
Max Noetherbegan the systematic study of algebraic surfaces, andGuido Castelnuovoproved important parts of the classification.Federigo Enriques(1914,1949) described the classification of complex projective surfaces.Kunihiko Kodaira(1964,1966,1968a,1968b) later extended the classification to include non-algebraic compact surfaces. The analogous classification of surfaces in positive characteristic was begun byDavid Mumford(1969) and completed byEnrico Bombieriand David Mumford (1976,1977); it is similar to the characteristic 0 projective case, except that one also gets singular and supersingular Enriques surfaces in characteristic 2, and quasi-hyperelliptic surfaces in characteristics 2 and 3.
The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.
For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like (which for class VII depends on theglobal spherical shell conjecture, still unproved in 2024). For surfaces of general type not much is known about their explicit classification, though many examples have been found.
The classification of algebraic surfaces in positive characteristic (Mumford 1969, Mumford & Bombieri1976,1977) is similar to that of algebraic surfaces in characteristic 0, except that there are no Kodaira surfaces or surfaces of type VII, and there are some extra families of Enriques surfaces in characteristic 2, and hyperelliptic surfaces in characteristics 2 and 3, and in Kodaira dimension 1 in characteristics 2 and 3 one also allows quasielliptic fibrations. These extra families can be understood as follows: In characteristic 0 these surfaces are the quotients of surfaces by finite groups, but in finite characteristics it is also possible to take quotients by finitegroup schemesthat are notétale.
Oscar Zariskiconstructed some surfaces in positive characteristic that are unirational but not rational, derived frominseparable extensions(Zariski surfaces). In positive characteristic Serre showed thath0(Ω){\displaystyle h^{0}(\Omega )}may differ fromh1(O){\displaystyle h^{1}({\mathcal {O}})}, and Igusa showed that even when they are equal they may be greater than the irregularity (the dimension of thePicard variety).
The most important invariants of a compact complex surfaces used in the classification can be given in terms of the dimensions of variouscoherent sheaf cohomologygroups. The basic ones are theplurigeneraand the Hodge numbers defined as follows:
There are many invariants that (at least for complex surfaces) can be written as linear combinations of the Hodge numbers, as follows:
There are further invariants of compact complex surfaces that are not used so much in the classification. These include algebraic invariants such as thePicard groupPic(X) of divisors modulolinear equivalence, its quotient theNéron–Severi groupNS(X) with rank thePicard numberρ, topological invariants such as thefundamental groupπ1and the integral homology and cohomology groups, and invariants of the underlying smooth4-manifoldsuch as theSeiberg–Witten invariantsandDonaldson invariants.
Any surface is birational to a non-singular surface, so for most purposes it is enough to classify the non-singular surfaces.
Given any point on a surface, we can form a new surface byblowing upthis point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non-singular surfaceXis calledminimalif it cannot be obtained from another non-singular surface by blowing up a point. ByCastelnuovo's contraction theorem, this is equivalent to saying thatXhas no (−1)-curves (smooth rational curves with self-intersection number −1). (In the more modern terminology of theminimal model program, a smooth projective surfaceXwould be calledminimalif its canonical line bundleKXisnef. A smooth projective surface has a minimal model in that stronger sense if and only if its Kodaira dimension is nonnegative.)
Every surfaceXis birational to a minimal non-singular surface, and this minimal non-singular surface is unique ifXhas Kodaira dimension at least 0 or is not algebraic. Algebraic surfaces of Kodaira dimension−∞{\displaystyle -\infty }may be birational to more than one minimal non-singular surface, but it is easy to describe the relation between these minimal surfaces. For example,P1×P1blown up at a point is isomorphic toP2blown up twice. So to classify all compact complex surfaces up to birational isomorphism it is (more or less) enough to classify the minimal non-singular ones.
Algebraic surfaces of Kodaira dimension−∞{\displaystyle -\infty }can be classified as follows. Ifq> 0 then the map to the Albanese variety has fibers that are projective lines (if the surface is minimal) so the surface is a ruled surface. Ifq= 0 this argument does not work as the Albanese variety is a point, but in this caseCastelnuovo's theoremimplies that the surface is rational.
For non-algebraic surfaces Kodaira found an extra class of surfaces, called type VII, which are still not well understood.
Rational surfacemeans surface birational to thecomplex projective planeP2. These are all algebraic. The minimal rational surfaces areP2itself and theHirzebruch surfacesΣnforn= 0 orn≥ 2. (The Hirzebruch surface Σnis theP1bundle overP1associated to the sheaf O(0) + O(n). The surface Σ0is isomorphic toP1×P1, and Σ1is isomorphic toP2blown up at a point so is not minimal.)
Invariants:The plurigenera are all 0 and the fundamental group is trivial.
Hodge diamond:
Examples:P2,P1×P1= Σ0, Hirzebruch surfaces Σn,quadrics,cubic surfaces,del Pezzo surfaces,Veronese surface. Many of these examples are non-minimal.
Ruled surfaces of genusghave a smooth morphism to a curve of genusgwhose fibers are linesP1. They are all algebraic.
(The ones of genus 0 are the Hirzebruch surfaces and are rational.) Any ruled surface is birationally equivalent toP1×Cfor a unique curveC, so the classification of ruled surfaces up to birational equivalence is essentially the same as the classification of curves. A ruled surface not isomorphic toP1×P1has a unique ruling (P1×P1has two).
Invariants:The plurigenera are all 0.
Hodge diamond:
Examples:The product of any curve of genus > 0 withP1.
These surfaces are never algebraic orKähler. The minimal ones withb2= 0 have been classified by Bogomolov, and are eitherHopf surfacesorInoue surfaces. Examples with positive second Betti number includeInoue-Hirzebruch surfaces,Enoki surfaces, and more generallyKato surfaces. Theglobal spherical shell conjectureimplies that all minimal class VII surfaces with positive second Betti number are Kato surfaces, which would more or less complete the classification of the type VII surfaces.
Invariants:q= 1,h1,0= 0. All plurigenera are 0.
Hodge diamond:
These surfaces are classified by starting with Noether's formula12χ=c2+c12.{\displaystyle 12\chi =c_{2}+c_{1}^{2}.}For Kodaira dimension 0,Khas zerointersection number with itself, soc12=0.{\displaystyle c_{1}^{2}=0.}Using
we arrive at:
Moreover sinceκ= 0 we have:
combining this with the previous equation gives:
In general 2h0,1≥b1, so three terms on the left are non-negative integers and there are only a few solutions to this equation.
Most solutions to these conditions correspond to classes of surfaces, as in the following table:
These are the minimal compact complex surfaces of Kodaira dimension 0 withq= 0 and trivial canonical line bundle. They are allKähler manifolds. All K3 surfaces are diffeomorphic, and their diffeomorphism class is an important example of a smooth spin simply connected 4-manifold.
Invariants:The second cohomology groupH2(X,Z) is isomorphic to the unique evenunimodular latticeII3,19of dimension 22 and signature −16.
Hodge diamond:
Examples:
AmarkedK3 surface is a K3 surface together with an isomorphism from II3,19toH2(X,Z). The moduli space of marked K3 surfaces is connected non-Hausdorff smooth analytic space of dimension 20. The algebraic K3 surfaces form a countable collection of 19-dimensional subvarieties of it.
The two-dimensionalcomplex toriinclude theabelian surfaces. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are exactly the 2-dimensionalabelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Criteria to be a product of two elliptic curves (up toisogeny) were a popular study in the nineteenth century.
Invariants:The plurigenera are all 1. The surface is diffeomorphic toS1×S1×S1×S1so the fundamental group isZ4.
Hodge diamond:
Examples:A product of two elliptic curves. The Jacobian of a genus 2 curve. Any quotient ofC2by a lattice.
These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes:primary Kodaira surfaceswith trivial canonical bundle, andsecondary Kodaira surfaceswhich are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces.
Invariants: If the surface is the quotient of a primary Kodaira surface by a group of orderk= 1, 2, 3, 4, 6, then the plurigeneraPnare 1 ifnis divisible bykand 0 otherwise.
Hodge diamond:
Examples:Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers byZacting as multiplication by powers of somecomplex numberz. This gives a primary Kodaira surface.
These are the complex surfaces such thatq= 0 and the canonical line bundle is non-trivial, but has trivial square. Enriques surfaces are all algebraic (and thereforeKähler). They are quotients of K3 surfaces by a group of order 2 and their theory is similar to that of algebraic K3 surfaces.
Invariants:The plurigeneraPnare 1 ifnis even and 0 ifnis odd. The fundamental group has order 2. The second cohomology group H2(X,Z) is isomorphic to the sum of the unique evenunimodular latticeII1,9of dimension 10 and signature −8 and a group of order 2.
Hodge diamond:
Marked Enriques surfaces form a connected 10-dimensional family, which has been described explicitly.
In characteristic 2 there are some extra families of Enriques surfaces called singular and supersingular Enriques surfaces; see the article onEnriques surfacesfor details.
Over the complex numbers these are quotients of a product of two elliptic curves by a finite group of automorphisms. The finite group can beZ/2Z,Z/2Z+Z/2Z,Z/3Z,Z/3Z+Z/3Z,Z/4Z,Z/4Z+Z/2Z, orZ/6Z, giving seven families of such surfaces.
Hodge diamond:
Over fields of characteristics 2 or 3 there are some extra families given by taking quotients by a non-etale group scheme; see the article onhyperelliptic surfacesfor details.
Anelliptic surfaceis a surface equipped with an elliptic fibration (a surjective holomorphic map to a curveBsuch that all but finitely many fibers are smooth irreducible curves of genus 1). The generic fiber in such a fibration is a genus 1 curve over the function field ofB. Conversely, given a genus 1 curve over the function field of a curve, its relative minimal model is an elliptic surface. Kodaira and others have given a fairly complete description of all elliptic surfaces. In particular, Kodaira gave acomplete list of the possible singular fibers. The theory of elliptic surfaces is analogous to the theory of proper regular models of elliptic curves overdiscrete valuation rings(e.g., the ring ofp-adic integers) andDedekind domains(e.g., the ring of integers of a number field).
In finite characteristic 2 and 3 one can also getquasi-ellipticsurfaces, whose fibers may almost all be rational curves with a single node, which are "degenerate elliptic curves".
Every surface ofKodaira dimension1 is an elliptic surface (or a quasielliptic surface in characteristics 2 or 3), but the converse is not true: an elliptic surface can have Kodaira dimension−∞{\displaystyle -\infty }, 0, or 1. AllEnriques surfaces, allhyperelliptic surfaces, allKodaira surfaces, someK3 surfaces, someabelian surfaces, and somerational surfacesare elliptic surfaces, and these examples have Kodaira dimension less than 1. An elliptic surface whose base curveBis of genus at least 2 always has Kodaira dimension 1, but the Kodaira dimension can be 1 also for some elliptic surfaces withBof genus 0 or 1.
Invariants:c12=0,c2⩾0.{\displaystyle c_{1}^{2}=0,c_{2}\geqslant 0.}
Example:IfEis an elliptic curve andBis a curve of genus at least 2, thenE×Bis an elliptic surface of Kodaira dimension 1.
These are all algebraic, and in some sense most surfaces are in this class. Gieseker showed that there is acoarse moduli schemefor surfaces of general type; this means that for any fixed values of the Chern numbersc21andc2, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. However it is a very difficult problem to describe these schemes explicitly, and there are very few pairs of Chern numbers for which this has been done (except when the scheme is empty!)
Invariants:There are several conditions that the Chern numbers of a minimal complex surface of general type must satisfy:
Most pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type.
Examples:The simplest examples are the product of two curves of genus at least 2, and a hypersurface of degree at least 5 inP3. There are a large number of other constructions known. However, there is no known construction that can produce "typical" surfaces of general type for large Chern numbers; in fact it is not even known if there is any reasonable concept of a "typical" surface of general type. There are many other examples that have been found, including mostHilbert modular surfaces,fake projective planes,Barlow surfaces, and so on.
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Inalgebraic geometry, theNéron model(orNéron minimal model, orminimal model)
for anabelian varietyAKdefined over the field of fractionsKof a Dedekind domainRis the "push-forward" ofAKfrom Spec(K) to Spec(R), in other words the "best possible" group schemeARdefined overRcorresponding toAK.
They were introduced byAndré Néron(1961,1964) for abelian varieties over the quotient field of a Dedekind domainRwith perfect residue fields, andRaynaud (1966)extended this construction to semiabelian varieties over all Dedekind domains.
Suppose thatRis aDedekind domainwith field of fractionsK, and suppose thatAKis a smooth separated scheme overK(such as an abelian variety). Then aNéron modelofAKis defined to be asmoothseparatedschemeARoverRwith fiberAKthat is universal in the following sense.
In particular, the canonical mapAR(R)→AK(K){\displaystyle A_{R}(R)\to A_{K}(K)}is an isomorphism. If a Néron model exists then it is unique up to unique isomorphism.
In terms of sheaves, any schemeAover Spec(K) represents a sheaf on the category of schemes smooth over Spec(K) with the smooth Grothendieck topology, and this has a pushforward by the injection map from Spec(K) to Spec(R), which is a sheaf over Spec(R). If this pushforward is representable by a scheme, then this scheme is the Néron model ofA.
In general the schemeAKneed not have any Néron model.
For abelian varietiesAKNéron models exist and are unique (up to unique isomorphism) and are commutative quasi-projectivegroup schemesoverR. The fiber of a Néron model over aclosed pointof Spec(R) is a smooth commutativealgebraic group, but need not be an abelian variety: for example, it may be disconnected or a torus. Néron models exist as well for certain commutative groups other than abelian varieties such as tori, but these are only locally of finite type. Néron models do not exist for the additive group.
The Néron model of an elliptic curveAKoverKcan be constructed as follows. First form the minimal model overRin the sense of algebraic (or arithmetic) surfaces. This is a regular proper surface overRbut is not in general smooth overRor a group scheme overR. Its subscheme of smooth points overRis the Néron model, which is a smooth group scheme overRbut not necessarily proper overR. The fibers in general may have several irreducible components, and to form the Néron model one discards all multiple components, all points where two components intersect, and all singular points of the components.
Tate's algorithmcalculates thespecial fiberof the Néron model of an elliptic curve, or more precisely the fibers of the minimal surface containing the Néron model.
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https://en.wikipedia.org/wiki/N%C3%A9ron_minimal_model
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The following tables provide acomparison ofnumerical analysissoftware.
[Note 1]
[Note 2]
[Note 3]
Was:Inria
(Andrey Ivashov)
Theoperating systemsthe software can run on natively (withoutemulation).
Colors indicate features available as
Theoperating systemsthe software can run on natively (withoutemulation).
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https://en.wikipedia.org/wiki/Comparison_of_numerical-analysis_software
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The following tables compare general and technical information for manystatistical analysissoftware packages.
University of Amsterdam
Support for variousANOVAmethods
Support for variousregressionmethods.
Support for varioustime series analysismethods.
Support for variousstatisticalchartsanddiagrams.
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https://en.wikipedia.org/wiki/Comparison_of_statistical_packages
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This is a list ofnumerical libraries, which arelibrariesused insoftware developmentfor performingnumericalcalculations. It is not a complete listing but is instead a list of numerical libraries with articles on Wikipedia, with few exceptions.
The choice of a typical library depends on a range of requirements such as: desired features (e.g. large dimensional linear algebra, parallel computation, partial differential equations), licensing, readability of API, portability or platform/compiler dependence (e.g. Linux, Windows, Visual C++, GCC), performance, ease-of-use, continued support from developers, standard compliance, specialized optimization in code for specific application scenarios or even the size of the code-base to be installed.
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https://en.wikipedia.org/wiki/List_of_numerical_libraries
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The following is a list ofstatistical software.
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https://en.wikipedia.org/wiki/List_of_statistical_software
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Mathematical softwareissoftwareused tomodel, analyze or calculate numeric, symbolic or geometric data.[1]
Numerical analysisandsymbolic computationhad been in most important place of the subject, but other kind of them is also growing now. A useful mathematical knowledge of such asalgorismwhich exist before theinventionofelectronic computer, helped to mathematical software developing. On the other hand, by the growth ofcomputing power(such as seeing onMoore's law), the new treatment (for example, a new kind of technique such asdata assimilationwhich combined numerical analysis andstatistics) needing conversely the progress of themathematical scienceorapplied mathematics.The progress of mathematical information presentation such asTeXorMathML[2]will demand to evolution formformula manipulation languageto truemathematics manipulation language(notwithstanding the problem that whethermathematical theory is inconsistentor not). And popularization of general purpose mathematical software, special purpose mathematical software[3]so calledone purpose softwarewhich used special subject will alive with adapting for environment progress at normalization of platform. So the diversity of mathematical software will be kept.
A software calculator allows the user to perform simple mathematical operations, like addition, multiplication, exponentiation and trigonometry. Data input is typically manual, and the output is a text label.
Many mathematical suites arecomputer algebra systemsthat usesymbolic mathematics. They are designed to solve classical algebra equations and problems in human readable notation.
Many tools are available for statistical analysis of data. See alsoComparison of statistical packages.
TheNetlibrepository contains various collections of software routines for numerical problems, mostly inFortranandC. Commercial products implementing many different numerical algorithms include theIMSL,NMathandNAG libraries; a free alternative is theGNU Scientific Library. A different approach is taken by theNumerical Recipeslibrary, where emphasis is placed on clear understanding of algorithms.
Manycomputer algebra systems(listed above) can also be used for numerical computations.
Music mathematics software utilizes mathematics to analyze or synthesize musical symbols and patterns.
A growing number of mathematical software is available in web browsers, without the need to download or install any code.[5]
Low-level mathematical libraries intended for use within otherprogramming languages:
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https://en.wikipedia.org/wiki/Mathematical_software
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Web-based simulation(WBS) is the invocation ofcomputer simulationservices over theWorld Wide Web, specifically through aweb browser.[1][2][3][4]Increasingly, the web is being looked upon as an environment for providingmodeling and simulationapplications, and as such, is an emerging area of investigation within the simulation community.[4][5][6]
Web-based simulation is used in several contexts:
Web-based simulation can take place either on the server side or on the client side. Inserver-side simulation, the numerical calculations andvisualization(generation of plots and other computer graphics) is carried out on the web server, while the interactivegraphical user interface(GUI) often partly is provided by the client-side, for example usingserver-side scriptingsuch asPHPorCGI scripts, interactive services based onAjaxor a conventional application software remotely accessed through aVNCJava applet.
Inclient-side simulation, the simulation program is downloaded from the server side but completely executed on the client side, for example usingJava applets,Flash animations,JavaScript, or some mathematical software viewer plug-in. Server-side simulation is not scalable for many simultaneous users, but places fewer demands on the user computer performance and web-browser plug-ins than client-side simulation.
The termon-line simulationsometimes refers to server-side web-based simulation, sometimes tosymbioticsimulation, i.e. a simulation that interacts in real-time with a physical system.
The upcomingcloud-computingtechnologies can be used for new server-side simulation approaches. For instance, there are[example needed]multi-agent-simulationapplications which are deployed on cloud-computing instances and act independently. This allows simulations to be highly scalable.[clarification needed]
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https://en.wikipedia.org/wiki/Web-based_simulation
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Inmathematics,localization of a categoryconsists of adding to acategoryinversemorphismsfor some collection of morphisms, constraining them to becomeisomorphisms. This is formally similar to the process oflocalization of a ring; it in general makes objects isomorphic that were not so before. Inhomotopy theory, for example, there are many examples of mappings that are invertibleup tohomotopy; and so large classes ofhomotopy equivalentspaces[clarification needed].Calculus of fractionsis another name for working in a localized category.
AcategoryCconsists of objects andmorphismsbetween these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replaceCby another categoryC'in which certain morphisms are forced to be isomorphisms. This process is called localization.
For example, in the category ofR-modules(for some fixed commutative ringR) the multiplication by a fixed elementrofRis typically (i.e., unlessris aunit) not an isomorphism:
The category that is most closely related toR-modules, but where this mapisan isomorphism turns out to be the category ofR[S−1]{\displaystyle R[S^{-1}]}-modules. HereR[S−1]{\displaystyle R[S^{-1}]}is thelocalizationofRwith respect to the (multiplicatively closed) subsetSconsisting of all powers ofr,S={1,r,r2,r3,…}{\displaystyle S=\{1,r,r^{2},r^{3},\dots \}}The expression "most closely related" is formalized by two conditions: first, there is afunctor
sending anyR-module to itslocalizationwith respect toS. Moreover, given any categoryCand any functor
sending the multiplication map byron anyR-module (see above) to an isomorphism ofC, there is a unique functor
such thatF=G∘φ{\displaystyle F=G\circ \varphi }.
The above examples of localization ofR-modules is abstracted in the following definition. In this shape, it applies in many more examples, some of which are sketched below.
Given acategoryCand some classWofmorphismsinC, the localizationC[W−1] is another category which is obtained by inverting all the morphisms inW. More formally, it is characterized by auniversal property: there is a natural localization functorC→C[W−1] and given another categoryD, a functorF:C→Dfactors uniquely overC[W−1] if and only ifFsends all arrows inWto isomorphisms.
Thus, the localization of the category is unique up to unique isomorphism of categories, provided that it exists. One construction of the localization is done by declaring that its objects are the same as those inC, but the morphisms are enhanced by adding a formal inverse for each morphism inW. Under suitable hypotheses onW,[1]the morphisms from objectXto objectYare given byroofs
(whereX'is an arbitrary object ofCandfis in the given classWof morphisms), modulo certain equivalence relations. These relations turn the map going in the "wrong" direction into an inverse off. This "calculus of fractions" can be seen as a generalization of the construction of rational numbers as equivalence classes of pairs of integers.
This procedure, however, in general yields aproper classof morphisms betweenXandY. Typically, the morphisms in a category are only allowed to form a set. Some authors simply ignore such set-theoretic issues.
A rigorous construction of localization of categories, avoiding these set-theoretic issues, was one of the initial reasons for the development of the theory ofmodel categories: a model categoryMis a category in which there are three classes of maps; one of these classes is the class ofweak equivalences. Thehomotopy categoryHo(M) is then the localization with respect to the weak equivalences. The axioms of a model category ensure that this localization can be defined without set-theoretical difficulties.
Some authors also define alocalizationof a categoryCto be anidempotentand coaugmented functor. A coaugmented functor is a pair(L,l)whereL:C → Cis anendofunctorandl:Id → Lis a natural transformation from the identity functor toL(called the coaugmentation). A coaugmented functor is idempotent if, for everyX, both mapsL(lX),lL(X):L(X) → LL(X)are isomorphisms. It can be proven that in this case, both maps are equal.[2]
This definition is related to the one given above as follows: applying the first definition, there is, in many situations, not only a canonical functorC→C[W−1]{\displaystyle C\to C[W^{-1}]}, but also a functor in the opposite direction,
For example, modules over the localizationR[S−1]{\displaystyle R[S^{-1}]}of a ring are also modules overRitself, giving a functor
In this case, the composition
is a localization ofCin the sense of an idempotent and coaugmented functor.
Serreintroduced the idea of working inhomotopy theorymodulosome classCofabelian groups. This meant that groupsAandBwere treated as isomorphic, if for exampleA/Blay inC.
In the theory ofmodulesover acommutative ringR, whenRhasKrull dimension≥ 2, it can be useful to treat modulesMandNaspseudo-isomorphicifM/Nhassupportof codimension at least two. This idea is much used inIwasawa theory.
Thederived categoryof anabelian categoryis much used inhomological algebra. It is the localization of the category of chain complexes (up to homotopy) with respect to thequasi-isomorphisms.
Given anabelian categoryAand aSerre subcategoryB,one can define thequotient categoryA/B,which is an abelian category equipped with anexact functorfromAtoA/Bthat isessentially surjectiveand has kernelB.This quotient category can be constructed as a localization ofAby the class of morphisms whose kernel and cokernel are both inB.
Anisogenyfrom anabelian varietyAto another oneBis a surjective morphism with finitekernel. Some theorems on abelian varieties require the idea ofabelian variety up to isogenyfor their convenient statement. For example, given an abelian subvarietyA1ofA, there is another subvarietyA2ofAsuch that
isisogenoustoA(Poincaré's reducibility theorem: see for exampleAbelian VarietiesbyDavid Mumford). To call this adirect sumdecomposition, we should work in the category of abelian varieties up to isogeny.
Thelocalization of a topological space, introduced byDennis Sullivan, produces another topological space whose homology is a localization of the homology of the original space.
A much more general concept fromhomotopical algebra, including as special cases both the localization of spaces and of categories, is theBousfield localizationof amodel category. Bousfield localization forces certain maps to becomeweak equivalences, which is in general weaker than forcing them to become isomorphisms.[3]
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https://en.wikipedia.org/wiki/Abelian_varieties_up_to_isogeny
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Inarithmetic geometry, theSelmer group, named in honor of the work ofErnst Sejersted Selmer(1951) byJohn William Scott Cassels(1962), is agroupconstructed from anisogenyofabelian varieties.
The Selmer group of an abelian varietyAwith respect to anisogenyf:A→Bof abelian varieties can be defined in terms ofGalois cohomologyas
whereAv[f] denotes thef-torsionofAvandκv{\displaystyle \kappa _{v}}is the localKummer mapBv(Kv)/f(Av(Kv))→H1(GKv,Av[f]){\displaystyle B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])}. Note thatH1(GKv,Av[f])/im(κv){\displaystyle H^{1}(G_{K_{v}},A_{v}[f])/\operatorname {im} (\kappa _{v})}isisomorphictoH1(GKv,Av)[f]{\displaystyle H^{1}(G_{K_{v}},A_{v})[f]}. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group haveKv-rational points for all placesvofK. The Selmer group isfinite. This implies that the part of theTate–Shafarevich groupkilled byfis finite due to the followingexact sequence
The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weakMordell–Weil theoremthat itssubgroupB(K)/f(A(K)) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is someprimepsuch that thep-component of the Tate–Shafarevich group is finite. It isconjecturedthat theTate–Shafarevich groupis in fact finite, in which case any primepwould work. However, if (as seems unlikely) the Tate–Shafarevich group has an infinitep-component for every primep, then the procedure may never terminate.
Ralph Greenberg(1994) has generalized the notion of Selmer group to more generalp-adicGalois representationsand top-adic variations ofmotivesin the context ofIwasawa theory.
More generally one can define the Selmer group of a finite Galois moduleM(such as the kernel of an isogeny) as the elements ofH1(GK,M) that have images inside certain given subgroups ofH1(GKv,M).
In his 1954 paperA Conjecture Concerning Rational Points On Cubic Curves,[1]Selmer investigates generators for the rational points on certain cubic curves using two descents. He notes that a method used by Cassels[2]points to an insufficiency in the methods of detecting generators used previously by Selmer. However, the method of Cassels is also insufficient to detect all generators. Selmer examines the situation numerically, and formulates the conjecture:[1]
When a second descent exists, the number of generators found is an even number less than what is indicated by the first descent.
Cassels explores the situation in a series of eight papers, beginning in 1959 withArithmetic on curves of genus 1: I. On a conjecture of Selmer.[3]In the (1962) third paper in the series,Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups,[4]Cassels remarks:
We shall call it a Selmer group because Selmer initiated the present work.
And thus we have the Selmer groups.
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https://en.wikipedia.org/wiki/Selmer_group
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Inmathematics,Siegel modular formsare a major type ofautomorphic form. These generalize conventionalellipticmodular formswhich are closely related toelliptic curves. The complex manifolds constructed in the theory of Siegel modular forms areSiegel modular varieties, which are basic models for what amoduli spacefor abelian varieties (with some extralevel structure) should be and are constructed as quotients of theSiegel upper half-spacerather than theupper half-planebydiscrete groups.
Siegel modular forms areholomorphic functionson the set ofsymmetricn×nmatrices withpositive definiteimaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. asspecial functionsofseveral complex variables.
Siegel modular forms were first investigated byCarl Ludwig Siegel(1939) for the purpose of studyingquadratic formsanalytically. These primarily arise in various branches ofnumber theory, such asarithmetic geometryandelliptic cohomology. Siegel modular forms have also been used in some areas ofphysics, such asconformal field theoryandblack hole thermodynamicsinstring theory.
Letg,N∈N{\displaystyle g,N\in \mathbb {N} }and define
theSiegel upper half-space. Define thesymplectic groupof levelN{\displaystyle N}, denoted byΓg(N),{\displaystyle \Gamma _{g}(N),}as
whereIg{\displaystyle I_{g}}is theg×g{\displaystyle g\times g}identity matrix. Finally, let
be arational representation, whereV{\displaystyle V}is a finite-dimensional complexvector space.
Given
and
define the notation
Then aholomorphic function
is aSiegel modular formof degreeg{\displaystyle g}(sometimes called the genus), weightρ{\displaystyle \rho }, and levelN{\displaystyle N}if
for allγ∈Γg(N){\displaystyle \gamma \in \Gamma _{g}(N)}.
In the case thatg=1{\displaystyle g=1}, we further require thatf{\displaystyle f}be holomorphic 'at infinity'. This assumption is not necessary forg>1{\displaystyle g>1}due to the Koecher principle, explained below. Denote the space of weightρ{\displaystyle \rho }, degreeg{\displaystyle g}, and levelN{\displaystyle N}Siegel modular forms by
Some methods for constructing Siegel modular forms include:
For degree 1, the level 1 Siegel modular forms are the same as level 1 modular forms. The ring of such forms is a polynomial ringC[E4,E6] in the (degree 1) Eisenstein seriesE4andE6.
For degree 2, (Igusa1962,1967) showed that the ring of level 1 Siegel modular forms is generated by the (degree 2) Eisenstein seriesE4andE6and 3 more forms of weights 10, 12, and 35. The ideal of relations between them is generated by the square of the weight 35 form minus a certain polynomial in the others.
For degree 3,Tsuyumine (1986)described the ring of level 1 Siegel modular forms, giving a set of 34 generators.
For degree 4, the level 1 Siegel modular forms of small weights have been found. There are no cusp forms of weights 2, 4, or 6. The space of cusp forms of weight 8 is 1-dimensional, spanned by theSchottky form. The space of cusp forms of weight 10 has dimension 1, the space of cusp forms of weight 12 has dimension 2, the space of cusp forms of weight 14 has dimension 3, and the space of cusp forms of weight 16 has dimension 7 (Poor & Yuen 2007)harv error: no target: CITEREFPoorYuen2007 (help).
For degree 5, the space of cusp forms has dimension 0 for weight 10, dimension 2 for weight 12. The space of forms of weight 12 has dimension 5.
For degree 6, there are no cusp forms of weights 0, 2, 4, 6, 8. The space of Siegel modular forms of weight 2 has dimension 0, and those of weights 4 or 6 both have dimension 1.
For small weights and level 1,Duke & Imamoḡlu (1998)give the following results (for any positive degree):
The following table combines the results above with information fromPoor & Yuen (2006)harvtxt error: no target: CITEREFPoorYuen2006 (help)andChenevier & Lannes (2014)andTaïbi (2014).
The theorem known as theKoecher principlestates that iff{\displaystyle f}is a Siegel modular form of weightρ{\displaystyle \rho }, level 1, and degreeg>1{\displaystyle g>1}, thenf{\displaystyle f}is bounded on subsets ofHg{\displaystyle {\mathcal {H}}_{g}}of the form
whereϵ>0{\displaystyle \epsilon >0}. Corollary to this theorem is the fact that Siegel modular forms of degreeg>1{\displaystyle g>1}haveFourier expansionsand are thus holomorphic at infinity.[1]
In the D1D5P system ofsupersymmetric black holesin string theory, the function that naturally captures the microstates of black hole entropy is a Siegel modular form.[2]In general, Siegel modular forms have been described as having the potential to describe black holes or other gravitational systems.[2]
Siegel modular forms also have uses as generating functions for families of CFT2 with increasing central charge inconformal field theory, particularly the hypotheticalAdS/CFT correspondence.[3]
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Inmathematics, arigidcollectionCofmathematical objects(for instance sets or functions) is one in which everyc∈Cis uniquely determined by less information aboutcthan one would expect.
The above statement does not define amathematical property; instead, it describes in what sense the adjective "rigid" is typically used in mathematics, by mathematicians.
Some examples include:
Incombinatorics, the term rigid is also used to define the notion of arigid surjection, which is asurjectionf:n→m{\displaystyle f:n\to m}for which the following equivalent conditions hold:[1]
This relates to the above definition of rigid, in that each rigid surjectionf{\displaystyle f}uniquely defines, and is uniquely defined by, apartitionofn{\displaystyle n}intom{\displaystyle m}pieces. Given a rigid surjectionf{\displaystyle f}, the partition is defined byn=f−1(0)⊔⋯⊔f−1(m−1){\displaystyle n=f^{-1}(0)\sqcup \cdots \sqcup f^{-1}(m-1)}. Conversely, given a partition ofn=A0⊔⋯⊔Am−1{\displaystyle n=A_{0}\sqcup \cdots \sqcup A_{m-1}}, order theAi{\displaystyle A_{i}}by lettingAi≺Aj⟺minAi<minAj{\displaystyle A_{i}\prec A_{j}\iff \min A_{i}<\min A_{j}}. Ifn=B0⊔⋯⊔Bm−1{\displaystyle n=B_{0}\sqcup \cdots \sqcup B_{m-1}}is now the≺{\displaystyle \prec }-ordered partition, the functionf:n→m{\displaystyle f:n\to m}defined byf(i)=j⟺i∈Bj{\displaystyle f(i)=j\iff i\in B_{j}}is a rigid surjection.
This article incorporates material from rigid onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
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https://en.wikipedia.org/wiki/Rigidity_(mathematics)
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Local rigiditytheorems in the theory of discrete subgroups ofLie groupsare results which show that small deformations of certain such subgroups are always trivial. It is different fromMostow rigidityand weaker (but holds more frequently) thansuperrigidity.
The first such theorem was proven byAtle Selbergfor co-compact discrete subgroups of the unimodular groupsSLn(R){\displaystyle \mathrm {SL} _{n}(\mathbb {R} )}.[1]Shortly afterwards a similar statement was proven byEugenio Calabiin the setting of fundamental groups of compact hyperbolic manifolds. Finally, the theorem was extended to all co-compact subgroups of semisimple Lie groups byAndré Weil.[2][3]The extension to non-cocompact lattices was made later by Howard Garland andMadabusi Santanam Raghunathan.[4]The result is now sometimes referred to as Calabi—Weil (or just Weil) rigidity.
LetΓ{\displaystyle \Gamma }be a groupgeneratedby a finite number of elementsg1,…,gn{\displaystyle g_{1},\ldots ,g_{n}}andG{\displaystyle G}a Lie group. Then the mapHom(Γ,G)→Gn{\displaystyle \mathrm {Hom} (\Gamma ,G)\to G^{n}}defined byρ↦(ρ(g1),…,ρ(gn)){\displaystyle \rho \mapsto (\rho (g_{1}),\ldots ,\rho (g_{n}))}is injective and this endowsHom(Γ,G){\displaystyle \mathrm {Hom} (\Gamma ,G)}with a topologyinducedby that ofGn{\displaystyle G^{n}}. IfΓ{\displaystyle \Gamma }is a subgroup ofG{\displaystyle G}then adeformationofΓ{\displaystyle \Gamma }is any element inHom(Γ,G){\displaystyle \mathrm {Hom} (\Gamma ,G)}. Two representationsϕ,ψ{\displaystyle \phi ,\psi }are said to be conjugated if there exists ag∈G{\displaystyle g\in G}such thatϕ(γ)=gψ(γ)g−1{\displaystyle \phi (\gamma )=g\psi (\gamma )g^{-1}}for allγ∈Γ{\displaystyle \gamma \in \Gamma }. See alsocharacter variety.
The simplest statement is whenΓ{\displaystyle \Gamma }is a lattice in a simple Lie groupG{\displaystyle G}and the latter is not locally isomorphic toSL2(R){\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}orSL2(C){\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}andΓ{\displaystyle \Gamma }(this means that its Lie algebra is not that of one of these two groups).
Whenever such a statement holds for a pairG⊃Γ{\displaystyle G\supset \Gamma }we will say that local rigidity holds.
Local rigidity holds for cocompact lattices inSL2(C){\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}. A latticeΓ{\displaystyle \Gamma }inSL2(C){\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}which is not cocompact has nontrivial deformations coming from Thurston'shyperbolic Dehn surgerytheory. However, if one adds the restriction that a representation must send parabolic elements inΓ{\displaystyle \Gamma }to parabolic elements then local rigidity holds.
In this case local rigidity never holds (exceptcocompacttriangle groups). For cocompact lattices a small deformation remains a cocompact lattice but it may not be conjugated to the original one (seeTeichmüller spacefor more detail). Non-cocompact lattices are virtually free and hence have non-lattice deformations.
Local rigidity holds for lattices insemisimple Lie groupsproviding the latter have no factor of type A1 (i.e. locally isomorphic toSL2(R){\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}orSL2(C){\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}) or the former is irreducible.
There are also local rigidity results where the ambient group is changed, even in case where superrigidity fails. For example, ifΓ{\displaystyle \Gamma }is a lattice in theunitary groupSU(n,1){\displaystyle \mathrm {SU} (n,1)}andn≥2{\displaystyle n\geq 2}then the inclusionΓ⊂SU(n,1)⊂SU(n+1,1){\displaystyle \Gamma \subset \mathrm {SU} (n,1)\subset \mathrm {SU} (n+1,1)}is locally rigid.[5]
A uniform latticeΓ{\displaystyle \Gamma }in any compactly generated topological groupG{\displaystyle G}istopologically locally rigid, in the sense that any sufficiently small deformationφ{\displaystyle \varphi }of the inclusioni:Γ⊂G{\displaystyle i:\Gamma \subset G}is injective andφ(Γ){\displaystyle \varphi (\Gamma )}is a uniform lattice inG{\displaystyle G}. An irreducible uniform lattice in the isometry group of any proper geodesically completeCAT(0){\displaystyle \mathrm {CAT} (0)}-space not isometric to the hyperbolic plane and without Euclidean factors is locally rigid.[6]
Weil's original proof is by relating deformations of a subgroupΓ{\displaystyle \Gamma }inG{\displaystyle G}to the firstcohomologygroup ofΓ{\displaystyle \Gamma }with coefficients in the Lie algebra ofG{\displaystyle G}, and then showing that this cohomology vanishes for cocompact lattices whenG{\displaystyle G}has no simple factor of absolute type A1. A more geometric proof which also work in the non-compact cases usesCharles Ehresmann(andWilliam Thurston's) theory of(G,X){\displaystyle (G,X)}structures.[7]
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Given atopological spaceand agroupactingon it, the images of a single point under the group action form anorbitof the action. Afundamental domainorfundamental regionis a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be aconnectedsubset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action thentilethe space. One general construction of fundamental domains usesVoronoi cells.
Given anactionof agroupGon atopological spaceXbyhomeomorphisms, a fundamental domain for this action is a setDof representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. One typical condition is thatDisalmostan open set, in the sense thatDis thesymmetric differenceof an open set inXwith a set ofmeasure zero, for a certain (quasi)invariantmeasureonX. A fundamental domain always contains afree regular setU, anopen setmoved around byGintodisjointcopies, and nearly as good asDin representing the orbits. FrequentlyDis required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero. This is a typical situation inergodic theory. If a fundamental domain is used to calculate anintegralonX/G, sets of measure zero do not matter.
For example, whenXisEuclidean spaceRnof dimensionn, andGis thelatticeZnacting on it by translations, the quotientX/Gis then-dimensionaltorus. A fundamental domainDhere can be taken to be [0,1)n, which differs from the open set (0,1)nby a set of measure zero, or theclosedunit cube [0,1]n, whoseboundaryconsists of the points whose orbit has more than one representative inD.
Examples in the three-dimensional Euclidean spaceR3.
In the case of translational symmetry combined with other symmetries, the fundamental domain is part of the primitive cell. For example, forwallpaper groupsthe fundamental domain is a factor 1, 2, 3, 4, 6, 8, or 12 smaller than the primitive cell.
The diagram to the right shows part of the construction of the fundamental domain for the action of themodular groupΓ on theupper half-planeH.
This famous diagram appears in all classical books onmodular functions. (It was probably well known toC. F. Gauss, who dealt with fundamental domains in the guise of thereduction theoryofquadratic forms.) Here, each triangular region (bounded by the blue lines) is afree regular setof the action of Γ onH. The boundaries (the blue lines) are not a part of the free regular sets. To construct a fundamental domain ofH/Γ, one must also consider how to assign points on the boundary, being careful not to double-count such points. Thus, the free regular set in this example is
The fundamental domain is built by adding the boundary on the left plus half the arc on the bottom including the point in the middle:
The choice of which points of the boundary to include as a part of the fundamental domain is arbitrary, and varies from author to author.
The core difficulty of defining the fundamental domain lies not so much with the definition of the setper se, but rather with how to treat integrals over the fundamental domain, when integrating functions with poles and zeros on the boundary of the domain.
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Inmathematics, ahomothety(orhomothecy, orhomogeneous dilation) is atransformationof anaffine spacedetermined by a pointScalled itscenterand a nonzero numberkcalled itsratio, which sends pointXto a pointX′by the rule,[1]
Using position vectors:
In case ofS=O{\displaystyle S=O}(Origin):
which is auniform scalingand shows the meaning of special choices fork{\displaystyle k}:
For1/k{\displaystyle 1/k}one gets theinversemapping defined byk{\displaystyle k}.
InEuclidean geometryhomotheties are thesimilaritiesthat fix a point and either preserve (ifk>0{\displaystyle k>0}) or reverse (ifk<0{\displaystyle k<0}) the direction of all vectors. Together with thetranslations, all homotheties of an affine (or Euclidean) space form agroup, the group ofdilationsorhomothety-translations. These are precisely theaffine transformationswith the property that the image of every linegis a lineparalleltog.
Inprojective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwiseinvariant.[2]
In Euclidean geometry, a homothety of ratiok{\displaystyle k}multipliesdistancesbetween points by|k|{\displaystyle |k|},areasbyk2{\displaystyle k^{2}}and volumes by|k|3{\displaystyle |k|^{3}}. Herek{\displaystyle k}is theratio of magnificationordilation factororscale factororsimilitude ratio. Such a transformation can be called anenlargementif the scale factor exceeds 1. The above-mentioned fixed pointSis calledhomothetic centerorcenter of similarityorcenter of similitude.
The term, coined by French mathematicianMichel Chasles, is derived from twoGreekelements: the prefixhomo-(όμο'similar'}; andtransl.grc– transl.thesis(Θέσις)'position'). It describes the relationship between two figures of the same shape and orientation. For example, twoRussian dollslooking in the same direction can be considered homothetic.
Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops.
The following properties hold in any dimension.
A homothety has the following properties:
Both properties show:
Derivation of the properties:In order to make calculations easy it is assumed that the centerS{\displaystyle S}is the origin:x→kx{\displaystyle \mathbf {x} \to k\mathbf {x} }. A lineg{\displaystyle g}with parametric representationx=p+tv{\displaystyle \mathbf {x} =\mathbf {p} +t\mathbf {v} }is mapped onto the point setg′{\displaystyle g'}with equationx=k(p+tv)=kp+tkv{\displaystyle \mathbf {x} =k(\mathbf {p} +t\mathbf {v} )=k\mathbf {p} +tk\mathbf {v} }, which is a line parallel tog{\displaystyle g}.
The distance of two pointsP:p,Q:q{\displaystyle P:\mathbf {p} ,\;Q:\mathbf {q} }is|p−q|{\displaystyle |\mathbf {p} -\mathbf {q} |}and|kp−kq|=|k||p−q|{\displaystyle |k\mathbf {p} -k\mathbf {q} |=|k||\mathbf {p} -\mathbf {q} |}the distance between their images. Hence, theratio(quotient) of two line segments remains unchanged.
In case ofS≠O{\displaystyle S\neq O}the calculation is analogous but a little extensive.
Consequences: A triangle is mapped on asimilarone. The homothetic image of acircleis a circle. The image of anellipseis a similar one. i.e. the ratio of the two axes is unchanged.
If for a homothety with centerS{\displaystyle S}the imageQ1{\displaystyle Q_{1}}of a pointP1{\displaystyle P_{1}}is given (see diagram) then the imageQ2{\displaystyle Q_{2}}of a second pointP2{\displaystyle P_{2}}, which lies not on lineSP1{\displaystyle SP_{1}}can be constructed graphically using the intercept theorem:Q2{\displaystyle Q_{2}}is the common point th two linesP1P2¯{\displaystyle {\overline {P_{1}P_{2}}}}andSP2¯{\displaystyle {\overline {SP_{2}}}}. The image of a point collinear withP1,Q1{\displaystyle P_{1},Q_{1}}can be determined usingP2,Q2{\displaystyle P_{2},Q_{2}}.
Before computers became ubiquitous, scalings of drawings were done by using apantograph, a tool similar to acompass.
Construction and geometrical background:
Because of|SQ0|/|SP0|=|Q0Q|/|PP0|{\displaystyle |SQ_{0}|/|SP_{0}|=|Q_{0}Q|/|PP_{0}|}(see diagram) one gets from theintercept theoremthat the pointsS,P,Q{\displaystyle S,P,Q}are collinear (lie on a line) and equation|SQ|=k|SP|{\displaystyle |SQ|=k|SP|}holds. That shows: the mappingP→Q{\displaystyle P\to Q}is a homothety with centerS{\displaystyle S}and ratiok{\displaystyle k}.
Derivation:
For the compositionσ2σ1{\displaystyle \sigma _{2}\sigma _{1}}of the two homothetiesσ1,σ2{\displaystyle \sigma _{1},\sigma _{2}}with centersS1,S2{\displaystyle S_{1},S_{2}}with
one gets by calculation for the image of pointX:x{\displaystyle X:\mathbf {x} }:
Hence, the composition is
is afixpoint(is not moved) and the composition
is ahomothetywith centerS3{\displaystyle S_{3}}and ratiok1k2{\displaystyle k_{1}k_{2}}.S3{\displaystyle S_{3}}lies on lineS1S2¯{\displaystyle {\overline {S_{1}S_{2}}}}.
Derivation:
The composition of the homothety
which is a homothety with centers′=s+v1−k{\displaystyle \mathbf {s} '=\mathbf {s} +{\frac {\mathbf {v} }{1-k}}}and ratiok{\displaystyle k}.
The homothetyσ:x→s+k(x−s){\displaystyle \sigma :\mathbf {x} \to \mathbf {s} +k(\mathbf {x} -\mathbf {s} )}with centerS=(u,v){\displaystyle S=(u,v)}can be written as the composition of a homothety with centerO{\displaystyle O}and a translation:
Henceσ{\displaystyle \sigma }can be represented inhomogeneous coordinatesby the matrix:
A pure homothetylinear transformationis alsoconformalbecause it is composed of translation and uniform scale.
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Abelian varietiesare a natural generalization ofelliptic curvesto higher dimensions. However, unlike the case of elliptic curves, there is no well-behaved stack playing the role of amoduli stackfor higher-dimensional abelian varieties.[1]One can solve this problem by constructing a moduli stack of abelian varieties equipped with extra structure, such as aprincipal polarisation. Just as there is amoduli stackof elliptic curves overC{\displaystyle \mathbb {C} }constructed as a stacky quotient of theupper-half planeby the action ofSL2(Z){\displaystyle SL_{2}(\mathbb {Z} )},[2]there is a moduli space of principally polarised abelian varieties given as a stacky quotient ofSiegel upper half-spaceby thesymplectic groupSp2g(Z){\displaystyle \operatorname {Sp} _{2g}(\mathbb {Z} )}.[3]By adding even more extra structure, such as a levelnstructure, one can go further and obtain afine moduli space.
Recall that theSiegel upper half-spaceHg{\displaystyle H_{g}}is the set of symmetricg×g{\displaystyle g\times g}complex matrices whose imaginary part is positive definite.[4]This an open subset in the space ofg×g{\displaystyle g\times g}symmetric matrices. Notice that ifg=1{\displaystyle g=1},Hg{\displaystyle H_{g}}consists of complex numbers with positive imaginary part, and is thus the upper half plane, which appears prominently in the study of elliptic curves. In general, any pointΩ∈Hg{\displaystyle \Omega \in H_{g}}gives a complex torus
XΩ=Cg/(ΩZg+Zg){\displaystyle X_{\Omega }=\mathbb {C} ^{g}/(\Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g})}
with a principal polarizationHΩ{\displaystyle H_{\Omega }}from the matrixΩ−1{\displaystyle \Omega ^{-1}}[3]page 34. It turns out all principally polarized Abelian varieties arise this way, givingHg{\displaystyle H_{g}}the structure of a parameter space for all principally polarized Abelian varieties. But, there exists an equivalence where
XΩ≅XΩ′⟺Ω=MΩ′{\displaystyle X_{\Omega }\cong X_{\Omega '}\iff \Omega =M\Omega '}forM∈Sp2g(Z){\displaystyle M\in \operatorname {Sp} _{2g}(\mathbb {Z} )}
hence the moduli space of principally polarized abelian varieties is constructed from thestack quotient
Ag=[Sp2g(Z)∖Hg]{\displaystyle {\mathcal {A}}_{g}=[\operatorname {Sp} _{2g}(\mathbb {Z} )\backslash H_{g}]}
which gives aDeligne-Mumford stackoverSpec(C){\displaystyle \operatorname {Spec} (\mathbb {C} )}. If this is instead given by aGIT quotient, then it gives the coarse moduli spaceAg{\displaystyle A_{g}}.
In many cases, it is easier to work with principally polarized Abelian varieties equipped with leveln-structure because this breaks the symmetries and gives a moduli space instead of a moduli stack.[5][6]This means the functor is representable by an algebraic manifold, such as avarietyorscheme, instead of a stack. Aleveln-structureis given by a fixed basis of
whereL{\displaystyle L}is the latticeΩZg+Zg⊂C2g{\displaystyle \Omega \mathbb {Z} ^{g}+\mathbb {Z} ^{g}\subset \mathbb {C} ^{2g}}. Fixing such a basis removes the automorphisms of an abelian variety at a point in the moduli space, hence there exists a bona fide algebraic manifold without a stabilizer structure. Denote
Γ(n)=ker[Sp2g(Z)→Sp2g(Z/n)]{\displaystyle \Gamma (n)=\ker[\operatorname {Sp} _{2g}(\mathbb {Z} )\to \operatorname {Sp} _{2g}(\mathbb {Z} /n)]}
and define
Ag,n=Γ(n)∖Hg{\displaystyle A_{g,n}=\Gamma (n)\backslash H_{g}}
as a quotient variety.
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Innumber theory, aShimura varietyis a higher-dimensional analogue of amodular curvethat arises as a quotientvarietyof aHermitian symmetric spaceby acongruence subgroupof areductive algebraic groupdefined overQ. Shimura varieties are notalgebraic varietiesbut are families of algebraic varieties.Shimura curvesare the one-dimensional Shimura varieties.Hilbert modular surfacesandSiegel modular varietiesare among the best known classes of Shimura varieties.
Special instances of Shimura varieties were originally introduced byGoro Shimurain the course of his generalization of thecomplex multiplicationtheory. Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit modelsdefinedover anumber field, thereflex fieldof the Shimura variety. In the 1970s,Pierre Delignecreated an axiomatic framework for the work of Shimura. In 1979,Robert Langlandsremarked that Shimura varieties form a natural realm of examples for which equivalence betweenmotivicandautomorphicL-functionspostulated in theLanglands programcan be tested.Automorphic formsrealized in thecohomologyof a Shimura variety are more amenable to study than general automorphic forms; in particular, there is a construction attachingGalois representationsto them.[1]
LetS= ResC/RGmbe theWeil restrictionof the multiplicative group fromcomplex numberstoreal numbers. It is a realalgebraic group, whose group ofR-points,S(R), isC*and group ofC-points isC*×C*. AShimura datumis a pair (G,X) consisting of a (connected)reductive algebraic groupGdefined over the fieldQofrational numbersand aG(R)-conjugacy classXofhomomorphismsh:S→GRsatisfying the following axioms:
It follows from these axioms thatXhas a unique structure of acomplex manifold(possibly, disconnected) such that for every representationρ:GR→GL(V), the family (V,ρ⋅h) is a holomorphic family ofHodge structures; moreover, it forms a variation of Hodge structure, andXis a finite disjoint union ofhermitian symmetric domains.
LetAƒbe thering of finite adelesofQ. For every sufficiently small compact open subgroupKofG(Aƒ), thedouble cosetspace
is a finite disjoint union oflocally symmetric varietiesof the formΓi∖X+{\displaystyle \Gamma _{i}\backslash X^{+}}, where the plus superscript indicates aconnected component. The varieties ShK(G,X) are complex algebraic varieties and they form aninverse systemover all sufficiently small compact open subgroupsK. This inverse system
admits a natural right action ofG(Aƒ). It is called theShimura varietyassociated with the Shimura datum (G,X) and denoted Sh(G,X).
For special types of hermitian symmetric domains andcongruence subgroupsΓ,algebraic varietiesof the form Γ \X= ShK(G,X) and theircompactificationswere introduced in a series of papers ofGoro Shimuraduring the 1960s. Shimura's approach, later presented in his monograph, was largely phenomenological, pursuing the widest generalizations of the reciprocity law formulation ofcomplex multiplicationtheory. In retrospect, the name "Shimura variety" was introduced byDeligne, who proceeded to isolate the abstract features that played a role in Shimura's theory. In Deligne's formulation, Shimura varieties are parameter spaces of certain types ofHodge structures. Thus they form a natural higher-dimensional generalization ofmodular curvesviewed asmoduli spacesofelliptic curveswith level structure. In many cases, the moduli problems to which Shimura varieties are solutions have been likewise identified.
LetFbe a totally real number field andDaquaterniondivision algebraoverF. The multiplicative groupD×gives rise to a canonical Shimura variety. Its dimensiondis the number of infinite places over whichDsplits. In particular, ifd= 1 (for example, ifF=QandD⊗R≅ M2(R)), fixing a sufficiently smallarithmetic subgroupofD×, one gets a Shimura curve, and curves arising from this construction are already compact (i.e.projective).
Some examples of Shimura curves with explicitly known equations are given by theHurwitz curvesof low genus:
and by theFermat curveof degree 7.[2]
Other examples of Shimura varieties includePicard modular surfacesandHilbert modular surfaces, also known as Hilbert–Blumenthal varieties.
Each Shimura variety can be defined over a canonicalnumber fieldEcalled thereflex field. This important result due to Shimura shows that Shimura varieties, whicha prioriare only complex manifolds, have an algebraicfield of definitionand, therefore, arithmetical significance. It forms the starting point in his formulation of the reciprocity law, where an important role is played by certain arithmetically definedspecial points.
The qualitative nature of theZariski closureof sets of special points on a Shimura variety is described by theAndré–Oort conjecture. Conditional results have been obtained on this conjecture, assuming ageneralized Riemann hypothesis.[3]
Shimura varieties play an outstanding role in theLanglands program. The prototypical theorem, theEichler–Shimura congruence relation, implies that theHasse–Weil zeta functionof a modular curve is a product of L-functions associated to explicitly determinedmodular formsof weight 2. Indeed, it was in the process of generalization of this theorem that Goro Shimura introduced his varieties and proved his reciprocity law. Zeta functions of Shimura varieties associated with the groupGL2over other number fields and its inner forms (i.e. multiplicative groups of quaternion algebras) were studied by Eichler, Shimura, Kuga, Sato, and Ihara. On the basis of their results,Robert Langlandsmade a prediction that the Hasse-Weil zeta function of anyalgebraic varietyWdefined over a number field would be a product of positive and negative powers of automorphic L-functions, i.e. it should arise from a collection ofautomorphic representations.[1]However philosophically natural it may be to expect such a description, statements of this type have only been proved whenWis a Shimura variety.[4]In the words of Langlands:
To show that all L-functions associated to Shimura varieties – thus to any motive defined by a Shimura variety – can be expressed in terms of the automorphic L-functions of [his paper of 1970] is weaker, even very much weaker, than to show that all motivic L-functions are equal to such L-functions. Moreover, although the stronger statement is expected to be valid, there is, so far as I know, no very compelling reason to expect that all motivic L-functions will be attached to Shimura varieties.[5]
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Innumber theoryandalgebraic geometry, amodular curveY(Γ) is aRiemann surface, or the correspondingalgebraic curve, constructed as aquotientof the complexupper half-planeHby theactionof acongruence subgroupΓ of themodular groupof integral 2×2 matrices SL(2,Z). The term modular curve can also be used to refer to thecompactified modular curvesX(Γ) which arecompactificationsobtained by adding finitely many points (called thecusps of Γ) to this quotient (via an action on theextended complex upper-half plane). The points of a modular curveparametrizeisomorphism classes ofelliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference tocomplex numbers, and, moreover, prove that modular curves aredefinedeither over the field ofrational numbersQor acyclotomic fieldQ(ζn). The latter fact and its generalizations are of fundamental importance in number theory.
The modular group SL(2,Z) acts on the upper half-plane byfractional linear transformations. The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2,Z), i.e. a subgroup containing theprincipal congruence subgroup of levelNfor some positive integerN, which is defined to be
The minimal suchNis called thelevel of Γ. Acomplex structurecan be put on the quotient Γ\Hto obtain anoncompactRiemann surface called amodular curve, and commonly denotedY(Γ).
A common compactification ofY(Γ) is obtained by adding finitely many points called the cusps of Γ. Specifically, this is done by considering the action of Γ on theextended complex upper-half planeH* =H∪Q∪ {∞}. We introduce a topology onH* by taking as a basis:
This turnsH* into a topological space which is a subset of theRiemann sphereP1(C). The group Γ acts on the subsetQ∪ {∞}, breaking it up into finitely manyorbitscalled thecusps of Γ. If Γ acts transitively onQ∪ {∞}, the space Γ\H* becomes theAlexandroff compactificationof Γ\H. Once again, a complex structure can be put on the quotient Γ\H* turning it into a Riemann surface denotedX(Γ) which is nowcompact. This space is a compactification ofY(Γ).[1]
The most common examples are the curvesX(N),X0(N), andX1(N) associated with the subgroups Γ(N), Γ0(N), and Γ1(N).
The modular curveX(5) has genus 0: it is the Riemann sphere with 12 cusps located at the vertices of a regularicosahedron. The coveringX(5) →X(1) is realized by the action of theicosahedral groupon the Riemann sphere. This group is a simple group of order 60 isomorphic toA5and PSL(2, 5).
The modular curveX(7) is theKlein quarticof genus 3 with 24 cusps. It can be interpreted as a surface with three handles tiled by 24 heptagons, with a cusp at the center of each face. These tilings can be understood viadessins d'enfantsandBelyi functions– the cusps are the points lying over ∞ (red dots), while the vertices and centers of the edges (black and white dots) are the points lying over 0 and 1. The Galois group of the coveringX(7) →X(1) is a simple group of order 168 isomorphic toPSL(2, 7).
There is an explicit classical model forX0(N), theclassical modular curve; this is sometimes calledthemodular curve. The definition of Γ(N) can be restated as follows: it is the subgroup of the modular group which is the kernel of the reductionmoduloN. Then Γ0(N) is the larger subgroup of matrices which are upper triangular moduloN:
and Γ1(N) is the intermediate group defined by:
These curves have a direct interpretation asmoduli spacesforelliptic curveswithlevel structureand for this reason they play an important role inarithmetic geometry. The levelNmodular curveX(N) is the moduli space for elliptic curves with a basis for theN-torsion. ForX0(N) andX1(N), the level structure is, respectively, a cyclic subgroup of orderNand a point of orderN. These curves have been studied in great detail, and in particular, it is known thatX0(N) can be defined overQ.
The equations defining modular curves are the best-known examples ofmodular equations. The "best models" can be very different from those taken directly fromelliptic functiontheory.Hecke operatorsmay be studied geometrically, ascorrespondencesconnecting pairs of modular curves.
Quotients ofHthatarecompact do occur forFuchsian groupsΓ other than subgroups of the modular group; a class of them constructed fromquaternion algebrasis also of interest in number theory.
The coveringX(N) →X(1) is Galois, with Galois group SL(2,N)/{1, −1}, which is equal to PSL(2,N) ifNis prime. Applying theRiemann–Hurwitz formulaandGauss–Bonnet theorem, one can calculate the genus ofX(N). For aprimelevelp≥ 5,
where χ = 2 − 2gis theEuler characteristic, |G| = (p+1)p(p−1)/2 is the order of the group PSL(2,p), andD= π − π/2 − π/3 − π/pis theangular defectof the spherical (2,3,p) triangle. This results in a formula
ThusX(5) has genus 0,X(7) has genus 3, andX(11) has genus 26. Forp= 2 or 3, one must additionally take into account the ramification, that is, the presence of orderpelements in PSL(2,Z), and the fact that PSL(2, 2) has order 6, rather than 3. There is a more complicated formula for the genus of the modular curveX(N) of any levelNthat involves divisors ofN.
In general amodular function fieldis afunction fieldof a modular curve (or, occasionally, of some othermoduli spacethat turns out to be anirreducible variety).Genuszero means such a function field has a singletranscendental functionas generator: for example thej-functiongenerates the function field ofX(1) = PSL(2,Z)\H*. The traditional name for such a generator, which is unique up to aMöbius transformationand can be appropriately normalized, is aHauptmodul(mainorprincipal modular function, pluralHauptmoduln).
The spacesX1(n) have genus zero forn= 1, ..., 10 andn= 12. Since each of these curves is defined overQand has aQ-rational point, it follows that there are infinitely many rational points on each such curve, and hence infinitely many elliptic curves defined overQwithn-torsion for these values ofn. The converse statement, that only these values ofncan occur, isMazur's torsion theorem.
The modular curvesX0(N){\displaystyle \textstyle X_{0}(N)}are of genus one if and only ifN{\displaystyle \textstyle N}equals one of the 12 values listed in the following table.[2]Aselliptic curvesoverQ{\displaystyle \mathbb {Q} }, they have minimal, integral Weierstrass modelsy2+a1xy+a3y=x3+a2x2+a4x+a6{\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}. This is,aj∈Z{\displaystyle \textstyle a_{j}\in \mathbb {Z} }and the absolute value of the discriminantΔ{\displaystyle \Delta }is minimal among all integral Weierstrass models for the same curve. The following table contains the uniquereduced, minimal, integral Weierstrass models, which meansa1,a3∈{0,1}{\displaystyle \textstyle a_{1},a_{3}\in \{0,1\}}anda2∈{−1,0,1}{\displaystyle \textstyle a_{2}\in \{-1,0,1\}}.[3]The last column of this table refers to the home page of the respective elliptic modular curveX0(N){\displaystyle \textstyle X_{0}(N)}onThe L-functions and modular forms database (LMFDB).
Modular curves of genus 0, which are quite rare, turned out to be of major importance in relation with themonstrous moonshineconjectures. First several coefficients ofq-expansions of their Hauptmoduln were computed already in the 19th century, but it came as a shock that the same large integers show up as dimensions of representations of the largest sporadic simple group Monster.
Another connection is that the modular curve corresponding to thenormalizerΓ0(p)+ofΓ0(p) in SL(2,R) has genus zero if and only ifpis 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71, and these are preciselysupersingular primes in moonshine theory, i.e. the prime factors of the order of themonster group. The result about Γ0(p)+is due toJean-Pierre Serre,Andrew OggandJohn G. Thompsonin the 1970s, and the subsequent observation relating it to the monster group is due to Ogg, who wrote up a paper offering a bottle ofJack Daniel'swhiskey to anyone who could explain this fact, which was a starting point for the theory of monstrous moonshine.[4]
The relation runs very deep and, as demonstrated byRichard Borcherds, it also involvesgeneralized Kac–Moody algebras. Work in this area underlined the importance ofmodularfunctionsthat are meromorphic and can have poles at the cusps, as opposed tomodularforms, that are holomorphic everywhere, including the cusps, and had been the main objects of study for the better part of the 20th century.
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Inmathematics,elliptic cohomologyis acohomology theoryin the sense ofalgebraic topology. It is related toelliptic curvesandmodular forms.
Historically, elliptic cohomology arose from the study ofelliptic genera. It was known by Atiyah and Hirzebruch that ifS1{\displaystyle S^{1}}acts smoothly and non-trivially on a spin manifold, then the index of theDirac operatorvanishes. In 1983,Wittenconjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerningS1{\displaystyle S^{1}}-actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. In turn, Witten related these to (conjectural) index theory onfree loopspaces. Elliptic cohomology, invented in its original form byLandweber,StongandRavenelin the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differential operators on free loop spaces. In some sense it can be seen as an approximation to theK-theoryof the free loop space.
Call a cohomology theoryA∗{\displaystyle A^{*}}even periodic ifAi=0{\displaystyle A^{i}=0}for i odd and there is an invertible elementu∈A2{\displaystyle u\in A^{2}}. These theories possess acomplex orientation, which gives aformal group law. A particularly rich source for formal group laws areelliptic curves. A cohomology theoryA{\displaystyle A}with
is calledellipticif it is even periodic and its formal group law is isomorphic to a formal group law of an elliptic curveE{\displaystyle E}overR{\displaystyle R}. The usual construction of such elliptic cohomology theories uses theLandweber exact functor theorem. If the formal group law ofE{\displaystyle E}is Landweber exact, one can define an elliptic cohomology theory (on finite complexes) by
Franke has identified the condition needed to fulfill Landweber exactness:
These conditions can be checked in many cases related to elliptic genera. Moreover, the conditions are fulfilled in the universal case in the sense that the map from themoduli stack of elliptic curvesto the moduli stack offormal groups
isflat. This gives then apresheafofcohomology theories
Oeℓℓpre:Aff/(M1,1)flat→Spectra{\displaystyle {\mathcal {O}}_{e\ell \ell }^{pre}:{\text{Aff}}/({\mathcal {M}}_{1,1})_{flat}\to {\textbf {Spectra}}}
over the site of affineschemesflat over the moduli stack of elliptic curves. The desire to get a universal elliptic cohomology theory by taking global sections has led to the construction of thetopological modular forms[1]pg 20
Tmf=HolimX→M1,1Oeℓℓpre(X){\displaystyle \mathbf {Tmf} ={\underset {X\to {\mathcal {M}}_{1,1}}{\textbf {Holim}}}{\text{ }}{\mathcal {O}}_{e\ell \ell }^{pre}(X)}
as the homotopy limit of this presheaf over the previous site.
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Inmathematics, theMordell–Weil theoremstates that for anabelian varietyA{\displaystyle A}over anumber fieldK{\displaystyle K}, the groupA(K){\displaystyle A(K)}ofK-rational pointsofA{\displaystyle A}is afinitely-generated abelian group, called theMordell–Weil group. The case withA{\displaystyle A}anelliptic curveE{\displaystyle E}andK{\displaystyle K}the field ofrational numbersisMordell's theorem, answering a question apparently posed byHenri Poincaréaround 1901; it was proved byLouis Mordellin 1922. It is a foundational theorem ofDiophantine geometryand thearithmetic of abelian varieties.
Thetangent-chord process(one form ofaddition theoremon acubic curve) had been known as far back as the seventeenth century. The process ofinfinite descentofFermatwas well known, but Mordell succeeded in establishing the finiteness of thequotient groupE(Q)/2E(Q){\displaystyle E(\mathbb {Q} )/2E(\mathbb {Q} )}which forms a major step in the proof. Certainly the finiteness of this group is anecessary conditionforE(Q){\displaystyle E(\mathbb {Q} )}to be finitely generated; and it shows that therankis finite. This turns out to be the essential difficulty. It can be proved by direct analysis of the doubling of a point onE.
Some years laterAndré Weiltook up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation[1]published in 1928. More abstract methods were required, to carry out a proof with the same basic structure. The second half of the proof needs some type ofheight function, in terms of which to bound the 'size' of points ofA(K){\displaystyle A(K)}. Some measure of the co-ordinates will do; heights are logarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set ofhomogeneous coordinates. For an abelian variety, there is noa prioripreferred representation, though, as aprojective variety.
Both halves of the proof have been improved significantly by subsequent technical advances: inGalois cohomologyas applied to descent, and in the study of the best height functions (which arequadratic forms).
The theorem leaves a number of questions still unanswered:
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https://en.wikipedia.org/wiki/Mordell%E2%80%93Weil_theorem
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Inmathematics, more specificallyalgebra,abstract algebraormodern algebrais the study ofalgebraic structures, which aresetswith specificoperationsacting on their elements.[1]Algebraic structures includegroups,rings,fields,modules,vector spaces,lattices, andalgebras over a field. The termabstract algebrawas coined in the early 20th century to distinguish it from older parts of algebra, and more specifically fromelementary algebra, the use ofvariablesto represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except inpedagogy.
Algebraic structures, with their associatedhomomorphisms, formmathematical categories.Category theorygives a unified framework to study properties and constructions that are similar for various structures.
Universal algebrais a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called thevarietyof groups.
Before the nineteenth century,algebrawas defined as the study ofpolynomials.[2]Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions ofalgebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formalaxiomaticdefinitions of variousalgebraic structuressuch as groups, rings, and fields.[3]This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden'sModerne Algebra,[4]which start each chapter with a formal definition of a structure and then follow it with concrete examples.[5]
The study of polynomial equations oralgebraic equationshas a long history.c.1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified asrhetorical algebraand was the dominant approach up to the 16th century.Al-Khwarizmioriginated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear untilFrançois Viète's 1591New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637La Géométrie.[6]The formal study of solving symbolic equations ledLeonhard Eulerto accept what were then considered "nonsense" roots such asnegative numbersandimaginary numbers, in the late 18th century.[7]However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century.[8]
George Peacock's 1830Treatise of Algebrawas the first attempt to place algebra on a strictly symbolic basis. He distinguished a newsymbolical algebra, distinct from the oldarithmetical algebra. Whereas in arithmetical algebraa−b{\displaystyle a-b}is restricted toa≥b{\displaystyle a\geq b}, in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as(−a)(−b)=ab{\displaystyle (-a)(-b)=ab}, by lettinga=0,c=0{\displaystyle a=0,c=0}in(a−b)(c−d)=ac+bd−ad−bc{\displaystyle (a-b)(c-d)=ac+bd-ad-bc}. Peacock used what he termed theprinciple of the permanence of equivalent formsto justify his argument, but his reasoning suffered from theproblem of induction.[9]For example,ab=ab{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}}holds for the nonnegativereal numbers, but not for generalcomplex numbers.
Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic equation led to theGalois group of a polynomial. Gauss's 1801 study ofFermat's little theoremled to thering of integers modulo n, themultiplicative group of integers modulo n, and the more general concepts ofcyclic groupsandabelian groups. Klein's 1872Erlangen programstudied geometry and led tosymmetry groupssuch as theEuclidean groupand the group ofprojective transformations. In 1874 Lie introduced the theory ofLie groups, aiming for "the Galois theory of differential equations". In 1876 Poincaré and Klein introduced the group ofMöbius transformations, and its subgroups such as themodular groupandFuchsian group, based on work on automorphic functions in analysis.[10]
The abstract concept of group emerged slowly over the middle of the nineteenth century. Galois in 1832 was the first to use the term "group",[11]signifying a collection of permutations closed under composition.[12]Arthur Cayley's 1854 paperOn the theory of groupsdefined a group as a set with an associative composition operation and the identity 1, today called amonoid.[13]In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the leftcancellation propertyb≠c→a⋅b≠a⋅c{\displaystyle b\neq c\to a\cdot b\neq a\cdot c},[14]similar to the modern laws for a finiteabelian group.[15]Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation.[16]Walther von Dyckin 1882 was the first to require inverse elements as part of the definition of a group.[17]
Once this abstract group concept emerged, results were reformulated in this abstract setting. For example,Sylow's theoremwas reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a subgroup of a permutation group.[18][19]Otto Hölderwas particularly prolific in this area, defining quotient groups in 1889, group automorphisms in 1893, as well as simple groups. He also completed theJordan–Hölder theorem. Dedekind and Miller independently characterizedHamiltonian groupsand introduced the notion of thecommutatorof two elements. Burnside, Frobenius, and Molien created therepresentation theoryof finite groups at the end of the nineteenth century.[18]J. A. de Séguier's 1905 monographElements of the Theory of Abstract Groupspresented many of these results in an abstract, general form, relegating "concrete" groups to an appendix, although it was limited to finite groups. The first monograph on both finite and infinite abstract groups was O. K. Schmidt's 1916Abstract Theory of Groups.[20]
Noncommutative ring theory began with extensions of the complex numbers tohypercomplex numbers, specificallyWilliam Rowan Hamilton'squaternionsin 1843. Many other number systems followed shortly. In 1844, Hamilton presentedbiquaternions, Cayley introducedoctonions, and Grassman introducedexterior algebras.[21]James Cocklepresentedtessarinesin 1848[22]andcoquaternionsin 1849.[23]William Kingdon Cliffordintroducedsplit-biquaternionsin 1873. In addition Cayley introducedgroup algebrasover the real and complex numbers in 1854 andsquare matricesin two papers of 1855 and 1858.[24]
Once there were sufficient examples, it remained to classify them. In an 1870 monograph,Benjamin Peirceclassified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of anassociative algebra. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined thePeirce decomposition. Frobenius in 1878 andCharles Sanders Peircein 1881 independently proved that the only finite-dimensional division algebras overR{\displaystyle \mathbb {R} }were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimpleLie algebrascould be decomposed into simple ones, and classified all simple Lie algebras. Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra overR{\displaystyle \mathbb {R} }orC{\displaystyle \mathbb {C} }uniquely decomposes into thedirect sumsof a nilpotent algebra and a semisimple algebra that is the product of some number ofsimple algebras, square matrices over division algebras. Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called theWedderburn principal theoremandArtin–Wedderburn theorem.[25]
For commutative rings, several areas together led to commutative ring theory.[26]In two papers in 1828 and 1832, Gauss formulated theGaussian integersand showed that they form aunique factorization domain(UFD) and proved thebiquadratic reciprocitylaw. Jacobi and Eisenstein at around the same time proved acubic reciprocitylaw for theEisenstein integers.[25]The study ofFermat's last theoremled to thealgebraic integers. In 1847,Gabriel Laméthought he had proven FLT, but his proof was faulty as he assumed all thecyclotomic fieldswere UFDs, yet as Kummer pointed out,Q(ζ23)){\displaystyle \mathbb {Q} (\zeta _{23}))}was not a UFD.[27]In 1846 and 1847 Kummer introducedideal numbersand proved unique factorization into ideal primes for cyclotomic fields.[28]Dedekind extended this in 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product ofprime ideals, a precursor of the theory ofDedekind domains. Overall, Dedekind's work created the subject ofalgebraic number theory.[29]
In the 1850s, Riemann introduced the fundamental concept of aRiemann surface. Riemann's methods relied on an assumption he calledDirichlet's principle,[30]which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing thedirect method in the calculus of variations.[31]In the 1860s and 1870s, Clebsch, Gordan, Brill, and especiallyM. Noetherstudiedalgebraic functionsand curves. In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ringR[x,y]{\displaystyle \mathbb {R} [x,y]}, although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory ofalgebraic function fieldswhich allowed the first rigorous definition of a Riemann surface and a rigorous proof of theRiemann–Roch theorem. Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and Macauley in 1913 further investigated the ideals of polynomial rings implicit inE. Noether's work. Lasker proved a special case of theLasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection ofprimary ideals. Macauley proved the uniqueness of this decomposition.[32]Overall, this work led to the development ofalgebraic geometry.[26]
In 1801 Gauss introducedbinary quadratic formsover the integers and defined theirequivalence. He further defined thediscriminantof these forms, which is aninvariant of a binary form. Between the 1860s and 1890sinvariant theorydeveloped and became a major field of algebra. Cayley, Sylvester, Gordan and others found theJacobianand theHessianfor binary quartic forms and cubic forms.[33]In 1868 Gordan proved that thegraded algebraof invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis.[34]Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 toHilbert's basis theorem.[35]
Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given byAbraham Fraenkelin 1914.[35]His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element.[36]In addition, he had two axioms on "regular elements" inspired by work on thep-adic numbers, which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative.[37]Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one.[38]
In 1920,Emmy Noether, in collaboration with W. Schmeidler, published a paper about thetheory of idealsin which they definedleft and right idealsin aring. The following year she published a landmark paper calledIdealtheorie in Ringbereichen(Ideal theory in rings'), analyzingascending chain conditionswith regard to (mathematical) ideals. The publication gave rise to the term "Noetherian ring", and several other mathematical objects being calledNoetherian.[39][40]Noted algebraistIrving Kaplanskycalled this work "revolutionary";[39]results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom.[41]Artin, inspired by Noether's work, came up with thedescending chain condition. These definitions marked the birth of abstract ring theory.[42]
In 1801 Gauss introduced theintegers mod p, where p is a prime number. Galois extended this in 1830 tofinite fieldswithpn{\displaystyle p^{n}}elements.[43]In 1871Richard Dedekindintroduced, for a set of real or complex numbers that is closed under the four arithmetic operations,[44]theGermanwordKörper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893.[45]In 1881Leopold Kroneckerdefined what he called adomain of rationality, which is a field ofrational fractionsin modern terms.[46]The first clear definition of an abstract field was due toHeinrich Martin Weberin 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry.[47]In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by theircharacteristic, and proved many theorems commonly seen today.[48]
The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the namemodern algebra. Its study was part of the drive for moreintellectual rigorin mathematics. Initially, the assumptions in classicalalgebra, on which the whole of mathematics (and major parts of thenatural sciences) depend, took the form ofaxiomatic systems. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certainalgebraic structuresbegan to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of anabstract group. Questions of structure and classification of various mathematical objects came to the forefront.[50]
These processes were occurring throughout all of mathematics but became especially pronounced in algebra. Formal definitions through primitive operations and axioms were proposed for many basic algebraic structures, such asgroups,rings, andfields. Hence such things asgroup theoryandring theorytook their places inpure mathematics. The algebraic investigations of general fields byErnst Steinitzand of commutative and then general rings byDavid Hilbert,Emil ArtinandEmmy Noether, building on the work ofErnst Kummer,Leopold KroneckerandRichard Dedekind, who had considered ideals in commutative rings, and ofGeorg FrobeniusandIssai Schur, concerningrepresentation theoryof groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of the 20th century were systematically exposed inBartel van der Waerden'sModerne Algebra, the two-volumemonographpublished in 1930–1931 that reoriented the idea of algebra fromthe theory of equationstothetheory of algebraic structures.[51]
By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied aresets, to which the theorems ofset theoryapply. Those sets that have a certain binary operation defined on them formmagmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to formsemigroups); identity, and inverses (to formgroups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems ofgroup theorymay be used when studyingrings(algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewernontrivialtheorems and fewer applications.[citation needed]
Examples of algebraic structures with a singlebinary operationare:
Examples involving several operations include:
A group is a setG{\displaystyle G}together with a "group product", a binary operation⋅:G×G→G{\displaystyle \cdot :G\times G\rightarrow G}. The group satisfies the following defining axioms (cf.Group (mathematics) § Definition):
Identity: there exists an elemente{\displaystyle e}such that, for each elementa{\displaystyle a}inG{\displaystyle G}, it holds thate⋅a=a⋅e=a{\displaystyle e\cdot a=a\cdot e=a}.
Inverse: for each elementa{\displaystyle a}ofG{\displaystyle G}, there exists an elementb{\displaystyle b}so thata⋅b=b⋅a=e{\displaystyle a\cdot b=b\cdot a=e}.
Associativity: for each triplet of elementsa,b,c{\displaystyle a,b,c}inG{\displaystyle G}, it holds that(a⋅b)⋅c=a⋅(b⋅c){\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}.
A ring is a setR{\displaystyle R}with twobinary operations, addition:(x,y)↦x+y,{\displaystyle (x,y)\mapsto x+y,}and multiplication:(x,y)↦xy{\displaystyle (x,y)\mapsto xy}satisfying the followingaxioms.
Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance,algebraic topologyuses algebraic objects to study topologies. ThePoincaré conjecture, proved in 2003, asserts that thefundamental groupof a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not.Algebraic number theorystudies various numberringsthat generalize the set of integers. Using tools of algebraic number theory,Andrew WilesprovedFermat's Last Theorem.[citation needed]
In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. Ingauge theory, the requirement oflocal symmetrycan be used to deduce the equations describing a system. The groups that describe those symmetries areLie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number offorce carriersin a theory is equal to the dimension of the Lie algebra, and thesebosonsinteract with the force they mediate if the Lie algebra is nonabelian.[52]
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https://en.wikipedia.org/wiki/Abstract_algebra
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Innumber theory, given aprime numberp,[note 1]thep-adic numbersform an extension of therational numberswhich is distinct from thereal numbers, though with some similar properties;p-adic numbers can be written in a form similar to (possiblyinfinite)decimals, but with digits based on a prime numberprather than ten, and extending to the left rather than to the right.
For example, comparing the expansion of the rational number15{\displaystyle {\tfrac {1}{5}}}inbase3vs. the3-adic expansion,
Formally, given a prime numberp, ap-adic number can be defined as aseries
wherekis aninteger(possibly negative), and eachai{\displaystyle a_{i}}is an integer such that0≤ai<p.{\displaystyle 0\leq a_{i}<p.}Ap-adic integeris ap-adic number such thatk≥0.{\displaystyle k\geq 0.}
In general the series that represents ap-adic number is notconvergentin the usual sense, but it is convergent for thep-adic absolute value|s|p=p−k,{\displaystyle |s|_{p}=p^{-k},}wherekis the least integerisuch thatai≠0{\displaystyle a_{i}\neq 0}(if allai{\displaystyle a_{i}}are zero, one has the zerop-adic number, which has0as itsp-adic absolute value).
Every rational number can be uniquely expressed as the sum of a series as above, with respect to thep-adic absolute value. This allows considering rational numbers as specialp-adic numbers, and alternatively defining thep-adic numbers as thecompletionof the rational numbers for thep-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
p-adic numbers were first described byKurt Henselin 1897,[1]though, with hindsight, some ofErnst Kummer'searlier work can be interpreted as implicitly usingp-adic numbers.[note 2]
Roughly speaking,modular arithmeticmodulo a positive integernconsists of "approximating" every integer by the remainder of itsdivisionbyn, called itsresidue modulon. The main property of modular arithmetic is that the residue modulonof the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulon. If one knows that the absolute value of the result is less thann/2, this allows a computation of the result which does not involve any integer larger thann.
For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying theChinese remainder theoremfor recovering the result modulo the product of the moduli.
Another method discovered byKurt Henselconsists of using a prime modulusp, and applyingHensel's lemmafor recovering iteratively the result modulop2,p3,…,pn,…{\displaystyle p^{2},p^{3},\ldots ,p^{n},\ldots }If the process is continued infinitely, this provides eventually a result which is ap-adic number.
The theory ofp-adic numbers is fundamentally based on the two following lemmas:
Every nonzero rational number can be writtenpvmn,{\textstyle p^{v}{\frac {m}{n}},}wherev,m, andnare integers and neithermnornis divisible byp.The exponentvis uniquely determined by the rational number and is called itsp-adic valuation(this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from thefundamental theorem of arithmetic.
Every nonzero rational numberrof valuationvcan be uniquely writtenr=apv+s,{\displaystyle r=ap^{v}+s,}wheresis a rational number of valuation greater thanv, andais an integer such that0<a<p.{\displaystyle 0<a<p.}
The proof of this lemma results frommodular arithmetic: By the above lemma,r=pvmn,{\textstyle r=p^{v}{\frac {m}{n}},}wheremandnare integerscoprimewithp.
ByBézout's lemma, there exist integersaandb, with0≤a<p{\displaystyle 0\leq a<p}, such thatm=an+bp.{\displaystyle m=an+bp.}Settings=b/n{\displaystyle s=b/n}(henceval(s)≥0{\displaystyle {\rm {val}}(s)\geq 0}), we have
To show the uniqueness of this representation, observe that ifr=a′pv+pv+1s′,{\displaystyle r=a'p^{v}+p^{v+1}s',}with0≤a′<p{\displaystyle 0\leq a'<p}andval(s′)≥0{\displaystyle {\rm {val}}(s')\geq 0},
there holds by difference(a−a′)+p(s−s′)=0,{\displaystyle (a-a')+p(s-s')=0,}with|a−a′|<p{\displaystyle |a-a'|<p}andval(s−s′)≥0{\displaystyle {\rm {val}}(s-s')\geq 0}.
Writes−s′=c/d{\displaystyle s-s'=c/d}, wheredis coprime top; then(a−a′)d+pc=0{\displaystyle (a-a')d+pc=0}, which is possible only ifa−a′=0{\displaystyle a-a'=0}andc=0{\displaystyle c=0}.
Hencea=a′{\displaystyle a=a'}ands=s′{\displaystyle s=s'}.
The above process can be iterated starting fromsinstead ofr, giving the following.
Given a nonzero rational numberrof valuationvand a positive integerk, there are a rational numbersk{\displaystyle s_{k}}of nonnegative valuation andkuniquely defined nonnegative integersa0,…,ak−1{\displaystyle a_{0},\ldots ,a_{k-1}}less thanpsuch thata0>0{\displaystyle a_{0}>0}and
Thep-adic numbers are essentially obtained by continuing this infinitely to produce aninfinite series.
Thep-adic numbers are commonly defined by means ofp-adic series.
Ap-adic seriesis aformal power seriesof the form
wherev{\displaystyle v}is an integer and theri{\displaystyle r_{i}}are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator ofri{\displaystyle r_{i}}is not divisible byp).
Every rational number may be viewed as ap-adic series with a single nonzero term, consisting of its factorization of the formpknd,{\displaystyle p^{k}{\tfrac {n}{d}},}withnanddboth coprime withp.
Twop-adic series∑i=v∞ripi{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i}}and∑i=w∞sipi{\textstyle \sum _{i=w}^{\infty }s_{i}p^{i}}areequivalentif there is an integerNsuch that, for every integern>N,{\displaystyle n>N,}the rational number
is zero or has ap-adic valuation greater thann.
Ap-adic series∑i=v∞aipi{\textstyle \sum _{i=v}^{\infty }a_{i}p^{i}}isnormalizedif either allai{\displaystyle a_{i}}are integers such that0≤ai<p,{\displaystyle 0\leq a_{i}<p,}andav>0,{\displaystyle a_{v}>0,}or allai{\displaystyle a_{i}}are zero. In the latter case, the series is called thezero series.
Everyp-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see§ Normalization of ap-adic series, below.
In other words, the equivalence ofp-adic series is anequivalence relation, and eachequivalence classcontains exactly one normalizedp-adic series.
The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence ofp-adic series. That is, denoting the equivalence with~, ifS,TandUare nonzerop-adic series such thatS∼T,{\displaystyle S\sim T,}one has
Thep-adic numbers are often defined as the equivalence classes ofp-adic series, in a similar way as the definition of the real numbers as equivalence classes ofCauchy sequences. The uniqueness property of normalization, allows uniquely representing anyp-adic number by the corresponding normalizedp-adic series. The compatibility of the series equivalence leads almost immediately to basic properties ofp-adic numbers:
Starting with the series∑i=v∞ripi,{\textstyle \sum _{i=v}^{\infty }r_{i}p^{i},}the first above lemma allows getting an equivalent series such that thep-adic valuation ofrv{\displaystyle r_{v}}is zero. For that, one considers the first nonzerori.{\displaystyle r_{i}.}If itsp-adic valuation is zero, it suffices to changevintoi, that is to start the summation fromv. Otherwise, thep-adic valuation ofri{\displaystyle r_{i}}isj>0,{\displaystyle j>0,}andri=pjsi{\displaystyle r_{i}=p^{j}s_{i}}where the valuation ofsi{\displaystyle s_{i}}is zero; so, one gets an equivalent series by changingri{\displaystyle r_{i}}to0andri+j{\displaystyle r_{i+j}}tori+j+si.{\displaystyle r_{i+j}+s_{i}.}Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation ofrv{\displaystyle r_{v}}is zero.
Then, if the series is not normalized, consider the first nonzerori{\displaystyle r_{i}}that is not an integer in the interval[0,p−1].{\displaystyle [0,p-1].}The second above lemma allows writing itri=ai+psi;{\displaystyle r_{i}=a_{i}+ps_{i};}one gets n equivalent series by replacingri{\displaystyle r_{i}}withai,{\displaystyle a_{i},}and addingsi{\displaystyle s_{i}}tori+1.{\displaystyle r_{i+1}.}Iterating this process, possibly infinitely many times, provides eventually the desired normalizedp-adic series.
There are several equivalent definitions ofp-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions usecompletionof adiscrete valuation ring(see§ p-adic integers),completion of a metric space(see§ Topological properties), orinverse limits(see§ Modular properties).
Ap-adic number can be defined as anormalizedp-adic series. Since there are other equivalent definitions that are commonly used, one says often that a normalizedp-adic seriesrepresentsap-adic number, instead of saying that itisap-adic number.
One can say also that anyp-adic series represents ap-adic number, since everyp-adic series is equivalent to a unique normalizedp-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) ofp-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations onp-adic numbers, since the series operations are compatible with equivalence ofp-adic series.
With these operations,p-adic numbers form afieldcalled thefield ofp-adic numbersand denotedQp{\displaystyle \mathbb {Q} _{p}}orQp.{\displaystyle \mathbf {Q} _{p}.}There is a uniquefield homomorphismfrom the rational numbers into thep-adic numbers, which maps a rational number to itsp-adic expansion. Theimageof this homomorphism is commonly identified with the field of rational numbers. This allows considering thep-adic numbers as anextension fieldof the rational numbers, and the rational numbers as asubfieldof thep-adic numbers.
Thevaluationof a nonzerop-adic numberx, commonly denotedvp(x),{\displaystyle v_{p}(x),}is the exponent ofpin the first nonzero term of everyp-adic series that representsx. By convention,vp(0)=∞;{\displaystyle v_{p}(0)=\infty ;}that is, the valuation of zero is∞.{\displaystyle \infty .}This valuation is adiscrete valuation. The restriction of this valuation to the rational numbers is thep-adic valuation ofQ,{\displaystyle \mathbb {Q} ,}that is, the exponentvin the factorization of a rational number asndpv,{\displaystyle {\tfrac {n}{d}}p^{v},}with bothnanddcoprimewithp.
Thep-adic integersare thep-adic numbers with a nonnegative valuation.
Ap{\displaystyle p}-adic integer can be represented as a sequence
of residuesxe{\displaystyle x_{e}}modpe{\displaystyle p^{e}}for each integere{\displaystyle e}, satisfying the compatibility relationsxi≡xj(modpi){\displaystyle x_{i}\equiv x_{j}~(\operatorname {mod} p^{i})}fori<j{\displaystyle i<j}.
Everyintegeris ap{\displaystyle p}-adic integer (including zero, since0<∞{\displaystyle 0<\infty }). The rational numbers of the formndpk{\textstyle {\tfrac {n}{d}}p^{k}}withd{\displaystyle d}coprime withp{\displaystyle p}andk≥0{\displaystyle k\geq 0}are alsop{\displaystyle p}-adic integers (for the reason thatd{\displaystyle d}has an inverse modpe{\displaystyle p^{e}}for everye{\displaystyle e}).
Thep-adic integers form acommutative ring, denotedZp{\displaystyle \mathbb {Z} _{p}}orZp{\displaystyle \mathbf {Z} _{p}}, that has the following properties.
The last property provides a definition of thep-adic numbers that is equivalent to the above one: the field of thep-adic numbers is thefield of fractionsof the completion of the localization of the integers at the prime ideal generated byp.
Thep-adic valuation allows defining anabsolute valueonp-adic numbers: thep-adic absolute value of a nonzerop-adic numberxis
wherevp(x){\displaystyle v_{p}(x)}is thep-adic valuation ofx. Thep-adic absolute value of0{\displaystyle 0}is|0|p=0.{\displaystyle |0|_{p}=0.}This is an absolute value that satisfies thestrong triangle inequalitysince, for everyxandyone has
Moreover, if|x|p≠|y|p,{\displaystyle |x|_{p}\neq |y|_{p},}one has|x+y|p=max(|x|p,|y|p).{\displaystyle |x+y|_{p}=\max(|x|_{p},|y|_{p}).}
This makes thep-adic numbers ametric space, and even anultrametric space, with thep-adic distance defined bydp(x,y)=|x−y|p.{\displaystyle d_{p}(x,y)=|x-y|_{p}.}
As a metric space, thep-adic numbers form thecompletionof the rational numbers equipped with thep-adic absolute value. This provides another way for defining thep-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from everyCauchy sequencea subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of thepartial sumsof ap-adic series, and thus a unique normalizedp-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalizedp-adic series instead of equivalence classes of Cauchy sequences).
As the metric is defined from a discrete valuation, everyopen ballis alsoclosed. More precisely, the open ballBr(x)={y∣dp(x,y)<r}{\displaystyle B_{r}(x)=\{y\mid d_{p}(x,y)<r\}}equals the closed ballBp−v[x]={y∣dp(x,y)≤p−v},{\displaystyle B_{p^{-v}}[x]=\{y\mid d_{p}(x,y)\leq p^{-v}\},}wherevis the least integer such thatp−v<r.{\displaystyle p^{-v}<r.}Similarly,Br[x]=Bp−w(x),{\displaystyle B_{r}[x]=B_{p^{-w}}(x),}wherewis the greatest integer such thatp−w>r.{\displaystyle p^{-w}>r.}
This implies that thep-adic numbers form alocally compact space(locally compact field), and thep-adic integers—that is, the ballB1[0]=Bp(0){\displaystyle B_{1}[0]=B_{p}(0)}—form acompact space.
Thedecimal expansionof a positiverational numberr{\displaystyle r}is its representation as aseries
wherek{\displaystyle k}is an integer and eachai{\displaystyle a_{i}}is also anintegersuch that0≤ai<10.{\displaystyle 0\leq a_{i}<10.}This expansion can be computed bylong divisionof the numerator by the denominator, which is itself based on the following theorem: Ifr=nd{\displaystyle r={\tfrac {n}{d}}}is a rational number such that10k≤r<10k+1,{\displaystyle 10^{k}\leq r<10^{k+1},}there is an integera{\displaystyle a}such that0<a<10,{\displaystyle 0<a<10,}andr=a10k+r′,{\displaystyle r=a\,10^{k}+r',}withr′<10k.{\displaystyle r'<10^{k}.}The decimal expansion is obtained by repeatedly applying this result to the remainderr′{\displaystyle r'}which in the iteration assumes the role of the original rational numberr{\displaystyle r}.
Thep-adic expansionof a rational number is defined similarly, but with a different division step. More precisely, given a fixedprime numberp{\displaystyle p}, every nonzero rational numberr{\displaystyle r}can be uniquely written asr=pknd,{\displaystyle r=p^{k}{\tfrac {n}{d}},}wherek{\displaystyle k}is a (possibly negative) integer,n{\displaystyle n}andd{\displaystyle d}arecoprime integersboth coprime withp{\displaystyle p}, andd{\displaystyle d}is positive. The integerk{\displaystyle k}is thep-adic valuationofr{\displaystyle r}, denotedvp(r),{\displaystyle v_{p}(r),}andp−k{\displaystyle p^{-k}}is itsp-adic absolute value, denoted|r|p{\displaystyle |r|_{p}}(the absolute value is small when the valuation is large). The division step consists of writing
wherea{\displaystyle a}is an integer such that0≤a<p,{\displaystyle 0\leq a<p,}andr′{\displaystyle r'}is either zero, or a rational number such that|r′|p<p−k{\displaystyle |r'|_{p}<p^{-k}}(that is,vp(r′)>k{\displaystyle v_{p}(r')>k}).
Thep{\displaystyle p}-adic expansionofr{\displaystyle r}is theformal power series
obtained by repeating indefinitely theabovedivision step on successive remainders. In ap-adic expansion, allai{\displaystyle a_{i}}are integers such that0≤ai<p.{\displaystyle 0\leq a_{i}<p.}
Ifr=pkn1{\displaystyle r=p^{k}{\tfrac {n}{1}}}withn>0{\displaystyle n>0}, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation ofr{\displaystyle r}inbase-p.
The existence and the computation of thep-adic expansion of a rational number results fromBézout's identityin the following way. If, as above,r=pknd,{\displaystyle r=p^{k}{\tfrac {n}{d}},}andd{\displaystyle d}andp{\displaystyle p}are coprime, there exist integerst{\displaystyle t}andu{\displaystyle u}such thattd+up=1.{\displaystyle td+up=1.}So
Then, theEuclidean divisionofnt{\displaystyle nt}byp{\displaystyle p}gives
with0≤a<p.{\displaystyle 0\leq a<p.}This gives the division step as
so that in the iteration
is the new rational number.
The uniqueness of the division step and of the wholep-adic expansion is easy: ifpka1+pk+1s1=pka2+pk+1s2,{\displaystyle p^{k}a_{1}+p^{k+1}s_{1}=p^{k}a_{2}+p^{k+1}s_{2},}one hasa1−a2=p(s2−s1).{\displaystyle a_{1}-a_{2}=p(s_{2}-s_{1}).}This meansp{\displaystyle p}dividesa1−a2.{\displaystyle a_{1}-a_{2}.}Since0≤a1<p{\displaystyle 0\leq a_{1}<p}and0≤a2<p,{\displaystyle 0\leq a_{2}<p,}the following must be true:0≤a1{\displaystyle 0\leq a_{1}}anda2<p.{\displaystyle a_{2}<p.}Thus, one gets−p<a1−a2<p,{\displaystyle -p<a_{1}-a_{2}<p,}and sincep{\displaystyle p}dividesa1−a2{\displaystyle a_{1}-a_{2}}it must be thata1=a2.{\displaystyle a_{1}=a_{2}.}
Thep-adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of aconvergent serieswith thep-adic absolute value.
In the standardp-adic notation, the digits are written in the same order as in astandard base-psystem, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side.
Thep-adic expansion of a rational number is eventuallyperiodic.Conversely, a series∑i=k∞aipi,{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i},}with0≤ai<p{\displaystyle 0\leq a_{i}<p}converges (for thep-adic absolute value) to a rational numberif and only ifit is eventually periodic; in this case, the series is thep-adic expansion of that rational number. Theproofis similar to that of the similar result forrepeating decimals.
Let us compute the 5-adic expansion of13.{\displaystyle {\tfrac {1}{3}}.}Bézout's identity for 5 and the denominator 3 is2⋅3+(−1)⋅5=1{\displaystyle 2\cdot 3+(-1)\cdot 5=1}(for larger examples, this can be computed with theextended Euclidean algorithm). Thus
For the next step, one has to expand−1/3{\displaystyle -1/3}(the factor 5 has to be viewed as a "shift" of thep-adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded). To expand−1/3{\displaystyle -1/3}, we start from the same Bézout's identity and multiply it by−1{\displaystyle -1}, giving
The "integer part"−2{\displaystyle -2}is not in the right interval. So, one has to useEuclidean divisionby5{\displaystyle 5}for getting−2=3−1⋅5,{\displaystyle -2=3-1\cdot 5,}giving
and the expansion in the first step becomes
Similarly, one has
and
As the "remainder"−13{\displaystyle -{\tfrac {1}{3}}}has already been found, the process can be continued easily, giving coefficients3{\displaystyle 3}foroddpowers of five, and1{\displaystyle 1}forevenpowers.
Or in the standard 5-adic notation
with theellipsis…{\displaystyle \ldots }on the left hand side.
It is possible to use apositional notationsimilar to that which is used to represent numbers inbasep.
Let∑i=k∞aipi{\textstyle \sum _{i=k}^{\infty }a_{i}p^{i}}be a normalizedp-adic series, i.e. eachai{\displaystyle a_{i}}is an integer in the interval[0,p−1].{\displaystyle [0,p-1].}One can suppose thatk≤0{\displaystyle k\leq 0}by settingai=0{\displaystyle a_{i}=0}for0≤i<k{\displaystyle 0\leq i<k}(ifk>0{\displaystyle k>0}), and adding the resulting zero terms to the series.
Ifk≥0,{\displaystyle k\geq 0,}the positional notation consists of writing theai{\displaystyle a_{i}}consecutively, ordered by decreasing values ofi, often withpappearing on the right as an index:
So, the computation of theexample aboveshows that
and
Whenk<0,{\displaystyle k<0,}a separating dot is added before the digits with negative index, and, if the indexpis present, it appears just after the separating dot. For example,
and
If ap-adic representation is finite on the left (that is,ai=0{\displaystyle a_{i}=0}for large values ofi), then it has the value of a nonnegative rational number of the formnpv,{\displaystyle np^{v},}withn,v{\displaystyle n,v}integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation inbasep. For these rational numbers, the two representations are the same.
Thequotient ringZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}may be identified with theringZ/pnZ{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }of the integersmodulopn.{\displaystyle p^{n}.}This can be shown by remarking that everyp-adic integer, represented by its normalizedp-adic series, is congruent modulopn{\displaystyle p^{n}}with itspartial sum∑i=0n−1aipi,{\textstyle \sum _{i=0}^{n-1}a_{i}p^{i},}whose value is an integer in the interval[0,pn−1].{\displaystyle [0,p^{n}-1].}A straightforward verification shows that this defines aring isomorphismfromZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}toZ/pnZ.{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} .}
Theinverse limitof the ringsZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}is defined as the ring formed by the sequencesa0,a1,…{\displaystyle a_{0},a_{1},\ldots }such thatai∈Z/piZ{\displaystyle a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z} }andai≡ai+1(modpi){\textstyle a_{i}\equiv a_{i+1}{\pmod {p^{i}}}}for everyi.
The mapping that maps a normalizedp-adic series to the sequence of its partial sums is a ring isomorphism fromZp{\displaystyle \mathbb {Z} _{p}}to the inverse limit of theZp/pnZp.{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}.}This provides another way for definingp-adic integers (up toan isomorphism).
This definition ofp-adic integers is specially useful for practical computations, as allowing buildingp-adic integers by successive approximations.
For example, for computing thep-adic (multiplicative) inverse of an integer, one can useNewton's method, starting from the inverse modulop; then, each Newton step computes the inverse modulopn2{\textstyle p^{n^{2}}}from the inverse modulopn.{\textstyle p^{n}.}
The same method can be used for computing thep-adicsquare rootof an integer that is aquadratic residuemodulop. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found inZp/pnZp{\displaystyle \mathbb {Z} _{p}/p^{n}\mathbb {Z} _{p}}. Applying Newton's method to find the square root requirespn{\textstyle p^{n}}to be larger than twice the given integer, which is quickly satisfied.
Hensel liftingis a similar method that allows to "lift" the factorization modulopof a polynomial with integer coefficients to a factorization modulopn{\textstyle p^{n}}for large values ofn. This is commonly used bypolynomial factorizationalgorithms.
There are several different conventions for writingp-adic expansions. So far this article has used a notation forp-adic expansions in whichpowersofpincrease from right to left. With this right-to-left notation the 3-adic expansion of15,{\displaystyle {\tfrac {1}{5}},}for example, is written as
When performing arithmetic in this notation, digits arecarriedto the left. It is also possible to writep-adic expansions so that the powers ofpincrease from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of15{\displaystyle {\tfrac {1}{5}}}is
p-adic expansions may be written withother sets of digitsinstead of{0, 1, ...,p− 1}. For example, the3-adic expansion of15{\displaystyle {\tfrac {1}{5}}}can be written usingbalanced ternarydigits{1, 0, 1}, with1representing negative one, as
In fact any set ofpintegers which are in distinctresidue classesmodulopmay be used asp-adic digits. In number theory,Teichmüller representativesare sometimes used as digits.[2]
Quote notationis a variant of thep-adic representation ofrational numbersthat was proposed in 1979 byEric HehnerandNigel Horspoolfor implementing on computers the (exact) arithmetic with these numbers.[3]
BothZp{\displaystyle \mathbb {Z} _{p}}andQp{\displaystyle \mathbb {Q} _{p}}areuncountableand have thecardinality of the continuum.[4]ForZp,{\displaystyle \mathbb {Z} _{p},}this results from thep-adic representation, which defines abijectionofZp{\displaystyle \mathbb {Z} _{p}}on thepower set{0,…,p−1}N.{\displaystyle \{0,\ldots ,p-1\}^{\mathbb {N} }.}ForQp{\displaystyle \mathbb {Q} _{p}}this results from its expression as acountably infiniteunionof copies ofZp{\displaystyle \mathbb {Z} _{p}}:
Qp{\displaystyle \mathbb {Q} _{p}}containsQ{\displaystyle \mathbb {Q} }and is a field ofcharacteristic0.
Because0can be written as sum of squares,[5]Qp{\displaystyle \mathbb {Q} _{p}}cannot be turned into anordered field.
The field ofreal numbersR{\displaystyle \mathbb {R} }has only a single properalgebraic extension: thecomplex numbersC{\displaystyle \mathbb {C} }. In other words, thisquadratic extensionis alreadyalgebraically closed. By contrast, thealgebraic closureofQp{\displaystyle \mathbb {Q} _{p}}, denotedQp¯,{\displaystyle {\overline {\mathbb {Q} _{p}}},}has infinite degree,[6]that is,Qp{\displaystyle \mathbb {Q} _{p}}has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of thep-adic valuation toQp¯,{\displaystyle {\overline {\mathbb {Q} _{p}}},}the latter is not (metrically) complete.[7][8]Its (metric) completion is calledCp{\displaystyle \mathbb {C} _{p}}orΩp{\displaystyle \Omega _{p}}.[8][9]Here an end is reached, asCp{\displaystyle \mathbb {C} _{p}}is algebraically closed.[8][10]However unlikeC{\displaystyle \mathbb {C} }this field is notlocally compact.[9]
Cp{\displaystyle \mathbb {C} _{p}}andC{\displaystyle \mathbb {C} }are isomorphic as rings,[11]so we may regardCp{\displaystyle \mathbb {C} _{p}}asC{\displaystyle \mathbb {C} }endowed with an exotic metric. The proof of existence of such a field isomorphism relies on theaxiom of choice, and does not provide an explicit example of such an isomorphism (that is, it is notconstructive).
IfK{\displaystyle K}is any finiteGalois extensionofQp,{\displaystyle \mathbb {Q} _{p},}theGalois groupGal(K/Qp){\displaystyle \operatorname {Gal} \left(K/\mathbb {Q} _{p}\right)}issolvable. Thus, the Galois groupGal(Qp¯/Qp){\displaystyle \operatorname {Gal} \left({\overline {\mathbb {Q} _{p}}}/\mathbb {Q} _{p}\right)}isprosolvable.
Qp{\displaystyle \mathbb {Q} _{p}}contains then-thcyclotomic field(n> 2) if and only ifn|p− 1.[12]For instance, then-th cyclotomic field is a subfield ofQ13{\displaystyle \mathbb {Q} _{13}}if and only ifn= 1, 2, 3, 4, 6, or12. In particular, there is no multiplicativep-torsioninQp{\displaystyle \mathbb {Q} _{p}}ifp> 2. Also,−1is the only non-trivial torsion element inQ2{\displaystyle \mathbb {Q} _{2}}.
Given anatural numberk, theindexof the multiplicative group of thek-th powers of the non-zero elements ofQp{\displaystyle \mathbb {Q} _{p}}inQp×{\displaystyle \mathbb {Q} _{p}^{\times }}is finite.
The numbere, defined as the sum ofreciprocalsoffactorials, is not a member of anyp-adic field; butep∈Qp{\displaystyle e^{p}\in \mathbb {Q} _{p}}forp≠2{\displaystyle p\neq 2}. Forp= 2one must take at least the fourth power.[13](Thus a number with similar properties ase— namely ap-th root ofep— is a member ofQp{\displaystyle \mathbb {Q} _{p}}for allp.)
Helmut Hasse'slocal–global principleis said to hold for an equation if it can be solved over the rational numbersif and only ifit can be solved over the real numbers and over thep-adic numbers for every primep. This principle holds, for example, for equations given byquadratic forms, but fails for higher polynomials in several indeterminates.
The reals and thep-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance generalalgebraic number fields, in an analogous way. This will be described now.
SupposeDis aDedekind domainandEis itsfield of fractions. Pick a non-zeroprime idealPofD. Ifxis a non-zero element ofE, thenxDis afractional idealand can be uniquely factored as a product of positive and negative powers of non-zero prime ideals ofD. We write ordP(x) for the exponent ofPin this factorization, and for any choice of numbercgreater than 1 we can set
Completing with respect to this absolute value|⋅|Pyields a fieldEP, the proper generalization of the field ofp-adic numbers to this setting. The choice ofcdoes not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when theresidue fieldD/Pis finite, to take forcthe size ofD/P.
For example, whenEis anumber field,Ostrowski's theoremsays that every non-trivialnon-Archimedean absolute valueonEarises as some|⋅|P. The remaining non-trivial absolute values onEarise from the different embeddings ofEinto the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings ofEinto the fieldsCp, thus putting the description of all
the non-trivial absolute values of a number field on a common footing.)
Often, one needs to simultaneously keep track of all the above-mentioned completions whenEis a number field (or more generally aglobal field), which are seen as encoding "local" information. This is accomplished byadele ringsandidele groups.
p-adic integers can be extended top-adic solenoidsTp{\displaystyle \mathbb {T} _{p}}. There is a map fromTp{\displaystyle \mathbb {T} _{p}}to thecircle groupwhose fibers are thep-adic integersZp{\displaystyle \mathbb {Z} _{p}}, in analogy to how there is a map fromR{\displaystyle \mathbb {R} }to the circle whose fibers areZ{\displaystyle \mathbb {Z} }.
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https://en.wikipedia.org/wiki/P-adic_number
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Inalgebraic geometry, astable curveis analgebraic curvethat is asymptotically stable in the sense ofgeometric invariant theory.
This is equivalent to the condition that it is acompleteconnected curve whose only singularities are ordinarydouble pointsand whoseautomorphism groupis finite.
The condition that the automorphism group is finite can be replaced by the condition that it is not ofarithmetic genusone and every non-singularrationalcomponent meets the other components in at least 3 points (Deligne & Mumford 1969).
Asemi-stable curveis one satisfying similar conditions, except that the automorphism group is allowed to bereductiverather than finite (or equivalently its connected component may be atorus). Alternatively the condition that non-singular rational components meet the other components in at least three points is replaced by the condition that they meet in at least two points.
Similarly a curve with a finite number of marked points is called stable if it is complete, connected, has only ordinary double points as singularities, and has finite automorphism group. For example, anelliptic curve(a non-singular genus 1 curve with 1 marked point) is stable.
Over the complex numbers, a connected curve is stable if and only if, after removing all singular and marked points, theuniversal coversof all its components are isomorphic to the unit disk.
Given an arbitraryschemeS{\displaystyle S}and settingg≥2{\displaystyle g\geq 2}astablegenus g curve overS{\displaystyle S}is defined as aproperflat morphismπ:C→S{\displaystyle \pi :C\to S}such that the geometric fibers are reduced, connected 1-dimensional schemesCs{\displaystyle C_{s}}such that
These technical conditions are necessary because (1) reduces the technical complexity (also Picard-Lefschetz theory can be used here), (2) rigidifies the curves so that there are no infinitesimal automorphisms of the moduli stack constructed later on, and (3) guarantees that the arithmetic genus of every fiber is the same. Note that for (1) the types of singularities found inelliptic surfacescan be completely classified.
One classical example of a family of stable curves is given by the Weierstrass family of curves
where the fibers over every point≠0,1{\displaystyle \neq 0,1}are smooth and the degenerate points only have one double-point singularity. This example can be generalized to the case of a one-parameter family of smoothhyperelliptic curvesdegenerating at finitely many points.
In the general case of more than one parameter care has to be taken to remove curves which have worse than double-point singularities. For example, consider the family overAs,t2{\displaystyle \mathbb {A} _{s,t}^{2}}constructed from the polynomials
since along the diagonals=t{\displaystyle s=t}there are non-double-point singularities. Another non-example is the family overAt1{\displaystyle \mathbb {A} _{t}^{1}}given by the polynomials
which are a family of elliptic curves degenerating to a rational curve with a cusp.
One of the most important properties of stable curves is the fact that they arelocal complete intersections. This implies that standardSerre dualitytheory can be used. In particular, it can be shown that for every stable curveωC/S⊗3{\displaystyle \omega _{C/S}^{\otimes 3}}is a relativelyvery ample sheaf; it can be used to embed the curve intoPS5g−6{\displaystyle \mathbb {P} _{S}^{5g-6}}. Using the standardHilbert schemetheory we can construct amoduli schemeof curves of genusg{\displaystyle g}embedded in some projective space. TheHilbert polynomialis given by
There is a sublocus of stable curves contained in the Hilbert scheme
Thisrepresentsthe functor
where∼{\displaystyle \sim }are isomorphisms of stable curves. In order to make this the moduli space of curves without regard to the embedding (which is encoded by the isomorphism of projective spaces) we have to mod out byPGL(5g−6){\displaystyle PGL(5g-6)}. This gives us the moduli stack
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https://en.wikipedia.org/wiki/Semistable_curve
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Incryptography,Curve448orCurve448-Goldilocksis anelliptic curvepotentially offering 224 bits of security and designed for use with theelliptic-curve Diffie–Hellman(ECDH) key agreement scheme.
Developed by Mike Hamburg ofRambusCryptography Research, Curve448 allows fast performance compared with other proposed curves with comparable security.[1]Thereference implementationis available under anMIT license.[2]The curve was favored by theInternet Research Task ForceCrypto Forum Research Group (IRTF CFRG) for inclusion inTransport Layer Security(TLS) standards along withCurve25519.
In 2017, NIST announced that Curve25519 and Curve448 would be added to "Special Publication 800-186", which specifies approvedelliptic curvesfor use by theUS Federal Government,[3]and in 2023 it was approved for use in FIPS 186-5.[4]Both are described inRFC7748. The nameX448is used for the DH function. X448 support was added toOpenSSLin version 1.1.1 (released on 11 September 2018).[5]
Hamburg chose theSolinas trinomial primebasep= 2448− 2224− 1, calling it a "Goldilocks" prime "because its form defines the golden ratioφ≡ 2224". The main advantage of agolden-ratioprime is fastKaratsuba multiplication.[6]
The curve Hamburg used is an untwistedEdwards curveEd:y2+x2= 1 − 39081x2y2. The constantd= −39081 was chosen as the smallest absolute value that had the required mathematical properties, thus anothing-up-my-sleeve number.
Curve448 is constructed such that it avoids many potentialimplementationpitfalls.[7]
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https://en.wikipedia.org/wiki/Curve448
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