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DNS over TLS(DoT) is a networksecurity protocolfor encrypting and wrappingDomain Name System(DNS) queries and answers via theTransport Layer Security(TLS) protocol. The goal of the method is to increase user privacy and security by preventing eavesdropping and manipulation of DNS data viaman-in-the-middle attacks. Thewell-known port numberfor DoT is 853.
While DNS over TLS is applicable to any DNS transaction, it was first standardized for use between stub or forwarding resolvers and recursive resolvers, inRFC7858in May of 2016. Subsequent IETF efforts specify the use of DoT between recursive and authoritative servers ("Authoritative DNS over TLS" or "ADoT")[1]and a related implementation between authoritative servers (Zone Transfer-over-TLS or "xfr-over-TLS").[2]
BINDsupports DoT connections as of version 9.17.[3]Earlier versions offered DoT capability by proxying throughstunnel.[4]Unboundhas supported DNS over TLS since 22 January 2023.[5][6]Unwind has supported DoT since 29 January 2023.[7][8]WithAndroid Pie's support for DNS over TLS, somead blockersnow support using the encrypted protocol as a relatively easy way to access their services versus any of the various work-around methods typically used such as VPNs and proxy servers.[9][10][11][12]
Androidclients runningAndroid Pieor newer support DNS over TLS and will use it by default if the network infrastructure, for example theISP, supports it.[13][14]
In April 2018, Google announced thatAndroid Piewill include support for DNS over TLS,[15]allowing users to set a DNS server phone-wide on both Wi-Fi and mobile connections, an option that was historically only possible onrooteddevices. DNSDist, fromPowerDNS, also announced support for DNS over TLS in version 1.3.0.[16]
LinuxandWindowsusers can use DNS over TLS as a client through theNLnet Labsstubby daemon or Knot Resolver.[17]Alternatively they may install getdns-utils[18]to use DoT directly with the getdns_query tool. TheunboundDNS resolver by NLnet Labs also supports DNS over TLS.[19]
Apple'siOS 14introduced OS-level support for DNS over TLS (and DNS over HTTPS). iOS does not allow manual configuration of DoT servers, and requires the use of a third-party application to make configuration changes.[20]
systemd-resolvedis a Linux-only implementation that can be configured to use DNS over TLS, by editing/etc/systemd/resolved.confand enabling the settingDNSOverTLS.[21][22]Most major Linux distributions have systemd installed by default.[23][circular reference]
DNS over TLS was first implemented in apublic recursive resolverbyQuad9in 2017.[24][25]Other recursive resolver operators such asGoogleandCloudflarefollowed suit in subsequent years, and now it is a broadly-supported feature generally available in most large recursive resolvers.[26][27][28][29][30][31][32][33][12]
DoT can impede analysis and monitoring of DNS traffic for cybersecurity purposes. DoT has been used to bypassparental controlswhich operate at the (unencrypted) standard DNS level; Circle, a parental control router which relies on DNS queries to check domains against a blocklist, blocks DoT by default due to this.[34]However, there are DNS providers that offer filtering and parental controls along with support for both DoT and DoH.[35][36][37][38][39][12]In that scenario, DNS queries are checked against block lists once they are received by the provider rather than prior to leaving the user's router.
As with any communication, encryption of DNS requests by itself does not protect privacy. It protects against third-party observers, but does not guarantee what the endpoints do with the (then decrypted) data.
DoT clients do not necessarily directly query anyauthoritative name servers. The client may rely on the DoT server using traditional (port 53 or 853) queries to finally reach authoritative servers. Thus, DoT does not qualify as anend-to-end encryptedprotocol, only hop-to-hop encrypted and only if DNS over TLS is used consistently.
DNS over HTTPS(DoH) is a similar protocol standard for encrypting DNS queries, differing only in the methods used for encryption and delivery from DoT. On the basis of privacy and security, whether or not a superior protocol exists among the two is a matter of controversial debate, while others argue the merits of either depend on the specific use case.[40]
DNSCryptis another network protocol that authenticates and encrypts DNS traffic, although it was never proposed to theInternet Engineering Task Force(IETF) with aRequest for Comments(RFC).
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DNS over HTTPS(DoH) is a protocol for performing remote Domain Name System (DNS) resolution via theHTTPSprotocol. A goal of the method is to increase user privacy and security by preventing eavesdropping and manipulation of DNS data byman-in-the-middle attacks[1]by using the HTTPS protocol toencryptthe data between the DoH client and the DoH-basedDNS resolver.[2]By March 2018,Googleand theMozilla Foundationhad started testing versions of DNS over HTTPS.[3][4]In February 2020,Firefoxswitched to DNS over HTTPS by default for users in the United States.[5]In May 2020,Chromeswitched to DNS over HTTPS by default.[6]
An alternative to DoH is theDNS over TLS(DoT) protocol, a similar standard for encryptingDNSqueries, differing only in the methods used for encryption and delivery. Based on privacy and security, whether either protocol is superior is a matter of controversial debate, while others argue that the merits of either depend on the specific use case.[7]
DoH is a proposed standard, published asRFC8484(October 2018) by theIETF. It usesHTTPS, and supports thewire formatDNS response data, as returned in existing UDP responses, in an HTTPS payload with theMIME typeapplication/dns-message.[1][8]: §4.1The underlying HTTP layer can be any version of HTTP, thoughHTTP/2is therecommendedminimum.[8]: §5.2If HTTP/2 is used, the server may also useHTTP/2 server pushto send values that it anticipates the client may find useful in advance.[8]: §5.3
DoH is a work in progress. Even though the IETF has published RFC 8484 as a proposed standard and companies are experimenting with it,[9][10]the IETF has yet to determine how it should best be implemented. The IETF is evaluating a number of approaches for how best to deploy DoH and has established a working group,Adaptive DNS Discovery (ADD), to do this work and develop a consensus. In addition, other industry working groups such as theEncrypted DNS Deployment Initiative, have been formed to "define and adopt DNS encryption technologies in a manner that ensures the continued high performance, resiliency, stability and security of the Internet's critical namespace and name resolution services, as well as ensuring the continued unimpaired functionality of security protections, parental controls, and other services that depend upon the DNS".[11]
Since DoH cannot be used under some circumstances, likecaptive portals, web browsers like Firefox can be configured to fall back to insecure DNS.[12]
Oblivious DNS over HTTPS (ODoH) is an experimental standard, published asRFC9230(June 2022) by theIETFproposing a protocol extension to ensure no single DoH server is aware of both the client's IP address and the content of their DNS queries and responses. Oblivious DoH was originally developed as Oblivious DNS (ODNS)[13]by researchers atPrinceton Universityand theUniversity of Chicagoas an extension to unencrypted DNS, before DoH itself was standardized and widely deployed. Apple and Cloudflare subsequently deployed the technology in the context of DoH, as Oblivious DoH (ODoH).[14]
In ODoH and ODNS, all DNS requests and responses are routed via a proxy, hiding the client's address from the resolver. Requests and responses are encrypted to hide their contents from the proxy, and only the resolver can decrypt the requests, and the client the responses. Thus, the proxy knows the client address and resolver but not the request, and the resolver knows the proxy and request but not the client address, preventing the client address being linked to the query, unless both the proxy and resolver servers collude.[15][16][17][18]
DoH is used for recursive DNS resolution byDNS resolvers. Resolvers (DoH clients) must have access to a DoH server hosting a query endpoint.[19]
Three usage scenarios are common:
Apple'siOS 14andmacOS 11released in late 2020 support both DoH andDoTprotocols.[20][21]In iOS, the protocols can be used via configuration profiles.
In November 2019,Microsoftannounced plans to implement support for encrypted DNS protocols inMicrosoft Windows, beginning with DoH.[22]In May 2020, Microsoft released Windows 10 Insider Preview Build 19628 that included initial support for DoH[23]along with instructions on how to enable it viaregistryandcommand line interface.[24]Windows 10 Insider Preview Build 20185 added a graphical user interface for specifying a DoH resolver.[25]DoH support is not included in Windows 10 21H2.[26]
Windows 11has DoH support.[27]
Android 11 onwards supports DNS overHTTP/3(DoH3) if a July 2022 system update is installed.[28]
BIND 9, an open source DNS resolver fromInternet Systems Consortiumadded native support for DoH in version 9.17.10.[29]
DNSdist, an open source DNS proxy/load balancer fromPowerDNS, added native support for DoH in version 1.4.0 in April 2019.[30]
Unbound, anopen sourceDNS resolver created byNLnet Labs, has supported DoH since version 1.12.0, released in October 2020.[31][32]It first implemented support forDNSencryption using the alternativeDoTprotocol much earlier, starting with version 1.4.14, released in December 2011.[33][34]Unboundruns on mostoperating systems, including distributions ofLinux,BSD,MacOS, andWindows.
DNS over HTTPS is available inGoogle Chrome83 or later for Windows, Linux, and macOS, configurable via the settings page. When enabled, and the operating system is configured with a supported DNS server, Chrome will upgrade DNS queries to be encrypted.[35]It is also possible to manually specify a preset or custom DoH server to use within the user interface.[36]
In September 2020, Google Chrome for Android began staged rollout of DNS over HTTPS. Users can configure a custom resolver or disable DNS over HTTPS in settings.[37]
Google Chrome has 5 DNS over HTTPS providers pre-configured which areGoogle Public DNS, Cloudflare's1.1.1.1, Quad9's9.9.9.9,NextDNS, and CleanBrowsing.[38]
Microsoft Edgesupports DNS over HTTPS, configurable via the settings page. When enabled, and the operating system is configured with a supported DNS server, Edge will upgrade DNS queries to be encrypted. It is also possible to manually specify a preset or custom DoH server to use within the user interface.[39]
In 2018,Mozillapartnered withCloudflareto deliver DoH forFirefoxusers that enable it (known as Trusted Recursive Resolver).[40]On February 25, 2020, Firefox started enabling DNS over HTTPS for all US-based users, relying on Cloudflare's resolver by default.[41]
Operasupports DoH, configurable via the browser settings page.[42]By default, DNS queries are sent to Cloudflare servers.[43]
DNS over HTTPS server implementations are already available free of charge by some public DNS providers.
Many issues with how to properly deploy DoH are still being resolved by the internet community including, but not limited to:
DoH can impede analysis and monitoring of DNS traffic for cybersecurity purposes; the 2019DDoSworm Godlua used DoH to mask connections to its command-and-control server.[44][45]
In January 2021,NSAwarned enterprises against using external DoH resolvers because they prevent DNS query filtering, inspection, and audit. Instead, NSA recommends configuring enterprise-owned DoH resolvers and blocking all known external DoH resolvers.[46]
DoH has been used to bypassparental controlswhich operate at the (unencrypted) standard DNS level; Circle, a parental control router which relies on DNS queries to check domains against a blocklist, blocks DoH by default due to this.[47]However, there are DNS providers that offer filtering and parental controls along with support for DoH by operating DoH servers.[48][49]
TheInternet Service Providers Association(ISPA)—a trade association representing British ISPs—and the also British bodyInternet Watch Foundationhave criticizedMozilla, developer of theFirefoxweb browser, for supporting DoH, as they believe that it will undermineweb blockingprograms in the country, including ISP default filtering of adult content, and mandatory court-ordered filtering of copyright violations. The ISPA nominated Mozilla for its "Internet Villain" award for 2019 (alongside the EUDirective on Copyright in the Digital Single Market, andDonald Trump), "for their proposed approach to introduce DNS-over-HTTPS in such a way as to bypass UK filtering obligations and parental controls, undermining internet safety standards in the UK." Mozilla responded to the allegations by the ISPA, arguing that it would not prevent filtering, and that they were "surprised and disappointed that an industry association for ISPs decided to misrepresent an improvement to decades-old internet infrastructure".[50][51]In response to the criticism, the ISPA apologized and withdrew the nomination.[52][53]Mozilla subsequently stated that DoH will not be used by default in the British market until further discussion with relevant stakeholders, but stated that it "would offer real security benefits to UK citizens".[54]
In July 2020,iYouPort, theUniversity of Maryland, and theGreat Firewall Report, reported that theGreat Firewall(GFW) by the Chinese government blocks TLS connections using the encrypted SNI extension in China.[55]
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DNSCryptis anetwork protocolthat authenticates and encryptsDomain Name System(DNS) traffic between the user'scomputerandrecursive name servers. DNSCrypt wraps unmodified DNS traffic between a client and a DNS resolver in a cryptographic construction, preventing eavesdropping and forgery by aman-in-the-middle.[1]
It also mitigatesUDP-based amplification attacks by requiring a question to be at least as large as the corresponding response. Thus, DNSCrypt helps to preventDNS amplification attacks.[2]: §9
DNSCrypt was originally designed by Frank Denis and Yecheng Fu. Multiple free and open source software implementations exist. It is available for a variety of operating systems, including Unix, Apple iOS, Linux, Android, and Microsoft Windows.[3]The free and open source software implementation dnscrypt-proxy[4]additionally integratesODoH.[5]
In addition to private deployments, the DNSCrypt protocol has been adopted by several public DNS resolvers, the vast majority being members of theOpenNICnetwork, as well asvirtual private network(VPN) services.
OpenDNS(now a part ofCisco) announced the first public DNS service supporting DNSCrypt on 6 December 2011, shortly followed by CloudNS Australia.[6]
On 29 March 2016,Yandexannounced support for the DNSCrypt protocol on their public DNS servers, as well as inYandex Browser.[citation needed]
On 14 October 2016,AdGuardadded DNSCrypt to their DNS filtering module so that users could move from their ISPs to custom or AdGuard's own DNS servers for online privacy andad blocking.[7][8]
On 10 September 2018, theQuad9nonprofit public recursive resolver service announced support for DNSCrypt.[9]
Other servers that support secure protocol are mentioned in the DNSCrypt creators' list.[10]
DNSCrypt can be used either overUDPor overTCP. In both cases,its default port is 443.[2]Even though the protocol radically differs fromHTTPS, both service types utilize the sameport. However, even thoughDNS over HTTPSand DNSCrypt are possible on the same port, they must still run separately on different servers. Two server applications cannot run simultaneously on the same server if both utilize the same port for communication; though a multiplexing approach is theoretically possible.
Instead of relying on trustedcertificate authoritiescommonly found in web browsers, the client has to explicitly trust the public signing key of the chosen provider. This public key is used to verify a set of certificates, retrieved using conventional DNS queries.[2]: §1These certificates contain short-term public keys used for key exchange, as well as an identifier of the cipher suite to use. Clients are encouraged to generate a new key for every query, while servers are encouraged to rotate short-term key pairs every 24 hours.[2]: §13
The DNSCrypt protocol can also be used for access control or accounting, by accepting only a predefined set of public keys. This can be used by commercial DNS services to identify customers without having to rely on IP addresses.[2]: §13
Queries and responses are encrypted using the same algorithm and padded to a multiple of 64 bytes in order to avoid leaking packet sizes. Over UDP, when a response would be larger than the question leading to it, a server can respond with a short packet whose TC (truncated) bit has been set. The client should then retry using TCP and increase the padding of subsequent queries.[2]: §9
Versions 1 and 2 of the protocol use theX25519algorithm for key exchange,EdDSAfor signatures, as well asXSalsa20-Poly1305orXChaCha20-Poly1305for authenticated encryption.[2]: §11
As of 2023, there are no known vulnerabilities in the DNSCrypt protocol nor practical attacks against its underlying cryptographic constructions.
Anonymized DNSCrypt is a protocol extension proposed in 2019 to further improve DNS privacy.[11]
Instead of directly responding to clients, a resolver can act as a transparent proxy to another resolver, hiding the real client IP to the latter. Anonymized DNSCrypt, specifically designed for DNS traffic, is a lightweight alternative to running DNSCrypt through Tor and SOCKS proxies.[11]
Deployment of Anonymized DNSCrypt started in October 2019, and the protocol adoption was fast, with 40 DNS relays being set up only two weeks after the public availability of client and server implementations.[12]
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Incomputer networking,linear network codingis a program in which intermediate nodes transmit data from source nodes to sink nodes by means oflinear combinations.
Linear network coding may be used to improve a network's throughput, efficiency, andscalability, as well as reducing attacks and eavesdropping. Thenodesof a network takeseveralpackets and combine for transmission. This process may be used to attain the maximum possibleinformationflowin anetwork.
It has been proven that, theoretically,linear codingis enough to achieve the upper bound in multicast problems with one source.[1]However linear coding is not sufficient in general; even for more general versions of linearity such asconvolutional codingandfilter-bank coding.[2]Finding optimal coding solutions for general network problems with arbitrary demands is a hard problem, which can beNP-hard[3][4]and evenundecidable.[5][6]
In a linear network coding problem, a group of nodesP{\displaystyle P}are involved in moving the data fromS{\displaystyle S}source nodes toK{\displaystyle K}sink nodes. Each node generates new packets which are linear combinations of past received packets by multiplying them bycoefficientschosen from afinite field, typically of sizeGF(2s){\displaystyle GF(2^{s})}.
More formally, each node,pk{\displaystyle p_{k}}withindegree,InDeg(pk)=S{\displaystyle InDeg(p_{k})=S}, generates a messageXk{\displaystyle X_{k}}from the linear combination of received messages{Mi}i=1S{\displaystyle \{M_{i}\}_{i=1}^{S}}by the formula:
Where the valuesgki{\displaystyle g_{k}^{i}}are coefficients selected fromGF(2s){\displaystyle GF(2^{s})}. Since operations are computed in a finite field, the generated message is of the same length as the original messages. Each node forwards the computed valueXk{\displaystyle X_{k}}along with the coefficients,gki{\displaystyle g_{k}^{i}}, used in thekth{\displaystyle k^{\text{th}}}level,gki{\displaystyle g_{k}^{i}}.
Sink nodes receive these network coded messages, and collect them in a matrix. The original messages can be recovered by performingGaussian eliminationon the matrix.[7]In reduced row echelon form, decoded packets correspond to the rows of the formei=[0...010...0]{\displaystyle e_{i}=[0...010...0]}
A network is represented by adirected graphG=(V,E,C){\displaystyle {\mathcal {G}}=(V,E,C)}.V{\displaystyle V}is the set of nodes or vertices,E{\displaystyle E}is the set of directed links (or edges), andC{\displaystyle C}gives the capacity of each link ofE{\displaystyle E}. LetT(s,t){\displaystyle T(s,t)}be the maximum possible throughput from nodes{\displaystyle s}to nodet{\displaystyle t}. By themax-flow min-cut theorem,T(s,t){\displaystyle T(s,t)}is upper bounded by the minimum capacity of allcuts, which is the sum of the capacities of the edges on a cut, between these two nodes.
Karl Mengerproved that there is always a set of edge-disjoint paths achieving the upper bound in aunicastscenario, known as themax-flow min-cut theorem. Later, theFord–Fulkerson algorithmwas proposed to find such paths in polynomial time. Then, Edmonds proved in the paper "Edge-Disjoint Branchings"[which?]the upper bound in the broadcast scenario is also achievable, and proposed a polynomial time algorithm.
However, the situation in themulticastscenario is more complicated, and in fact, such an upper bound can't be reached using traditionalroutingideas. Ahlswede et al. proved that it can be achieved if additional computing tasks (incoming packets are combined into one or several outgoing packets) can be done in the intermediate nodes.[8]
The butterfly network[8]is often used to illustrate how linear network coding can outperformrouting. Two source nodes (at the top of the picture) have information A and B that must be transmitted to the two destination nodes (at the bottom). Each destination node wants to know both A and B. Each edge can carry only a single value (we can think of an edge transmitting a bit in each time slot).
If only routing were allowed, then the central link would be only able to carry A or B, but not both. Supposing we send A through the center; then the left destination would receive A twice and not know B at all. Sending B poses a similar problem for the right destination. We say that routing is insufficient because no routing scheme can transmit both A and B to both destinations simultaneously. Meanwhile, it takes four time slots in total for both destination nodes to know A and B.
Using a simple code, as shown, A and B can be transmitted to both destinations simultaneously by sending the sum of the symbols through the two relay nodes – encoding A and B using the formula "A+B". The left destination receives A and A + B, and can calculate B by subtracting the two values. Similarly, the right destination will receive B and A + B, and will also be able to determine both A and B. Therefore, with network coding, it takes only three time slots and improves the throughput.
Random linear network coding[9](RLNC) is a simple yet powerful encoding scheme, which in broadcast transmission schemes allows close to optimal throughput using a decentralized algorithm. Nodes transmit random linear combinations of the packets they receive, with coefficients chosen randomly, with a uniform distribution from a Galois field. If the field size is sufficiently large, the probability that the receiver(s) will obtain linearly independent combinations (and therefore obtain innovative information) approaches 1. It should however be noted that, although random linear network coding has excellent throughput performance, if a receiver obtains an insufficient number of packets, it is extremely unlikely that they can recover any of the original packets. This can be addressed by sending additional random linear combinations until the receiver obtains the appropriate number of packets.
There are three key parameters in RLNC. The first one is the generation size. In RLNC, the original data transmitted over the network is divided into packets. The source and intermediate nodes in the network can combine and recombine the set of original and coded packets. The originalM{\displaystyle M}packets form a block, usually called a generation. The number of original packets combined and recombined together is the generation size. The second parameter is the packet size. Usually, the size of the original packets is fixed. In the case of unequally-sized packets, these can be zero-padded if they are shorter or split into multiple packets if they are longer. In practice, the packet size can be the size of themaximum transmission unit(MTU) of the underlying network protocol. For example, it can be around 1500 bytes in anEthernet frame. The third key parameter is the Galois field used. In practice, the most commonly used Galois fields are binary extension fields. And the most commonly used sizes for the Galois fields are the binary fieldGF(2){\displaystyle GF(2)}and the so-called binary-8 (GF(28){\displaystyle GF(2^{8})}). In the binary field, each element is one bit long, while in the binary-8, it is one byte long. Since the packet size is usually larger than the field size, each packet is seen as a set of elements from the Galois field (usually referred to as symbols) appended together. The packets have a fixed amount of symbols (Galois field elements), and since all the operations are performed over Galois fields, then the size of the packets does not change with subsequent linear combinations.
The sources and the intermediate nodes can combine any subset of the original and previously coded packets performing linear operations. To form a coded packet in RLNC, the original and previously coded packets are multiplied by randomly chosen coefficients and added together. Since each packet is just an appended set of Galois field elements, the operations of multiplication and addition are performed symbol-wise over each of the individual symbols of the packets, as shown in the picture from the example.
To preserve the statelessness of the code, the coding coefficients used to generate the coded packets are appended to the packets transmitted over the network. Therefore, each node in the network can see what coefficients were used to generate each coded packet. One novelty of linear network coding over traditional block codes is that it allows the recombination of previously coded packets into new and valid coded packets. This process is usually called recoding. After a recoding operation, the size of the appended coding coefficients does not change. Since all the operations are linear, the state of the recoded packet can be preserved by applying the same operations of addition and multiplication to the payload and the appended coding coefficients. In the following example, we will illustrate this process.
Any destination node must collect enough linearly independent coded packets to be able to reconstruct the original data. Each coded packet can be understood as a linear equation where the coefficients are known since they are appended to the packet. In these equations, each of the originalM{\displaystyle M}packets is the unknown. To solve the linear system of equations, the destination needs at leastM{\displaystyle M}linearly independent equations (packets).
In the figure, we can see an example of two packets linearly combined into a new coded packet. In the example, we have two packets, namely packetf{\displaystyle f}and packete{\displaystyle e}. The generation size of our example is two. We know this because each packet has two coding coefficients (Cij{\displaystyle C_{ij}}) appended. The appended coefficients can take any value from the Galois field. However, an original, uncoded data packet would have appended the coding coefficients[0,1]{\displaystyle [0,1]}or[1,0]{\displaystyle [1,0]}, which means that they are constructed by a linear combination of zero times one of the packets plus one time the other packet. Any coded packet would have appended other coefficients. In our example, packetf{\displaystyle f}for instance has appended the coefficients[C11,C12]{\displaystyle [C_{11},C_{12}]}. Since network coding can be applied at any layter of the communication protocol, these packets can have a header from the other layers, which is ignored in the network coding operations.
Now, lets assume that the network node wants to produce a new coded packet combining packetf{\displaystyle f}and packete{\displaystyle e}. In RLNC, it will randomly choose two coding coefficients,d1{\displaystyle d_{1}}andd2{\displaystyle d_{2}}in the example. The node will multiply each symbol of packetf{\displaystyle f}byd1{\displaystyle d_{1}}, and each symbol of packete{\displaystyle e}byd2{\displaystyle d_{2}}. Then, it will add the results symbol-wise to produce the new coded data. It will perform the same operations of multiplication and addition to the coding coefficients of the coded packets.
Linear network coding is still a relatively new subject. However, the topic has been vastly researched over the last twenty years. Nevertheless, there are still some misconceptions that are no longer valid:
Decoding computational complexity:Network coding decoders have been improved over the years. Nowadays, the algorithms are highly efficient and parallelizable. In 2016, with Intel Core i5 processors withSIMDinstructions enabled, the decoding goodput of network coding was 750 MB/s for a generation size of 16 packets and 250 MB/s for a generation size of 64 packets.[10]Furthermore, today's algorithms can be vastly parallelizable, increasing the encoding and decoding goodput even further.[11]
Transmission Overhead:It is usually thought that the transmission overhead of network coding is high due to the need to append the coding coefficients to each coded packet. In reality, this overhead is negligible in most applications. The overhead due to coding coefficients can be computed as follows. Each packet has appendedM{\displaystyle M}coding coefficients. The size of each coefficient is the number of bits needed to represent one element of the Galois field. In practice, most network coding applications use a generation size of no more than 32 packets per generation and Galois fields of 256 elements (binary-8). With these numbers, each packet needsM∗log2(s)=32{\displaystyle M*log_{2}(s)=32}bytes of appended overhead. If each packet is 1500 bytes long (i.e. the Ethernet MTU), then 32 bytes represent an overhead of only 2%.
Overhead due to linear dependencies:Since the coding coefficients are chosen randomly in RLNC, there is a chance that some transmitted coded packets are not beneficial to the destination because they are formed using a linearly dependent combination of packets. However, this overhead is negligible in most applications. The linear dependencies depend on the Galois fields' size and are practically independent of the generation size used. We can illustrate this with the following example. Let us assume we are using a Galois field ofq{\displaystyle q}elements and a generation size ofM{\displaystyle M}packets. If the destination has not received any coded packet, we say it hasM{\displaystyle M}degrees of freedom, and then almost any coded packet will be useful and innovative. In fact, only the zero-packet (only zeroes in the coding coefficients) will be non-innovative. The probability of generating the zero-packet is equal to the probability of each of theM{\displaystyle M}coding coefficient to be equal to the zero-element of the Galois field. I.e., the probability of a non-innovative packet is of1qM{\displaystyle {\frac {1}{q^{M}}}}. With each successive innovative transmission, it can be shown that the exponent of the probability of a non innovative packet is reduced by one. When the destination has receivedM−1{\displaystyle M-1}innovative packets (i.e., it needs only one more packet to fully decode the data). Then the probability of a non innovative packet is of1q{\displaystyle {\frac {1}{q}}}. We can use this knowledge to calculate the expected number of linearly dependent packets per generation. In the worst-case scenario, when the Galois field used contains only two elements (q=2{\displaystyle q=2}), the expected number of linearly dependent packets per generation is of 1.6 extra packets. If our generation size if of 32 or 64 packets, this represents an overhead of 5% or 2.5%, respectively. If we use the binary-8 field (q=256{\displaystyle q=256}), then the expected number of linearly dependent packets per generation is practically zero. Since it is the last packets the major contributor to the overhead due to linear dependencies, there are RLNC-based protocols such as tunable sparse network coding[12]that exploit this knowledge. These protocols introduce sparsity (zero-elements) in the coding coefficients at the beginning of the transmission to reduce the decoding complexity, and reduce the sparsity at the end of the transmission to reduce the overhead due to linear dependencies.
Over the years, multiple researchers and companies have integrated network coding solutions into their applications.[13]We can list some of the applications of network coding in different areas:
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Inmathematics, theWeil pairingis apairing(bilinear form, though withmultiplicative notation) on the points of order dividingnof anelliptic curveE, taking values innthroots of unity. More generally there is a similar Weil pairing between points of ordernof an abelian variety and its dual. It was introduced byAndré Weil(1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results forelliptic functionswere known, and can be expressed simply by use of theWeierstrass sigma function.
Choose an elliptic curveEdefined over afieldK, and an integern> 0 (we requirento be coprime to char(K) if char(K) > 0) such thatKcontains aprimitive nth root of unity. Then then-torsion onE(K¯){\displaystyle E({\overline {K}})}is known to be aCartesian productof twocyclic groupsof ordern. The Weil pairing produces ann-th root of unity
by means ofKummer theory, for any two pointsP,Q∈E(K)[n]{\displaystyle P,Q\in E(K)[n]}, whereE(K)[n]={T∈E(K)∣n⋅T=O}{\displaystyle E(K)[n]=\{T\in E(K)\mid n\cdot T=O\}}andμn={x∈K∣xn=1}{\displaystyle \mu _{n}=\{x\in K\mid x^{n}=1\}}.
A down-to-earth construction of the Weil pairing is as follows.[citation needed]Choose a functionFin thefunction fieldofEover thealgebraic closureofKwithdivisor
SoFhas a simple zero at each pointP+kQ, and a simple pole at each pointkQif these points are all distinct. ThenFis well-defined up to multiplication by a constant. IfGis the translation ofFbyQ, then by constructionGhas the same divisor, so the functionG/Fis constant.
Therefore if we define
we shall have ann-th root of unity (as translatingntimes must give 1) other than 1. With this definition it can be shown thatwis alternating and bilinear,[1]giving rise to a non-degenerate pairing on then-torsion.
The Weil pairing does not extend to a pairing on all the torsion points (the direct limit ofn-torsion points) because the pairings for differentnare not the same. However
they do fit together to give a pairingTℓ(E) ×Tℓ(E) →Tℓ(μ) on theTate moduleTℓ(E) of the elliptic curveE(the inverse limit of the ℓn-torsion points) to the Tate moduleTℓ(μ) of the multiplicative group (the inverse limit of ℓnroots of unity).
Forabelian varietiesover an algebraically closed fieldK, the Weil pairing is a nondegenerate pairing
for allnprime to the characteristic ofK.[2]HereA∨{\displaystyle A^{\vee }}denotes thedual abelian varietyofA. This is the so-calledWeil pairingfor higher dimensions. IfAis equipped with apolarisation
then composition gives a (possibly degenerate) pairing
IfCis a projective, nonsingular curve of genus ≥ 0 overk, andJitsJacobian, then thetheta-divisorofJinduces a principal polarisation ofJ, which in this particular case happens to be an isomorphism (seeautoduality of Jacobians). Hence, composing the Weil pairing forJwith the polarisation gives a nondegenerate pairing
for allnprime to the characteristic ofk.
As in the case of elliptic curves, explicit formulae for this pairing can be given in terms ofdivisorsofC.
The pairing is used innumber theoryandalgebraic geometry, and has also been applied inelliptic curve cryptographyandidentity based encryption.
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https://en.wikipedia.org/wiki/Weil_pairing
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As Gafgyt
As QBot
As PinkSlip
BASHLITE(also known asGafgyt,Lizkebab,PinkSlip,Qbot,TorlusandLizardStresser) ismalwarewhich infectsLinuxsystems in order to launchdistributed denial-of-service attacks(DDoS).[1]Originally it was also known under the nameBashdoor,[2]but this term now refers to the exploit method used by the malware. It has been used to launch attacks of up to 400Gbps.[3]
The original version in 2014 exploited a flaw in thebashshell - theShellshocksoftware bug - to exploit devices runningBusyBox.[4][5][6][7]A few months later a variant was detected that could also infect other vulnerable devices in the local network.[8]In 2015 its source code was leaked, causing a proliferation of different variants,[9]and by 2016 it was reported that one million devices have been infected.[10][11][12][13]
Of the identifiable devices participating in these botnets in August 2016 almost 96 percent wereIoTdevices (of which 95 percent were cameras andDVRs), roughly 4 percent werehome routers- and less than 1 percent were compromisedLinux servers.[9]
BASHLITE is written inC, and designed to easily cross-compile to variouscomputer architectures.[9]
Exact capabilities differ between variants, but the most common features[9]generate several different types of DDoS attacks: it can hold openTCPconnections, send a random string of junk characters to a TCP or aUDPport, or repeatedly send TCPpacketswith specified flags. They may also have a mechanism to run arbitrary shell commands on the infected machine. There are no facilities forreflectedoramplification attacks.
BASHLITE uses aclient–server modelfor command and control. The protocol used for communication is essentially a lightweight version ofInternet Relay Chat(IRC).[14]Even though it supports multiple command and control servers, most variants only have a single command and control IP-addresshardcoded.
It propagates viabrute forcing, using a built-in dictionary of common usernames and passwords. The malware connects to random IP addresses and attempts to login, with successful logins reported back to the command and control server.
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https://en.wikipedia.org/wiki/BASHLITE
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Incomputer security, abillion laughs attackis a type ofdenial-of-service (DoS) attackwhich is aimed atparsersofXMLdocuments.[1]
It is also referred to as anXML bombor as an exponential entity expansion attack.[2]
The example attack consists of defining 10 entities, each defined as consisting of 10 of the previous entity, with the document consisting of a single instance of the largest entity, which expands to onebillioncopies of the first entity. Versions with larger amount of entries also exist.
In the most frequently cited example, the first entity is thestring"lol", hence the name "billion laughs". At the time this vulnerability was first reported, thecomputer memoryused by a billion instances of the string "lol" would likely exceed that available to the process parsing the XML.
While the original form of the attack was aimed specifically at XML parsers, the term may be applicable to similar subjects as well.[1]
The problem was first reported as early as 2002,[3]but began to be widely addressed in 2008.[4]
Defenses against this kind of attack include capping the memory allocated in an individual parser if loss of the document is acceptable, or treating entities symbolically and expanding them lazily only when (and to the extent) their content is to be used.
When an XML parser loads this document, it sees that it includes one root element, "lolz", that contains the text "&lol9;". However, "&lol9;" is a defined entity that expands to a string containing ten "&lol8;" strings. Each "&lol8;" string is a defined entity that expands to ten "&lol7;" strings, and so on. After all the entity expansions have been processed, this small (< 1 KB) block of XML will actually contain 109= a billion "lol"s, taking up almost 3gigabytesof memory.[5]
The billion laughs attack described above can take anexponentialamount of space or time. Thequadratic blowupvariation causesquadratic growthin resource requirements by simply repeating a large entity over and over again, to avoid countermeasures that detect heavily nested entities.[6](Seecomputational complexity theoryfor comparisons of different growth classes.)
A "billion laughs" attack could exist for any file format that can contain macro expansions, for example thisYAMLbomb:
This crashed earlier versions ofGobecause the Go YAML processor (contrary to the YAML spec) expands references as if they were macros. The Go YAML processor was modified to fail parsing if the result object becomes too large.
Enterprise software likeKuberneteshas been affected by this attack through its YAML parser.[7][8]For this reason, either a parser with intentionally limited capabilities is preferred (like StrictYAML) or file formats that do not allow references are often preferred for data arriving from untrusted sources.[9][failed verification]
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https://en.wikipedia.org/wiki/Billion_laughs_attack
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AsLovsan
AsMSBLAST
Blaster(also known asLovsan,Lovesan, orMSBlast) was acomputer wormthat spread on computers runningoperating systemsWindows XPandWindows 2000during August 2003.[1]
The worm was first noticed and started spreading on August 11, 2003. The rate that it spread increased until the number of infections peaked on August 13, 2003. Once a network (such as a company or university) was infected, it spread more quickly within the network because firewalls typically did not prevent internal machines from using a certain port.[2]Filtering by ISPs and widespread publicity about the worm curbed the spread of Blaster.
In September 2003, Jeffrey Lee Parson, an 18-year-old fromHopkins, Minnesota, was indicted for creating the B variant of the Blaster worm; he admitted responsibility and was sentenced to an 18-monthprisonterm in January 2005.[3]The author of the original A variant remains unknown.
According to court papers, the original Blaster was created after security researchers from the Chinese group Xfocusreverse engineeredthe original Microsoft patch that allowed for execution of the attack.[4]
The worm spreads by exploiting abuffer overflowdiscovered by the Polish security research group Last Stage of Delirium[5]in theDCOMRPCservice on the affected operating systems, for which a patch had been released one month earlier in MS03-026[6](CVE-2003-0352) and later in MS03-039.[7]This allowed the worm to spread without users opening attachments simply by spamming itself to large numbers of random IP addresses. Four versions have been detected in the wild.[8]These are the most well-known exploits of the original flaw in RPC, but there were in fact another 12 different vulnerabilities that did not see as much media attention.[9]
The worm was programmed to start aSYN floodagainst port 80 ofwindowsupdate.comif the system date is after August 15 and before December 31 and after the 15th day of other months, thereby creating adistributed denial of service attack(DDoS) against the site.[8]The damage to Microsoft was minimal as the site targeted was windowsupdate.com, rather than windowsupdate.microsoft.com, to which the former was redirected. Microsoft temporarily shut down the targeted site to minimize potential effects from the worm.[citation needed]
The worm's executable, MSBlast.exe,[10]contains two messages. The first reads:
I just want to say LOVE YOU SAN!!
This message gave the worm the alternative name of Lovesan. The second reads:
billy gates why do you make this possible ? Stop making moneyand fix your software!!
This is a message toBill Gates, theco-founderof Microsoft and the target of the worm.
The worm also creates the followingregistryentry so that it is launched every time Windows starts:
HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Run\ windows auto update=msblast.exe
Although the worm can only spread on systems runningWindows 2000orWindows XP, it can cause instability in theRPCservice on systems running other versions ofWindows NT, includingWindows Server 2003andWindows XP Professional x64 Edition. In particular, the worm does not spread in Windows Server 2003 because Windows Server 2003 was compiled with the /GS switch, which detected the buffer overflow and shut the RPCSS process down.[26]
When infection occurs, the buffer overflow causes the RPC service to crash, leading Windows to display the following message and then automatically reboot, usually after 60 seconds.[27]
System Shutdown:
This system is shutting down. Please save all work in progress and log off. Any unsaved changes will be lost. This shutdown was initiated by NT AUTHORITY\SYSTEM
Time before shutdown: hours:minutes:seconds
Message:
Windows must now restart because the Remote Procedure Call (RPC) Service terminated unexpectedly.
This was the first indication many users had an infection; it often occurred a few minutes after every startup on compromised machines. A simple resolution to stop countdown is to run the "shutdown /a" command,[28]causing some side effects such as an empty (without users) Welcome Screen.[29]TheWelchiaworm had a similar effect. Months later, theSasser wormsurfaced, which caused a similar message to appear.
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https://en.wikipedia.org/wiki/Blaster_(computer_worm)
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Aclear channel assessment attackorQueensland attackis aphysical layerDoS attackagainstWi-Finetworks. The attack focuses the need of a wireless network to receive the "clear channel assessment"; which is a function withinCSMA/CAto determine whether the wireless medium is ready and able to receive data, so that the transmitter may start sending it. The attack makes it appear that the airwaves are busy, which basically puts the entire system on hold.
The attack works only on802.11b, and is not effective on the OFDM-based protocols802.11gand802.11a. However, some hybrid 802.11b/g access points will hinder the 802.11g network when the 802.11b network is attacked.[1]
The attack was originally discovered by researchers atQueensland University of Technology'sInformation Security Research Center,[2]thus it is where the nameQueensland attackcomes from.
The signal telling the system the airwaves are busy is of course sent through the attacker'sNIC, by placing it in continuous transmit mode. The attack can be set up through the use of the Intersil'sPrism Test Utility(PrismTestUtil322.exe).
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https://en.wikipedia.org/wiki/Clear_channel_assessment_attack
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Dendroidismalwarethat affects Android OS and targets the mobile platform.[1]
It was first discovered in early of 2014 by Symantec and appeared in the underground for sale for $300.[2]Certain features were noted as being used in Dendroid, such as the ability to hide from emulators at the time.[3]When first discovered in 2014 it was one of the most sophisticated Androidremote administration toolsknown at that time.[4]It was one of the firstTrojan applicationsto get past Google's Bouncer and caused researchers to warn about it being easier to create Android malware due to it.[5]It also seems to have followed in the footsteps ofZeusand SpyEye by having simple-to-use command and control panels.[6]The code appeared to be leaked somewhere around 2014.[7]It was noted that anapk binderwas included in the leak, which provided a simple way to bind Dendroid to legitimate applications.
It is capable of:
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https://en.wikipedia.org/wiki/Dendroid_(malware)
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Distributed denial-of-service attacks on root nameserversareInternetevents in which distributeddenial-of-service attackstarget one or more of the thirteenDomain Name Systemroot nameserverclusters. The root nameservers arecritical infrastructure components of the Internet, mappingdomain namestoIP addressesand other resource record (RR) data.
Attacks against the root nameservers could, in theory, impact operation of the entire global Domain Name System, and thus all Internet services that use the global DNS, rather than just specific websites. However, in practice, the root nameserver infrastructure is highly resilient and distributed, using both the inherent features of DNS (result caching, retries, and multiple servers for the same zone with fallback if one or more fail), and, in recent years, a combination ofanycastandload balancertechniques used to implement most of the thirteen nominal individual root servers as globally distributed clusters of servers in multiple data centers.
In particular, the caching and redundancy features of DNS mean that it would require a sustained outage of all the major root servers for many days before any serious problems were created for most Internet users, and even then there are still numerous ways in which ISPs could set their systems up during that period to mitigate even a total loss of all root servers for an extended period of time: for example by installing their own copies of theglobal DNS root zonedata on nameservers within their network, and redirecting traffic to the root server IP addresses to those servers. Nevertheless, DDoS attacks on the root zone are taken seriously as a risk by the operators of the root nameservers, and they continue to upgrade the capacity andDDoS mitigationcapabilities of their infrastructure to resist any future attacks.
An effective attack against DNS might involve targetingtop-level domainservers (such as those servicing the.comdomain) instead of root name servers. Alternatively, aman-in-the-middle attackorDNS poisoningattack could be used, though they would be more difficult to carry out.
On October 21, 2002 an attack lasting for approximately one hour was targeted at all 13 DNS root name servers.[1]The attackers sent manyICMPping packets using abotnetto each of the servers. However, because the servers were protected by packet filters which were configured to block all incoming ICMP ping packets, they did not sustain much damage and there was little to no impact on Internet users.[2]
On February 6, 2007 an attack began at 10:00UTCand lasted twenty-four hours. At least two of the root servers (G-ROOT and L-ROOT) reportedly "suffered badly" while two others (F-ROOT and M-ROOT) "experienced heavy traffic". The latter two servers largely mitigated the damage by distributing requests to other root server instances withanycastaddressing.ICANNpublished a formal analysis shortly after the event.[3]
Due to a lack of detail, speculation about the incident proliferated in the press until details were released.[4]
During two intervals on November 30, 2015 and December 1, 2015, several of the root name servers received up to 5 million queries per second each, receiving valid queries for a single undisclosed domain name and then a different domain the next day. Source addresses were spread throughout IPv4 space, however these may have been spoofed. Some root server networks became saturated, resulting in timeouts, however redundancy among the root servers preventeddownstreamissues from occurring during this incident.[5][6]
On February 12, 2012, a statement[7]was posted onPastebincited to be fromAnonymous, threatening an attack on the root servers on March 31, 2012.[8]
"To protestSOPA,Wallstreet, our irresponsible leaders and the beloved bankers who are starving the world for their own selfish needs out of sheer sadistic fun, On March 31, anonymous will shut the Internet down," reads the statement. "Remember, this is a protest, we are not trying to ‘kill' the Internet, we are only temporarily shutting it down where it hurts the most…It may only last one hour, maybe more, maybe even a few days. No matter what, it will be global. It will be known."[9]
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https://en.wikipedia.org/wiki/Distributed_denial-of-service_attacks_on_root_nameservers
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DNS Floodis a type ofdenial-of-service attack. It is the process whereby the traffic on anetworkresource or machine is stopped for some time. The offender sends a great number of requests to the resource or machine so that it might become unavailable to those who might try to reach it. During aDNSflood the host that connects to the Internet is disrupted due to an overload of traffic. It can be referred to as a disruption that causes the work of the resource or machine to halt by not allowing the traffic to land on it.
This attack is mainly done byhackers[citation needed]to benefit from the attacked resource or machine. DDoS attacks have been perpetrated for many reasons, including blackmailing website owners and knocking out websites, including high-profile sites such as large bank websites.[1]
Many methods can and are being adopted to prevent these types of attacks some of which include dropping malformed packages, use filters to avoid receiving packages from sources having potential to attack, timing out half open connections with greater hostility. One can also setSYN,ICMP, andUDPat lower levels to prevent such DDoS attacks from harming one's network.[2][3]
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https://en.wikipedia.org/wiki/DNS_Flood
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Hit-and-run DDoSis a type ofdenial-of-service (DDoS) attackthat uses short bursts of high volume attacks in random intervals, spanning a time frame of days or weeks. The purpose of a hit-and-run DDoS is to prevent a user of a service from using that service by bringing down the hostserver.[1]This type of attack is to be distinguished from a persistent DDoS attack which continues until the attacker stops the attack or the host server is able to defend against it.[2]
A DDoS attack is characterized by an explicit attempt by attackers to prevent legitimate users of a service from using that service.[3]A hit-and-run DDoS is accomplished by using high volume network or application attacks in short bursts. The attacks only last long enough to bring down the server hosting the service, normally 20 to 60 minutes. The attack is then repeated every 12 to 24 hours over a period of days or weeks, causing issues for the company hosting the service.
Hit-and-run DDoS is sometimes used as a test DDoS attack. An attacker will inject a few badpacketsinto a network to test if it is online and functioning. Once the network is verified as functioning, an attacker will then use a persistent DDoS attack.[4]
Hit-and-run DDoS exploits anti-DDoS software and services which are used to defend against prolonged DDoS attacks. Activating such software can take longer than the actual attack, allowing a denial of service before DDoS protection can start to defend from the attack.
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https://en.wikipedia.org/wiki/Hit-and-run_DDoS
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Anintrusion detection system(IDS) is a device orsoftwareapplication that monitors a network or systems for malicious activity or policy violations.[1]Any intrusion activity or violation is typically either reported to an administrator or collected centrally using asecurity information and event management (SIEM)system. A SIEM system combines outputs from multiple sources and usesalarm filteringtechniques to distinguish malicious activity fromfalse alarms.[2]
IDS types range in scope from single computers to large networks.[3]The most common classifications arenetwork intrusion detection systems(NIDS) andhost-based intrusion detection systems(HIDS). A system that monitors important operating system files is an example of an HIDS, while a system that analyzes incoming network traffic is an example of an NIDS. It is also possible to classify IDS by detection approach. The most well-known variants aresignature-based detection(recognizing bad patterns, such asexploitation attempts) and anomaly-based detection (detecting deviations from a model of "good" traffic, which often relies onmachine learning). Another common variant is reputation-based detection (recognizing the potential threat according to the reputation scores). Some IDS products have the ability to respond to detected intrusions. Systems with response capabilities are typically referred to as anintrusion prevention system(IPS).[4]Intrusion detection systems can also serve specific purposes by augmenting them with custom tools, such as using a honeypot to attract and characterize malicious traffic.[5]
Although they both relate tonetwork security, an IDS differs from afirewallin that a conventional network firewall (distinct from anext-generation firewall) uses a static set of rules to permit or deny network connections. It implicitly prevents intrusions, assuming an appropriate set of rules have been defined. Essentially, firewalls limit access between networks to prevent intrusion and do not signal an attack from inside the network. An IDS describes a suspected intrusion once it has taken place and signals an alarm. An IDS also watches for attacks that originate from within a system. This is traditionally achieved by examining network communications, identifyingheuristicsand patterns (often known as signatures) of common computer attacks, and taking action to alert operators. A system that terminates connections is called an intrusion prevention system, and performs access control like anapplication layer firewall.[6]
IDS can be classified by where detection takes place (network orhost) or the detection method that is employed (signature or anomaly-based).[7]
Network intrusion detection systems (NIDS) are placed at a strategic point or points within the network to monitor traffic to and from all devices on the network.[8]It performs an analysis of passing traffic on the entiresubnet, and matches the traffic that is passed on the subnets to the library of known attacks. Once an attack is identified, or abnormal behavior is sensed, the alert can be sent to the administrator. NIDS function to safeguard every device and the entire network from unauthorized access.[9]
An example of an NIDS would be installing it on the subnet where firewalls are located in order to see if someone is trying to break into the firewall. Ideally one would scan all inbound and outbound traffic, however doing so might create a bottleneck that would impair the overall speed of the network.OPNETand NetSim are commonly used tools for simulating network intrusion detection systems. NID Systems are also capable of comparing signatures for similar packets to link and drop harmful detected packets which have a signature matching the records in the NIDS. When we classify the design of the NIDS according to the system interactivity property, there are two types: on-line and off-line NIDS, often referred to as inline and tap mode, respectively. On-line NIDS deals with the network in real time. It analyses theEthernet packetsand applies some rules, to decide if it is an attack or not. Off-line NIDS deals with stored data and passes it through some processes to decide if it is an attack or not.
NIDS can be also combined with other technologies to increase detection and prediction rates.Artificial Neural Network(ANN) based IDS are capable of analyzing huge volumes of data due to the hidden layers and non-linear modeling, however this process requires time due its complex structure.[10]This allows IDS to more efficiently recognize intrusion patterns.[11]Neural networks assist IDS in predicting attacks by learning from mistakes; ANN based IDS help develop an early warning system, based on two layers. The first layer accepts single values, while the second layer takes the first's layers output as input; the cycle repeats and allows the system to automatically recognize new unforeseen patterns in the network.[12]This system can average 99.9% detection and classification rate, based on research results of 24 network attacks, divided in four categories: DOS, Probe, Remote-to-Local, and user-to-root.[13]
Host intrusion detection systems (HIDS) run on individual hosts or devices on the network. A HIDS monitors the inbound and outbound packets from the device only and will alert the user or administrator if suspicious activity is detected. It takes a snapshot of existing system files and matches it to the previous snapshot. If the critical system files were modified or deleted, an alert is sent to the administrator to investigate. An example of HIDS usage can be seen on mission critical machines, which are not expected to change their configurations.[14][15]
Signature-based IDS is the detection of attacks by looking for specific patterns, such as byte sequences in network traffic, or known malicious instruction sequences used by malware.[16]This terminology originates fromanti-virus software, which refers to these detected patterns as signatures. Although signature-based IDS can easily detect known attacks, it is difficult to detect new attacks, for which no pattern is available.[17]
In signature-based IDS, the signatures are released by a vendor for all its products. On-time updating of the IDS with the signature is a key aspect.
Anomaly-based intrusion detection systemswere primarily introduced to detect unknown attacks, in part due to the rapid development of malware. The basic approach is to use machine learning to create a model of trustworthy activity, and then compare new behavior against this model. Since these models can be trained according to the applications and hardware configurations, machine learning based method has a better generalized property in comparison to traditional signature-based IDS. Although this approach enables the detection of previously unknown attacks, it may suffer fromfalse positives: previously unknown legitimate activity may also be classified as malicious. Most of the existing IDSs suffer from the time-consuming during detection process that degrades the performance of IDSs. Efficientfeature selectionalgorithm makes the classification process used in detection more reliable.[18]
New types of what could be called anomaly-based intrusion detection systems are being viewed byGartneras User and Entity Behavior Analytics (UEBA)[19](an evolution of theuser behavior analyticscategory) and networktraffic analysis(NTA).[20]In particular, NTA deals with malicious insiders as well as targeted external attacks that have compromised a user machine or account. Gartner has noted that some organizations have opted for NTA over more traditional IDS.[21]
Some systems may attempt to stop an intrusion attempt but this is neither required nor expected of a monitoring system. Intrusion detection and prevention systems (IDPS) are primarily focused on identifying possible incidents, logging information about them, and reporting attempts. In addition, organizations use IDPS for other purposes, such as identifying problems with security policies, documenting existing threats and deterring individuals from violating security policies. IDPS have become a necessary addition to the security infrastructure of nearly every organization.[22]
IDPS typically record information related to observed events, notify security administrators of important observed events and produce reports. Many IDPS can also respond to a detected threat by attempting to prevent it from succeeding. They use several response techniques, which involve the IDPS stopping the attack itself, changing the security environment (e.g. reconfiguring a firewall) or changing the attack's content.[22]
Intrusion prevention systems(IPS), also known asintrusion detection and prevention systems(IDPS), arenetwork securityappliances that monitor network or system activities for malicious activity. The main functions of intrusion prevention systems are to identify malicious activity, log information about this activity, report it and attempt to block or stop it.[23].
Intrusion prevention systems are considered extensions of intrusion detection systems because they both monitor network traffic and/or system activities for malicious activity. The main differences are, unlike intrusion detection systems, intrusion prevention systems are placed in-line and are able to actively prevent or block intrusions that are detected.[24]: 273[25]: 289IPS can take such actions as sending an alarm, dropping detected malicious packets, resetting a connection or blocking traffic from the offending IP address.[26]An IPS also can correctcyclic redundancy check(CRC)errors, defragment packet streams, mitigate TCP sequencing issues, and clean up unwantedtransportandnetwork layeroptions.[24]: 278[27]
Intrusion prevention systems can be classified into four different types:[23][28]
The majority of intrusion prevention systems utilize one of three detection methods: signature-based, statistical anomaly-based, and stateful protocol analysis.[25]: 301[29]
The correct placement of intrusion detection systems is critical and varies depending on the network. The most common placement is behind the firewall, on the edge of a network. This practice provides the IDS with high visibility of traffic entering your network and will not receive any traffic between users on the network. The edge of the network is the point in which a network connects to the extranet. Another practice that can be accomplished if more resources are available is a strategy where a technician will place their first IDS at the point of highest visibility and depending on resource availability will place another at the next highest point, continuing that process until all points of the network are covered.[33]
If an IDS is placed beyond a network's firewall, its main purpose would be to defend against noise from the internet but, more importantly, defend against common attacks, such as port scans and network mapper. An IDS in this position would monitor layers 4 through 7 of the OSI model and would be signature-based. This is a very useful practice, because rather than showing actual breaches into the network that made it through the firewall, attempted breaches will be shown which reduces the amount of false positives. The IDS in this position also assists in decreasing the amount of time it takes to discover successful attacks against a network.[34]
Sometimes an IDS with more advanced features will be integrated with a firewall in order to be able to intercept sophisticated attacks entering the network. Examples of advanced features would include multiple security contexts in the routing level and bridging mode. All of this in turn potentially reduces cost and operational complexity.[34]
Another option for IDS placement is within the actual network. These will reveal attacks or suspicious activity within the network. Ignoring the security within a network can cause many problems, it will either allow users to bring about security risks or allow an attacker who has already broken into the network to roam around freely. Intense intranet security makes it difficult for even those hackers within the network to maneuver around and escalate their privileges.[34]
There are a number of techniques which attackers are using, the following are considered 'simple' measures which can be taken to evade IDS:
The earliest preliminary IDS concept was delineated in 1980 by James Anderson at theNational Security Agencyand consisted of a set of tools intended to help administrators review audit trails.[38]User access logs, file access logs, and system event logs are examples of audit trails.
Fred Cohennoted in 1987 that it is impossible to detect an intrusion in every case, and that the resources needed to detect intrusions grow with the amount of usage.[39]
Dorothy E. Denning, assisted byPeter G. Neumann, published a model of an IDS in 1986 that formed the basis for many systems today.[40]Her model used statistics foranomaly detection, and resulted in an early IDS atSRI Internationalnamed the Intrusion Detection Expert System (IDES), which ran onSunworkstations and could consider both user and network level data.[41]IDES had a dual approach with a rule-basedExpert Systemto detect known types of intrusions plus a statistical anomaly detection component based on profiles of users, host systems, and target systems. The author of "IDES: An Intelligent System for Detecting Intruders", Teresa F. Lunt, proposed adding anartificial neural networkas a third component. She said all three components could then report to a resolver. SRI followed IDES in 1993 with the Next-generation Intrusion Detection Expert System (NIDES).[42]
TheMulticsintrusion detection and alerting system (MIDAS), an expert system using P-BEST andLisp, was developed in 1988 based on the work of Denning and Neumann.[43]Haystack was also developed in that year using statistics to reduce audit trails.[44]
In 1986 theNational Security Agencystarted an IDS research transfer program underRebecca Bace. Bace later published the seminal text on the subject,Intrusion Detection, in 2000.[45]
Wisdom & Sense (W&S) was a statistics-based anomaly detector developed in 1989 at theLos Alamos National Laboratory.[46]W&S created rules based on statistical analysis, and then used those rules for anomaly detection.
In 1990, the Time-based Inductive Machine (TIM) did anomaly detection using inductive learning of sequential user patterns inCommon Lispon aVAX3500 computer.[47]The Network Security Monitor (NSM) performed masking on access matrices for anomaly detection on a Sun-3/50 workstation.[48]The Information Security Officer's Assistant (ISOA) was a 1990 prototype that considered a variety of strategies including statistics, a profile checker, and an expert system.[49]ComputerWatch atAT&T Bell Labsused statistics and rules for audit data reduction and intrusion detection.[50]
Then, in 1991, researchers at theUniversity of California, Daviscreated a prototype Distributed Intrusion Detection System (DIDS), which was also an expert system.[51]The Network Anomaly Detection and Intrusion Reporter (NADIR), also in 1991, was a prototype IDS developed at the Los Alamos National Laboratory's Integrated Computing Network (ICN), and was heavily influenced by the work of Denning and Lunt.[52]NADIR used a statistics-based anomaly detector and an expert system.
TheLawrence Berkeley National LaboratoryannouncedBroin 1998, which used its own rule language for packet analysis fromlibpcapdata.[53]Network Flight Recorder (NFR) in 1999 also used libpcap.[54]
APE was developed as a packet sniffer, also using libpcap, in November, 1998, and was renamedSnortone month later. Snort has since become the world's largest used IDS/IPS system with over 300,000 active users.[55]It can monitor both local systems, and remote capture points using theTZSPprotocol.
The Audit Data Analysis and Mining (ADAM) IDS in 2001 usedtcpdumpto build profiles of rules for classifications.[56]In 2003,Yongguang Zhangand Wenke Lee argue for the importance of IDS in networks with mobile nodes.[57]
In 2015, Viegas and his colleagues[58]proposed an anomaly-based intrusion detection engine, aiming System-on-Chip (SoC) for applications in Internet of Things (IoT), for instance. The proposal applies machine learning for anomaly detection, providing energy-efficiency to a Decision Tree, Naive-Bayes, and k-Nearest Neighbors classifiers implementation in an Atom CPU and its hardware-friendly implementation in a FPGA.[59][60]In the literature, this was the first work that implement each classifier equivalently in software and hardware and measures its energy consumption on both. Additionally, it was the first time that was measured the energy consumption for extracting each features used to make the network packet classification, implemented in software and hardware.[61]
This article incorporatespublic domain materialfromKaren Scarfone, Peter Mell.Guide to Intrusion Detection and Prevention Systems, SP800-94(PDF).National Institute of Standards and Technology. Retrieved1 January2010.
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Incomputer jargon, akiller pokeis a method of inducing physicalhardwaredamage on a machine or itsperipheralsby the insertion of invalid values, via, for example,BASIC'sPOKEcommand, into amemory-mappedcontrolregister. The term is typically used to describe a family of fairly well known tricks that can overload theanalog electronicsin theCRTmonitorsof computers lacking hardwaresanity checking(notable examples being theIBM Portable[1]andCommodore PET.)
TheZ1(1938) andZ3(1941) computers built byKonrad Zusecontained illegal sequences of instructions which damaged the hardware if executed by accident.[2]
ThePET-specific killer poke is connected to the architecture of that machine's video rasterizer circuits. In early PETs, writing a certain value to the memory address of a certainI/Oregister (POKE 59458,62[3]) made the machine able to display text and graphics on the screen 106% faster. This was accomplished by disabling a "wait to print to screen" safeguard designed to reduce static/noise by preventing the shared VRAM from being read by the display at the same time as it was being written to by the CPU. With this safeguard disabled, graphics could appear on the screen twice as fast, but small bits of static would also appear. Despite the static, some games designed for early PETs included this POKE in their source code in order to benefit from the faster graphics.[1]
When the PET range was revamped with updated hardware, the video rasterizer circuits were redesigned to run at a faster speed and without the need for a "wait to print" safeguard. Thus, the old POKE trick no longer resulted in faster graphics. Instead, performing the old trick on the new hardware led to strange behavior by the new video chip, which could causesignal contentionand possibly damage the PET's integratedCRTmonitor.[4]This is because the exact pin targeted by the POKE command used to control display timing, but in the upgraded video chip, that pin controlled the vertical sync. Thus, running the POKE on the newer hardware caused graphics to compress vertically, sometimes down to an extremely bright horizontal line. Fears that this anomaly mightburn into the display led to the nickname "killer poke";[3]however, it is not known to have ever caused any permanent damage to the monitor.[5]
TheCommodore 64had an optional external 5-1/4" floppy drive. TheCommodore 1541contained a 6502 microprocessor which was used to runCommodore DOSand also to manage the drive mechanism. The drives stored data on 35 tracks (#0–34), and the stepper motor could be manually controlled through BASIC by PRINT#-ing "MEMORY-WRITE" commands to the drive (which correspond to the POKE command of BASIC, but write to the drive's internal memory and I/O registers, not those of the computer itself). If the drive was at either end of its range (track 0 or track 39) and it was commanded to continue moving, there was no software or firmware method to prevent drive damage. Continued "knocking" of the drive head against the stop would throw the mechanism out of alignment. The problem was exacerbated bycopy protectiontechniques that used non-standard disk formats with unusual track counts. TheCommodore 1571had an optical head stop instead of a mechanical one.
The originalTRS-80andTRS-80 Model IIIhad the ability to switch between a 32-character-wide display and a 64-character display. Doing so actuated a relay in the video hardware, accomplished by writing to a specific memory-mapped control register.[6]Programs that repeatedly switched between 32- and 64-character modes at high speed (either on purpose or accidentally) could permanently damage the video hardware.[citation needed]While this is not a single "killer poke", it demonstrates a softwarefailure modethat could permanently damage the hardware.
The TRS-80 Model I also has a similar cassette motor relay accessible via a memory poke command and could result in damaging the relay.
Certain models of LG CD-ROM drives with specific firmware used an abnormal command for "update firmware": the "clear buffer" command usually used on CD-RW drives. Linux uses this command to tell the difference between CD-ROM and CD-RW drives. Most CD-ROM drives dependably return an error for the unsupported CD-RW command, but the faulty drives interpreted it as "update firmware", causing them to stop working (or, in casual parlance, to be "bricked").[7]
The resource of flash memory is large, but limited. Since writing to storage is an essential operation, most applications have enough privileges to exhaust the resource of flash chips within 24 hours by filling the storage enough to causewrite amplificationand continuously rewriting a small file.[8]
Systemdmounts variables used byUnified Extensible Firmware InterfaceonLinuxsystem'ssysfsas writable by the root user of a system. As a result, it is possible for theroot userof a system to completely brick a system with a non-conforming UEFI implementation (specifically someMSilaptops) by using thermcommand to delete the/sys/firmware/efi/efivars/directory, or recursively delete theroot directory.[9][better source needed]
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Alace card(also called awhoopee card,ventilator card,flyswatter card, orIBM doily[citation needed]) is apunched cardwith all holes punched. They were mainly used as practical jokes to cause disruption incard readers. Card readers tended to jam when a lace card was inserted, as the resulting card had insufficient structural integrity to avoid buckling inside the mechanism. Card punchers could also jam trying to produce cards with all holes punched, owing to power-supply problems. When a lace card was fed through the reader, acard knifeorcard saw(a flat tool used with punched card readers and card punches) was needed to clear the jam.[1][2]
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Regardingcomputer security, amixed threat attackis an attack that uses several different tactics to infiltrate acomputeruser's environment. A mixedthreatattack might include an infected file that comes in by way ofspamor can be received by anInternetdownload. Mixed threat attacks try to exploit multiple vulnerabilities to get into a system. By launching multiple diverse attacks in parallel, the attacker can exploit more entry points than with just a single attack.
Because these threats are based on multiple single-attacks, they are much harder to detect.Firewallscan help with these types of attacks; if configured correctly, they are somewhat effective against this type of attack. However, if the attack is embedded inside an application, it is no longer able to prevent it. Typical techniques employed are to define the multiple access threat with a signature that can represent identification for the virus removal software. These types of techniques need to be employed on the host machine because sometimes the firewall orIntrusion Detection Systemis not able to detect the attack.[1]
NimdaandCode Redare examples of computer worms that utilized mixed threat attacks.[1]
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Anintrusion detection system(IDS) is a device orsoftwareapplication that monitors a network or systems for malicious activity or policy violations.[1]Any intrusion activity or violation is typically either reported to an administrator or collected centrally using asecurity information and event management (SIEM)system. A SIEM system combines outputs from multiple sources and usesalarm filteringtechniques to distinguish malicious activity fromfalse alarms.[2]
IDS types range in scope from single computers to large networks.[3]The most common classifications arenetwork intrusion detection systems(NIDS) andhost-based intrusion detection systems(HIDS). A system that monitors important operating system files is an example of an HIDS, while a system that analyzes incoming network traffic is an example of an NIDS. It is also possible to classify IDS by detection approach. The most well-known variants aresignature-based detection(recognizing bad patterns, such asexploitation attempts) and anomaly-based detection (detecting deviations from a model of "good" traffic, which often relies onmachine learning). Another common variant is reputation-based detection (recognizing the potential threat according to the reputation scores). Some IDS products have the ability to respond to detected intrusions. Systems with response capabilities are typically referred to as anintrusion prevention system(IPS).[4]Intrusion detection systems can also serve specific purposes by augmenting them with custom tools, such as using a honeypot to attract and characterize malicious traffic.[5]
Although they both relate tonetwork security, an IDS differs from afirewallin that a conventional network firewall (distinct from anext-generation firewall) uses a static set of rules to permit or deny network connections. It implicitly prevents intrusions, assuming an appropriate set of rules have been defined. Essentially, firewalls limit access between networks to prevent intrusion and do not signal an attack from inside the network. An IDS describes a suspected intrusion once it has taken place and signals an alarm. An IDS also watches for attacks that originate from within a system. This is traditionally achieved by examining network communications, identifyingheuristicsand patterns (often known as signatures) of common computer attacks, and taking action to alert operators. A system that terminates connections is called an intrusion prevention system, and performs access control like anapplication layer firewall.[6]
IDS can be classified by where detection takes place (network orhost) or the detection method that is employed (signature or anomaly-based).[7]
Network intrusion detection systems (NIDS) are placed at a strategic point or points within the network to monitor traffic to and from all devices on the network.[8]It performs an analysis of passing traffic on the entiresubnet, and matches the traffic that is passed on the subnets to the library of known attacks. Once an attack is identified, or abnormal behavior is sensed, the alert can be sent to the administrator. NIDS function to safeguard every device and the entire network from unauthorized access.[9]
An example of an NIDS would be installing it on the subnet where firewalls are located in order to see if someone is trying to break into the firewall. Ideally one would scan all inbound and outbound traffic, however doing so might create a bottleneck that would impair the overall speed of the network.OPNETand NetSim are commonly used tools for simulating network intrusion detection systems. NID Systems are also capable of comparing signatures for similar packets to link and drop harmful detected packets which have a signature matching the records in the NIDS. When we classify the design of the NIDS according to the system interactivity property, there are two types: on-line and off-line NIDS, often referred to as inline and tap mode, respectively. On-line NIDS deals with the network in real time. It analyses theEthernet packetsand applies some rules, to decide if it is an attack or not. Off-line NIDS deals with stored data and passes it through some processes to decide if it is an attack or not.
NIDS can be also combined with other technologies to increase detection and prediction rates.Artificial Neural Network(ANN) based IDS are capable of analyzing huge volumes of data due to the hidden layers and non-linear modeling, however this process requires time due its complex structure.[10]This allows IDS to more efficiently recognize intrusion patterns.[11]Neural networks assist IDS in predicting attacks by learning from mistakes; ANN based IDS help develop an early warning system, based on two layers. The first layer accepts single values, while the second layer takes the first's layers output as input; the cycle repeats and allows the system to automatically recognize new unforeseen patterns in the network.[12]This system can average 99.9% detection and classification rate, based on research results of 24 network attacks, divided in four categories: DOS, Probe, Remote-to-Local, and user-to-root.[13]
Host intrusion detection systems (HIDS) run on individual hosts or devices on the network. A HIDS monitors the inbound and outbound packets from the device only and will alert the user or administrator if suspicious activity is detected. It takes a snapshot of existing system files and matches it to the previous snapshot. If the critical system files were modified or deleted, an alert is sent to the administrator to investigate. An example of HIDS usage can be seen on mission critical machines, which are not expected to change their configurations.[14][15]
Signature-based IDS is the detection of attacks by looking for specific patterns, such as byte sequences in network traffic, or known malicious instruction sequences used by malware.[16]This terminology originates fromanti-virus software, which refers to these detected patterns as signatures. Although signature-based IDS can easily detect known attacks, it is difficult to detect new attacks, for which no pattern is available.[17]
In signature-based IDS, the signatures are released by a vendor for all its products. On-time updating of the IDS with the signature is a key aspect.
Anomaly-based intrusion detection systemswere primarily introduced to detect unknown attacks, in part due to the rapid development of malware. The basic approach is to use machine learning to create a model of trustworthy activity, and then compare new behavior against this model. Since these models can be trained according to the applications and hardware configurations, machine learning based method has a better generalized property in comparison to traditional signature-based IDS. Although this approach enables the detection of previously unknown attacks, it may suffer fromfalse positives: previously unknown legitimate activity may also be classified as malicious. Most of the existing IDSs suffer from the time-consuming during detection process that degrades the performance of IDSs. Efficientfeature selectionalgorithm makes the classification process used in detection more reliable.[18]
New types of what could be called anomaly-based intrusion detection systems are being viewed byGartneras User and Entity Behavior Analytics (UEBA)[19](an evolution of theuser behavior analyticscategory) and networktraffic analysis(NTA).[20]In particular, NTA deals with malicious insiders as well as targeted external attacks that have compromised a user machine or account. Gartner has noted that some organizations have opted for NTA over more traditional IDS.[21]
Some systems may attempt to stop an intrusion attempt but this is neither required nor expected of a monitoring system. Intrusion detection and prevention systems (IDPS) are primarily focused on identifying possible incidents, logging information about them, and reporting attempts. In addition, organizations use IDPS for other purposes, such as identifying problems with security policies, documenting existing threats and deterring individuals from violating security policies. IDPS have become a necessary addition to the security infrastructure of nearly every organization.[22]
IDPS typically record information related to observed events, notify security administrators of important observed events and produce reports. Many IDPS can also respond to a detected threat by attempting to prevent it from succeeding. They use several response techniques, which involve the IDPS stopping the attack itself, changing the security environment (e.g. reconfiguring a firewall) or changing the attack's content.[22]
Intrusion prevention systems(IPS), also known asintrusion detection and prevention systems(IDPS), arenetwork securityappliances that monitor network or system activities for malicious activity. The main functions of intrusion prevention systems are to identify malicious activity, log information about this activity, report it and attempt to block or stop it.[23].
Intrusion prevention systems are considered extensions of intrusion detection systems because they both monitor network traffic and/or system activities for malicious activity. The main differences are, unlike intrusion detection systems, intrusion prevention systems are placed in-line and are able to actively prevent or block intrusions that are detected.[24]: 273[25]: 289IPS can take such actions as sending an alarm, dropping detected malicious packets, resetting a connection or blocking traffic from the offending IP address.[26]An IPS also can correctcyclic redundancy check(CRC)errors, defragment packet streams, mitigate TCP sequencing issues, and clean up unwantedtransportandnetwork layeroptions.[24]: 278[27]
Intrusion prevention systems can be classified into four different types:[23][28]
The majority of intrusion prevention systems utilize one of three detection methods: signature-based, statistical anomaly-based, and stateful protocol analysis.[25]: 301[29]
The correct placement of intrusion detection systems is critical and varies depending on the network. The most common placement is behind the firewall, on the edge of a network. This practice provides the IDS with high visibility of traffic entering your network and will not receive any traffic between users on the network. The edge of the network is the point in which a network connects to the extranet. Another practice that can be accomplished if more resources are available is a strategy where a technician will place their first IDS at the point of highest visibility and depending on resource availability will place another at the next highest point, continuing that process until all points of the network are covered.[33]
If an IDS is placed beyond a network's firewall, its main purpose would be to defend against noise from the internet but, more importantly, defend against common attacks, such as port scans and network mapper. An IDS in this position would monitor layers 4 through 7 of the OSI model and would be signature-based. This is a very useful practice, because rather than showing actual breaches into the network that made it through the firewall, attempted breaches will be shown which reduces the amount of false positives. The IDS in this position also assists in decreasing the amount of time it takes to discover successful attacks against a network.[34]
Sometimes an IDS with more advanced features will be integrated with a firewall in order to be able to intercept sophisticated attacks entering the network. Examples of advanced features would include multiple security contexts in the routing level and bridging mode. All of this in turn potentially reduces cost and operational complexity.[34]
Another option for IDS placement is within the actual network. These will reveal attacks or suspicious activity within the network. Ignoring the security within a network can cause many problems, it will either allow users to bring about security risks or allow an attacker who has already broken into the network to roam around freely. Intense intranet security makes it difficult for even those hackers within the network to maneuver around and escalate their privileges.[34]
There are a number of techniques which attackers are using, the following are considered 'simple' measures which can be taken to evade IDS:
The earliest preliminary IDS concept was delineated in 1980 by James Anderson at theNational Security Agencyand consisted of a set of tools intended to help administrators review audit trails.[38]User access logs, file access logs, and system event logs are examples of audit trails.
Fred Cohennoted in 1987 that it is impossible to detect an intrusion in every case, and that the resources needed to detect intrusions grow with the amount of usage.[39]
Dorothy E. Denning, assisted byPeter G. Neumann, published a model of an IDS in 1986 that formed the basis for many systems today.[40]Her model used statistics foranomaly detection, and resulted in an early IDS atSRI Internationalnamed the Intrusion Detection Expert System (IDES), which ran onSunworkstations and could consider both user and network level data.[41]IDES had a dual approach with a rule-basedExpert Systemto detect known types of intrusions plus a statistical anomaly detection component based on profiles of users, host systems, and target systems. The author of "IDES: An Intelligent System for Detecting Intruders", Teresa F. Lunt, proposed adding anartificial neural networkas a third component. She said all three components could then report to a resolver. SRI followed IDES in 1993 with the Next-generation Intrusion Detection Expert System (NIDES).[42]
TheMulticsintrusion detection and alerting system (MIDAS), an expert system using P-BEST andLisp, was developed in 1988 based on the work of Denning and Neumann.[43]Haystack was also developed in that year using statistics to reduce audit trails.[44]
In 1986 theNational Security Agencystarted an IDS research transfer program underRebecca Bace. Bace later published the seminal text on the subject,Intrusion Detection, in 2000.[45]
Wisdom & Sense (W&S) was a statistics-based anomaly detector developed in 1989 at theLos Alamos National Laboratory.[46]W&S created rules based on statistical analysis, and then used those rules for anomaly detection.
In 1990, the Time-based Inductive Machine (TIM) did anomaly detection using inductive learning of sequential user patterns inCommon Lispon aVAX3500 computer.[47]The Network Security Monitor (NSM) performed masking on access matrices for anomaly detection on a Sun-3/50 workstation.[48]The Information Security Officer's Assistant (ISOA) was a 1990 prototype that considered a variety of strategies including statistics, a profile checker, and an expert system.[49]ComputerWatch atAT&T Bell Labsused statistics and rules for audit data reduction and intrusion detection.[50]
Then, in 1991, researchers at theUniversity of California, Daviscreated a prototype Distributed Intrusion Detection System (DIDS), which was also an expert system.[51]The Network Anomaly Detection and Intrusion Reporter (NADIR), also in 1991, was a prototype IDS developed at the Los Alamos National Laboratory's Integrated Computing Network (ICN), and was heavily influenced by the work of Denning and Lunt.[52]NADIR used a statistics-based anomaly detector and an expert system.
TheLawrence Berkeley National LaboratoryannouncedBroin 1998, which used its own rule language for packet analysis fromlibpcapdata.[53]Network Flight Recorder (NFR) in 1999 also used libpcap.[54]
APE was developed as a packet sniffer, also using libpcap, in November, 1998, and was renamedSnortone month later. Snort has since become the world's largest used IDS/IPS system with over 300,000 active users.[55]It can monitor both local systems, and remote capture points using theTZSPprotocol.
The Audit Data Analysis and Mining (ADAM) IDS in 2001 usedtcpdumpto build profiles of rules for classifications.[56]In 2003,Yongguang Zhangand Wenke Lee argue for the importance of IDS in networks with mobile nodes.[57]
In 2015, Viegas and his colleagues[58]proposed an anomaly-based intrusion detection engine, aiming System-on-Chip (SoC) for applications in Internet of Things (IoT), for instance. The proposal applies machine learning for anomaly detection, providing energy-efficiency to a Decision Tree, Naive-Bayes, and k-Nearest Neighbors classifiers implementation in an Atom CPU and its hardware-friendly implementation in a FPGA.[59][60]In the literature, this was the first work that implement each classifier equivalently in software and hardware and measures its energy consumption on both. Additionally, it was the first time that was measured the energy consumption for extracting each features used to make the network packet classification, implemented in software and hardware.[61]
This article incorporatespublic domain materialfromKaren Scarfone, Peter Mell.Guide to Intrusion Detection and Prevention Systems, SP800-94(PDF).National Institute of Standards and Technology. Retrieved1 January2010.
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On October 21, 2016, three consecutivedistributed denial-of-service attackswere launched against theDomain Name System(DNS) providerDyn. The attack caused major Internet platforms and services to be unavailable to large swathes of users in Europe and North America.[3][4]The groupsAnonymousand New World Hackers claimed responsibility for the attack, but scant evidence was provided.[5]
As a DNS provider, Dyn provides to end-users the service of mapping an Internetdomain name—when, for instance, entered into aweb browser—to its correspondingIP address. Thedistributed denial-of-service(DDoS) attack was accomplished through numerous DNS lookup requests from tens of millions of IP addresses.[6]The activities are believed to have been executed through abotnetconsisting of manyInternet-connected devices—such asprinters,IP cameras,residential gatewaysandbaby monitors—that had been infected with theMiraimalware.
Services affected by the attack included:
TheUS Department of Homeland Securitystarted an investigation into the attacks, according to aWhite Housesource.[30][31][32]No group of hackers claimed responsibility during or in the immediate aftermath of the attack.[33]Dyn's chief strategist said in an interview that the assaults on the company's servers were very complex and unlike everyday DDoS attacks.[34]Barbara Simons, a member of the advisory board of the United StatesElection Assistance Commission, said such attacks could affectelectronic votingfor overseas military or civilians.[34]
Dyn disclosed that, according to business risk intelligence firm FlashPoint andAkamai Technologies, the attack was abotnetcoordinated through numerousInternet of Things-enabled (IoT) devices, includingcameras,residential gateways, andbaby monitors, that had been infected withMiraimalware. The attribution of the attack to the Mirai botnet had been previously reported by BackConnect Inc., another security firm.[35]Dyn stated that they were receiving malicious requests from tens of millions ofIP addresses.[6][36]Mirai is designed tobrute-forcethe security on an IoT device, allowing it to be controlled remotely.
Cybersecurity investigatorBrian Krebsnoted that the source code for Mirai had been released onto the Internet in anopen-sourcemanner some weeks prior, which made the investigation of the perpetrator more difficult.[37]
On 25 October 2016, US President Obama stated that the investigators still had no idea who carried out the cyberattack.[38]
On 13 December 2017, the Justice Department announced that three men (Paras Jha, 21, Josiah White, 20, and Dalton Norman, 21) had entered guilty pleas in cybercrime cases relating to the Mirai and clickfraud botnets.[39]
In correspondence with the websitePolitico,hacktivistgroups SpainSquad,Anonymous, and New World Hackers claimed responsibility for the attack in retaliation againstEcuador's rescinding Internet access toWikiLeaksfounderJulian Assange, at theirembassy in London, where he had been grantedasylum.[5]This claim has yet to be confirmed.[5]WikiLeaks alluded to the attack onTwitter, tweeting "Mr. Assange is still alive and WikiLeaks is still publishing. We ask supporters to stop taking down the US internet. You proved your point."[40]New World Hackers has claimed responsibility in the past for similar attacks targeting sites likeBBCandESPN.com.[41]
On October 26, FlashPoint stated that the attack was most likely done byscript kiddies.[42]
A November 17, 2016, aForbesarticle reported that the attack was likely carried out by "an angry gamer".[43]
On December 9, 2020, one of the perpetrators pleaded guilty to taking part in the attack. The perpetrator's name was withheld due to his or her age.[44]
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"Paper terrorism" is aneologismreferring to the use offalse liens,frivolous lawsuits, bogusletters of credit, and other legal orpseudolegaldocuments lacking sound factual basis as a method of harassment against an opponent on a scale described as evocative of conventionalarmed terrorism.[1]These methods are popular among some Americananti-governmentgroups[2]and those associated with theredemption movement.[3]
Mark Pitcavageof theAnti-Defamation Leaguestates that these methods were pioneered by thePosse Comitatus.[4]Some victims of paper terrorism have been forced to declarebankruptcy.[5]An article by theSouthern Poverty Law Centerstates that another tactic is filing reports with theInternal Revenue Servicefalsely accusing their political enemies of having unreported income.[6]
Such frivolous lawsuits also clog the court system making it more difficult to process other cases and including using challenges to the titles of property owned by government officials and others.[7]Another method of paper terrorism is filingbankruptcy petitionsagainst others in an effort to ruin theircredit ratings.[8]
In the late 1990s,[9]the "Republic of Texas", a militia group claiming thatTexaswas legally independent, carried out what it called "a campaign of paper terrorism" using bogus land claims and bad checks to try to congest Texas courts.[10]
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Project Shieldis an anti-distributed-denial-of-service(anti-DDoS) that is offered byJigsaw, a subsidiary ofGoogle, to websites that have "media, elections, and human rights related content."[1]The main goal of the project is to serve "small, under-resourced news sites that are vulnerable to the web's growing epidemic of DDOS attacks", according to team lead George Conard.[2]
Google initially announced Project Shield at their Ideas Conference on October 21, 2013.[1]The service was initially only offered to trusted testers, but on February 25, 2016, Google opened up the service to any qualifying website a Google-owned reverse proxy that identifies and filters malicious traffic.[3]In May 2018, Jigsaw announced that it would start offering free protection from distributed denial of service attacks to US political campaigns, candidates, and political action committees.[4][5]
In January 2019, Google's Jigsaw expanded Project Shield to offer free DDoS protection to political organizations and websites in Europe, ahead of the 2019 European Parliament elections. This was the first time the service was made available outside the US.[6]
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Aregular expression denial of service(ReDoS)[1]is analgorithmic complexity attackthat produces adenial-of-serviceby providing aregular expressionand/or an input that takes a long time to evaluate. The attack exploits the fact that many[2]regular expression implementationshave super-linearworst-case complexity; on certain regex-input pairs, the time taken can grow polynomially or exponentially in relation to the input size. An attacker can thus cause a program to spend substantial time by providing a specially crafted regular expression and/or input. The program will then slow down or become unresponsive.[3][4]
Regular expression("regex") matching can be done by building afinite-state automaton. Regex can be easily converted tonondeterministic automata(NFAs), in which for each state and input symbol, there may be several possible next states. After building the automaton, several possibilities exist:
Of the above algorithms, the first two are problematic. The first is problematic because a deterministic automaton could have up to2m{\displaystyle 2^{m}}states wherem{\displaystyle m}is the number of states in the nondeterministic automaton; thus, the conversion from NFA to DFA may takeexponential time. The second is problematic because a nondeterministic automaton could have an exponential number of paths of lengthn{\displaystyle n}, so that walking through an input of lengthn{\displaystyle n}will also take exponential time.[7]The last two algorithms, however, do not exhibit pathological behavior.
Note that for non-pathological regular expressions, the problematic algorithms are usually fast, and in practice, one can expect them to "compile" a regex in O(m) time and match it in O(n) time; instead, simulation of an NFA and lazy computation of the DFA haveO(m2n) worst-case complexity.[a]Regex denial of service occurs when these expectations are applied to a regex provided by the user, and malicious regular expressions provided by the user trigger the worst-case complexity of the regex matcher.
While regex algorithms can be written in an efficient way, most regex engines in existence extend the regex languages with additional constructs that cannot always be solved efficiently. Suchextended patternsessentially force the implementation of regex in mostprogramming languagesto use backtracking.
The most severe type of problem happens with backtracking regular expression matches, where some patterns have a runtime that is exponential in the length of the input string.[8]For strings ofn{\displaystyle n}characters, the runtime isO(2n){\displaystyle O(2^{n})}. This happens when a regular expression has three properties:
The second condition is best explained with two examples:
In both of these examples we used$to match the end of the string, satisfying the third condition, but it is also possible to use another character for this. For example(a|aa)*chas the same problematic structure.
All three of the above regular expressions will exhibit exponential runtime when applied to strings of the forma...ax{\displaystyle a...ax}. For example, if you try to match them againstaaaaaaaaaaaaaaaaaaaaaaaaxon a backtracking expression engine, it will take a significantly long time to complete, and the runtime will approximately double for each extraabefore thex.
It is also possible to have backtracking which is polynomial timeO(nx){\displaystyle O(n^{x})}, instead of exponential. This can also cause problems for long enough inputs, though less attention has been paid to this problem as malicious input must be much longer to have a significant effect. An example of such a pattern is "a*b?a*c", when the input is an arbitrarily long sequence of "a"s.
So-called "evil" or vulnerable regexes have been found in online regular expression repositories. Note that it is enough to find a vulnerablesubexpression in order to attack the full regex:
These two examples are also susceptible to the inputaaaaaaaaaaaaaaaaaaaaaaaa!.
If the regex itself is affected by user input, such as a web service permitting clients to provide a search pattern, then an attacker can inject a malicious regex to consume the server's resources. Therefore, in most cases, regular expression denial of service can be avoided by removing the possibility for the user to execute arbitrary patterns on the server. In this case, web applications and databases are the main vulnerable applications. Alternatively, a malicious page could hang the user's web browser or cause it to use arbitrary amounts of memory.
However, if a vulnerable regex exists on the server-side already, then an attacker may instead be able to provide an input that triggers its worst-case behavior. In this case,e-mail scannersandintrusion detection systemscould also be vulnerable.
In the case of a web application, the programmer may use the same regular expression to validate input on both the client and the server side of the system. An attacker could inspect the client code, looking for evil regular expressions, and send crafted input directly to the web server in order to hang it.[9]
ReDoS can be mitigated without changes to the regular expression engine, simply by setting a time limit for the execution of regular expressions when untrusted input is involved.[10]
ReDoS can be avoided entirely by using a non-vulnerable regular expression implementation. AfterCloudFlare'sweb application firewall(WAF) was brought down by a PCRE ReDoS in 2019, the company rewrote its WAF to use the non-backtracking Rust regex library, using an algorithm similar toRE2.[11][12]
Vulnerable regular expressions can be detected programmatically by alinter.[13]Methods range from purestatic analysis[14][15]tofuzzing.[16]In most cases, the problematic regular expressions can be rewritten as "non-evil" patterns. For example,(.*a)+can be rewritten to([^a]*a)+.Possessive matching and atomic grouping, which disable backtracking for parts of the expression,[17]can also be used to "pacify" vulnerable parts.[18][19]
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Resource exhaustion attacksare computer securityexploitsthatcrash,hang, or otherwise interfere with the targeted program or system. They are a form ofdenial-of-service attackbut are different fromdistributeddenial-of-service attacks, which involve overwhelming a network host such as a web server with requests from many locations.[1]
Resource exhaustion attacks generally exploit a software bug or design deficiency. In software withmanual memory management(most commonly written inCorC++),memory leaksare a very common bug exploited for resource exhaustion. Even if agarbage collectedprogramming language is used, resource exhaustion attacks are possible if the program uses memory inefficiently and does not impose limits on the amount of state used when necessary.
File descriptorleaks are another commonvector. Most general-purpose programming languages require the programmer to explicitly close file descriptors, so even particularly high-level languages allow the programmer to make such mistakes.
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Avirtual sit-inis a form ofelectronic civil disobediencederiving its name from thesit-inspopular during thecivil rights movementof the 1960s. The virtual sit-in attempts to recreate that same action digitally using adistributed denial-of-service attack(DDOS). During a virtual sit-in, hundreds ofactivistsattempt to access a targetwebsitesimultaneously and repetitively. If performed correctly, this will cause the target website to run slowly or even collapse entirely, preventing anyone from accessing it.[1][2]
On December 21, 1995, the first world Virtual sit-in, conceived byTommaso Tozzi, was created by the Florentine groupStrano Networkagainst the French government to protest against the nuclear tests in Mururoa and was defined as a"Netstrike".[3]On Thursday May 1, 1998,Ricardo Dominguez(co-founder ofElectronic Disturbance Theater) and Stefan Wray held a virtual sit-in in which they decided to attack theWorld Economic Forum(WEF). They did this to support their particular beliefs againstanti-globalization.[4]With over 160,000 people who attended the virtual sit-in for reasons that they could not take to the streets of New York City protest. More than 40,000 also downloaded software which made aDDOSattack easier was also recorded.[5]The attack lasted all of Thursday and Friday night.
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https://en.wikipedia.org/wiki/Virtual_sit-in
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Aweb shellis ashell-like interfacethat enables aweb serverto be remotely accessed, often for the purposes ofcyberattacks.[1]A web shell is unique in that aweb browseris used to interact with it.[2][3]
A web shell could be programmed in anyprogramming languagethat is supported on a server. Web shells are most commonly written inPHPdue to the widespread usage of PHP forweb applications. ThoughActive Server Pages,ASP.NET,Python,Perl,Ruby, andUnix shellscripts are also used.[1][2][3]
Usingnetwork monitoring tools, an attacker can findvulnerabilitiesthat can potentially allow delivery of a web shell. These vulnerabilities are often present in applications that are run on a web server.[2]
An attacker can use a web shell to issue shell commands, performprivilege escalationon the web server, and the ability toupload,delete,download, andexecutefiles to and from the web server.[2]
Web shells are used in attacks mostly because they are multi-purpose and difficult to detect.[4]They are commonly used for:
Web shells give hackers the ability to steal information, corrupt data, and uploadmalwaresthat are more damaging to a system. The issue increasingly escalates when hackers employ compromised servers to infiltrate a system and jeopardize additional machines. Web shells are also a way that malicious individuals target a variety of industries, including government, financial, and defense through cyber espionage. One of the very well known web shells used in this manner is known as “China Chopper.”[6]
Web shells are installed through vulnerabilities in web application or weak server security configuration including the following:[2][4]
An attacker may also modify (spoof) theContent-Typeheader to be sent by the attacker in a file upload to bypass improper file validation (validation using MIME type sent by the client), which will result in a successful upload of the attacker's shell.
The following is a simple example of a web shell written in PHP that executes and outputs the result of a shell command:
Assuming the filename isexample.php, an example that would output the contents of the/etc/passwdfile is shown below:
The above request will take the value of thexparameter of thequery string, sending the following shell command:
This could have been prevented if the shell functions of PHP were disabled so that arbitrary shell commands cannot be executed from PHP.
A web shell is usually installed by taking advantage of vulnerabilities present in the web server's software. That is why removal of these vulnerabilities is important to avoid the potential risk of a compromised web server.
The following are security measures for preventing the installation of a web shell:[2][3]
Web shells can be easily modified, so it's not easy to detect web shells andantivirussoftware are often not able to detect web shells.[2][9]
The following are common indicators that a web shell is present on a web server:[2][3]
For example, a file generating suspicious traffic (e.g. aPNGfile requesting withPOSTparameters).[2][10][11][12]Dubious logins from DMZ servers to internal sub-nets and vice versa.[2]
Web shells may also contain a login form, which is often disguised as anerror page.[2][13][14][15]
Using web shells, adversaries can modify the.htaccessfile (on servers running theApache HTTP Serversoftware) on web servers to redirectsearch enginerequests to theweb pagewithmalwareorspam. Often web shells detect theuser-agentand the content presented to thesearch engine spideris different from that presented to the user's browser. To find a web shell auser-agentchange of the crawler bot is usually required. Once the web shell is identified, it can be deleted easily.[2]
Analyzing the web server's log could specify the exact location of the web shell. Legitimate users/visitor usually have differentuser-agentsandreferers, on the other hand, a web shell is usually only visited by the attacker, therefore have very few variants of user-agent strings.[2]
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https://en.wikipedia.org/wiki/Web_shell
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Radio jammingis the deliberate blocking of or interference withwireless communications.[1][2]In some cases, jammers work by the transmission ofradio signalsthat disrupttelecommunicationsby decreasing thesignal-to-noise ratio.[3]
The concept can be used inwireless data networksto disrupt information flow.[4]It is a common form of censorship in totalitarian countries, in order to prevent foreign radio stations in border areas from reaching the country.[3]
Jamming is usually distinguished from interference that can occur due to device malfunctions or other accidental circumstances. Devices that simply cause interference are regulated differently. Unintentional "jamming" occurs when an operator transmits on a busyfrequencywithout first checking whether it is in use, or without being able to hear stations using the frequency. Another form of unintentional jamming occurs when equipment accidentallyradiatesa signal, such as acable televisionplant that accidentally emits on an aircraft emergency frequency.
Originally the terms were used interchangeably but nowadays most radio users use the term "jamming" to describe thedeliberateuse of radio noise or signals in an attempt to disrupt communications (or prevent listening to broadcasts) whereas the term "interference" is used to describeunintentionalforms of disruption (which are far more common). However, the distinction is still not universally applied. For inadvertent disruptions, seeelectromagnetic compatibility.
Intentional communications jamming is usually aimed at radio signals to disrupt control of a battle. Atransmitter, tuned to the same frequency as the opponents' receiving equipment and with the same type ofmodulation, can, with enough power, override any signal at thereceiver. Digital wireless jamming for signals such asBluetoothandWiFiis possible with very low power.
The most common types of this form of signal jamming arerandom noise, random pulse, stepped tones, warbler, random keyed modulatedCW, tone, rotary, pulse, spark, recorded sounds, gulls, and sweep-through. These can be divided into two groups: obvious and subtle.
Obvious jamming is easy to detect because it can be heard on the receiving equipment. It is usually some type of noise, such as stepped tones (bagpipes), random-keyed code, pulses, music (often distorted), erratically warbling tones, highly distorted speech, random noise (hiss), and recorded sounds. Various combinations of these methods may be used, often accompanied by regularMorseidentification signals to enable individual transmitters to be identified in order to assess their effectiveness. For example, China, which did and does use jamming extensively, plays a loop oftraditional Chinese musicwhile it is jamming channels (cf.Attempted jamming of numbers stations).
The purpose of this type of jamming is to block reception of transmitted signals and to cause a nuisance to the receiving operator. One early Soviet attempt at jamming Western broadcasters used the noise from thediesel generatorthat was powering the jamming transmitter.
Subtle jamming is jamming during which no sound is heard on the receiving equipment. The radio does not receive incoming signals; yet everything seems superficially normal to the operator. These are often technical attacks on modern equipment, such as "squelch capture". Thanks to the FMcapture effect,frequency modulatedbroadcasts may be jammed, unnoticed, by a simple unmodulated carrier. The receiver locks on to the larger carrier signal, and hence will ignore the FM signal that carries the information.
Digital signals use complex modulation techniques, such asQPSK. These signals are very robust in the presence of interfering signals. But the signal relies on hand shaking between the transmitter and receiver to identify and determine security settings and method of high-level transmission. If the jamming device sends initiation data packets, the receiver will begin its state machine to establish two-way data transmission. A jammer will loop back to the beginning instead of completing the handshake. This method jams the receiver in an infinite loop where it keeps trying to initiate a connection but never completes it, which effectively blocks all legitimate communication.
Bluetooth and other consumer radio protocols such as WiFi have built-in detectors, so that they transmit only when the channel is free. Simple continuous transmission on a given channel will continuously stop a transmitter transmitting, hence jamming the receiver from ever hearing from its intended transmitter. Other jammers work by analysing the packet headers and, depending on the source or destination, selectively transmitting over the end of the message, corrupting the packet.
DuringWorld War II, groundradio operatorswould attempt to misleadpilotsby false instructions in their ownlanguage, in what was more precisely aspoofing attackthan jamming.Radar jammingis also important to disrupt use ofradarused to guide an enemy'smissilesoraircraft. Modern secure communication techniques use such methods asspread spectrummodulation to resist the deleterious effects of jamming.
Jamming of foreign radiobroadcaststations has often been used in wartime (and during periods of tense international relations) to prevent or deter citizens from listening to broadcasts from enemy countries. However, such jamming is usually of limited effectiveness because the affected stations usually change frequencies, put on additional frequencies and/or increase transmission power.
Jamming has also occasionally been used by the governments of Germany (duringWorld War II),[6]Israel,[7]Cuba, Iraq, Iran (during theIran-Iraq War), China, North and South Korea and several Latin American countries, as well as byIrelandagainstpirate radiostations such asRadio Nova. The United Kingdom government used two coordinated, separately located transmitters to jam theoffshore radioship,Radio North Sea Internationaloff the coast of Britain in 1970.[8]
In occupied Europe theNazisattempted to jam broadcasts to the continent from theBBCand other allied stations. Along with increasingtransmitterpower and adding extra frequencies, attempts were made to counteract the jamming by droppingleafletsover cities instructing listeners to construct a directionalloop aerialthat would enable them to hear the stations through the jamming. In the Netherlands such aerials were nicknamed "moffenzeef" (English: "kraut sieve").[9][10]
During theContinuation War, after discovering the fact that the mines that the retreatingSovietforces had scattered throughout the city ofViipuriwere radio-triggered rather than timer- or pressure-triggered, the Finnish forces playedVesterinen's recording ofSäkkijärven Polkkawithout any pauses from September 4, 1941 to February 2, 1942, as they, to demine the city, needed to block the Soviets from activating the mines through the correct radio wave. The Soviets tried to trigger the mines by changing frequency; the mines had been set up to be able to be triggered by three different frequencies. The Finns countered this by playing Säkkijärven Polkka on all frequencies. During theBattle of the BeamsBritain jammed navigation signals used by German aircraft while the Soviets attempted to do likewise to American aircraft during theBerlin Airlift
Since the Soviet Union started jamming Western radio broadcasts to the Soviet Union in 1948 the primary targets have been theBBC External Broadcasting Services,Voice of America(VOA) and especiallyRFE/RL. Western nations had allowed jamming prior to World War II,[dubious–discuss]but in the post-War era the Western view has been that jamming violates thefreedom of informationwhile the Soviet view has been that under the international law principle ofnational sovereigntyjamming is an acceptable response to foreign radio broadcasts.[11]
During much of theCold War,Soviet(andEastern Bloc) jamming of some Western broadcasters led to a "power race" in which broadcasters and jammers alike repeatedly increased their transmission power, utilised highlydirectionalantennas and added extra frequencies (known as "barrage" or "frequency diversity" broadcasting) to the already heavily overcrowdedshortwavebands to such an extent that many broadcasters not directly targeted by the jammers (including pro-Soviet stations) suffered from the rising levels of noise and interference.[12][13]
There were also periods whenChinaand the Soviet Union jammed each other's programmes. The Soviet Union also jammedAlbanianprogrammes at times.
Some parts of the world were more affected by these broadcasting practices than others
Meanwhile, some listeners in the Soviet Union and Eastern Bloc devised ingenious methods (such as homemade directionalloop antennas) to hear the Western stations through the noise. Becauseradio propagationon shortwave can be difficult to predict reliably, listeners sometimes found that there were days/times when the jamming was particularly ineffective because radio fading (due toatmospheric conditions) was affecting the jamming signals but favouring the broadcasts (a phenomenon sometimes dubbed "twilight immunity"). On other days of course the reverse was the case. There were also times when jamming transmitters were (temporarily) off air due to breakdowns or maintenance. The Soviets (and most of their Eastern bloc allies) used two types of jamming transmitter.Skywavejamming covered a large area but for the reasons described was of limited effectiveness.Groundwavejamming was more effective but only over a small area and was thus used only in/near major cities throughout the Eastern Bloc. Both types of jamming were less effective on higher shortwave frequencies (above 15 MHz); however, many radios sold on the domestic market in the Soviet Union didn't tune these higher bands.[14]Skywave jamming was usually accompanied bymorsesignals in order to enable (coded) identification of the jamming station in order that Soviet monitoring posts could assess the effectiveness of each station.
In 1987 after decades of generally refusing to acknowledge that such jamming was even taking place the Soviets finally stopped jamming western broadcasts with the exception ofRFE/RLwhich continued to be jammed for several months into 1988. Previously there had been periods when some individual Eastern bloc countries refrained from jamming Western broadcasts but this varied widely by time and country. In general outside of the Soviet Union itselfBulgariawas one of the most prolific operators of jamming transmitters in the Eastern bloc withEast GermanyandYugoslaviathe least.
Whilewestern governmentsmay have occasionally considered jamming broadcasts from Eastern Bloc stations, it was generally accepted that doing so would be a pointless exercise. Ownership of shortwave radios was less common in western countries than in the Soviet Union where, due to the vast physical size of the country, manydomestic stationswere relayed on shortwave as it was the only practical way to cover remote areas. Additionally, western governments were generally less afraid of intellectual competition from the Eastern Bloc.
InFrancoist Spainthe dictatorship jammed for decadesRadio España Independiente, the radio station of theCommunist Party of Spainwhich broadcast fromMoscow(1941–1955),Bucharest(1955–1977) and East Berlin. It was the most important clandestine broadcaster in Spain and the regime considered it a threat, since it allowed its citizens to skip the censorship of the local media.[15]Broadcasts from East Germany to South Africa were also jammed.
In Latin America there were instances of communist radio stations such asRadio Venceremosbeing jammed, allegedly by theCIA, while there were short lived instances where Britain jammed some Egyptian (during theSuez Crisis),Greek(prior toCyprusgaining independence) andRhodesianstations.[16]During the early years of the Northern Ireland troubles the British army regularly jammed broadcasts from both Republican and Loyalist paramilitary groups.
In 2002, China acquired standard short-wave radio-broadcasting equipment designed for general public radio-broadcasting and technical support from Thales Broadcast Multimedia, a former subsidiary of the French state-owned companyThales Group.
Debates have been raised in Iran regarding the possible health hazards of satellite jamming. Iranian officials including the health minister have claimed that jamming has no health risk for humans. However, the minister of communication has recently admitted that satellite jamming has 'serious effects' and has called for identification of jamming stations so they can put a stop to this practice.[18][19][20]The government has generally denied any involvement in jamming and claimed they are sent from unknown sources.[18]According to some sources,IRGCis the organization behind satellite jamming in Iran.[21]
TheRussian Armed Forceshave, since the summer of 2015, begun using a multi-functionalEWweapon system inUkraine, known asBorisoglebsk 2.[22]It is postulated that this system has defeated communications in parts of that country, including mobile telephony andGPSsystems.[22][23][24]
Radio jamming (or "comm jamming") is a common plot element in theStar Warsfranchise. InStar Wars: Episode VI -Return of the Jedi, when the Rebel fleet approaches the Galactic Empire's force, believing themselves to be launching a surprise attack, GeneralLando Calrissianrealizes the Empire is jamming their signals, and therefore know they are approaching.
In the filmStar Trek II, after receiving a distress call from the space stationRegula I, Captain Kirk attempts to establish communications, but theEnterprise'scomm officer Lt. Uhura reports that further transmissions are "jammed at the source".
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XOR DDoSis a Linux Trojan malware with rootkit capabilities that was used to launch large-scale DDoS attacks. Its name stems from the heavy usage of XOR encryption in both malware and network communication to the C&Cs. It is built for multiple Linux architectures like ARM, x86 and x64. Noteworthy about XOR DDoS is the ability to hide itself with an embedded rootkit component which is obtained by multiple installation steps.[1]It was discovered in September 2014 byMalwareMustDie, awhite hatmalware research group.[2][3][4]From November 2014 it was involved in massive brute force campaign that lasted at least for three months.[5]
In order to gain access it launches a brute force attack in order to discover the password to Secure Shell services on Linux.[6]Once Secure Shell credentials are acquired and login is successful, it uses root privileges to run a script that downloads and installs XOR DDoS.[7]It is believed to be of Asian origin based on its targets, which tend to be located in Asia.[8]
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Zemrais a DDoS Bot which was first discovered in underground forums in May 2012.[1][2]
Zemra is capable of HTTP and SYN Flood flooding and also has a simpleCommand & Control panelthat is protected with 256-bit DES encryption for communicating with its command and control (C&C) server.[3]Zemra also sends information such as Computer name, Language settings, and Windows version. It will send this data to a remote location on a specific date and time.[4]It also opens a backdoor on TCP port 7710 to receive commands from a remote command-and-control server,[5]and it is able to monitor devices, collect system information, execute files, and even update or uninstall itself if necessary.[3][6]
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https://en.wikipedia.org/wiki/Zemra
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In computing, azip bomb, also known as adecompression bomborzip of death(ZOD), is a maliciousarchive filedesigned to crash or render useless the program or system reading it. The older the system or program, the less likely it is that the zip bomb will be detected. It is often employed to disableantivirus software, in order to create an opening for more traditionalmalware.[1]
A zip bomb allows a program to function normally, but, instead of hijacking the program's operation, it creates an archive that requires an excessive amount of time, disk space, computational power, or memory to unpack.[2]
Most modern antivirus programs can detect zip bombs and prevent the user from extracting anything from it.[3]
A zip bomb is usually a small file for ease of transport and to avoid suspicion. However, when the file is unpacked, its contents are more than the system can handle.
A famous example of a zip bomb is titled42.zip, which is azip fileof unknown authorship[4]consisting of 42kilobytesof compressed data, containing five layers of nested zip files in sets of 16, each bottom-layer archive containing a 4.3-gigabyte(4294967295bytes;4GiB−1 B) file for a total of4.5petabytes(4503599626321920bytes;4PiB−1MiB) of uncompressed data.[5]
In many anti-virus scanners, only a few layers ofrecursionare performed on archives to help prevent attacks that would cause abuffer overflow, anout-of-memorycondition, or exceed an acceptable amount of program execution time.[citation needed]Zip bombs often rely on repetition of identical files to achieve their extreme compression ratios.Dynamic programmingmethods can be employed to limit traversal of such files, so that only one file is followed recursively at each level, effectively converting their exponential growth to linear.[5]
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Shor's algorithmis aquantum algorithmfor finding theprime factorsof an integer. It was developed in 1994 by the American mathematicianPeter Shor.[1][2]It is one of the few known quantum algorithms with compelling potential applications and strong evidence of superpolynomial speedup compared to best known classical (non-quantum) algorithms.[3]On the other hand, factoring numbers of practical significance requires far morequbitsthan available in the near future.[4]Another concern is that noise in quantum circuits may undermine results,[5]requiring additional qubits forquantum error correction.
Shor proposed multiple similar algorithms for solving thefactoring problem, thediscrete logarithm problem, and the period-finding problem. "Shor's algorithm" usually refers to the factoring algorithm, but may refer to any of the three algorithms. The discrete logarithm algorithm and the factoring algorithm are instances of the period-finding algorithm, and all three are instances of thehidden subgroup problem.
On a quantum computer, to factor an integerN{\displaystyle N}, Shor's algorithm runs inpolynomial time, meaning the time taken is polynomial inlogN{\displaystyle \log N}.[6]It takesquantum gatesof orderO((logN)2(loglogN)(logloglogN)){\displaystyle O\!\left((\log N)^{2}(\log \log N)(\log \log \log N)\right)}using fast multiplication,[7]or evenO((logN)2(loglogN)){\displaystyle O\!\left((\log N)^{2}(\log \log N)\right)}utilizing the asymptotically fastest multiplication algorithm currently known due to Harvey andVan Der Hoven,[8]thus demonstrating that theinteger factorizationproblem can be efficiently solved on a quantum computer and is consequently in thecomplexity classBQP. This is significantly faster than the most efficient known classical factoring algorithm, thegeneral number field sieve, which works insub-exponential time:O(e1.9(logN)1/3(loglogN)2/3){\displaystyle O\!\left(e^{1.9(\log N)^{1/3}(\log \log N)^{2/3}}\right)}.[9]
If a quantum computer with a sufficient number ofqubitscould operate without succumbing toquantum noiseand otherquantum-decoherencephenomena, then Shor's algorithm could be used to breakpublic-key cryptographyschemes, such as
RSA can be broken if factoring large integers is computationally feasible. As far as is known, this is not possible using classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer. It was also a powerful motivator for the design and construction of quantum computers, and for the study of new quantum-computer algorithms. It has also facilitated research on new cryptosystems that are secure from quantum computers, collectively calledpost-quantum cryptography.
Given the high error rates of contemporary quantum computers and too few qubits to usequantum error correction, laboratory demonstrations obtain correct results only in a fraction of attempts.
In 2001, Shor's algorithm was demonstrated by a group atIBM, who factored15{\displaystyle 15}into3×5{\displaystyle 3\times 5}, using anNMR implementationof a quantum computer with seven qubits.[11]After IBM's implementation, two independent groups implemented Shor's algorithm usingphotonicqubits, emphasizing that multi-qubitentanglementwas observed when running the Shor's algorithm circuits.[12][13]In 2012, the factorization of15{\displaystyle 15}was performed with solid-state qubits.[14]Later, in 2012, the factorization of21{\displaystyle 21}was achieved.[15]In 2016, the factorization of15{\displaystyle 15}was performed again using trapped-ion qubits with a recycling technique.[16]In 2019, an attempt was made to factor the number35{\displaystyle 35}using Shor's algorithm on an IBMQ System One, but the algorithm failed because of accumulating errors.[17]However, all these demonstrations have compiled the algorithm by making use of prior knowledge of the answer, and some have even oversimplified the algorithm in a way that makes it equivalent to coin flipping.[18]Furthermore, attempts using quantum computers with other algorithms have been made.[19]However, these algorithms are similar to classical brute-force checking of factors, so unlike Shor's algorithm, they are not expected to ever perform better than classical factoring algorithms.[20]
Theoretical analyses of Shor's algorithm assume a quantum computer free of noise and errors. However, near-term practical implementations will have to deal with such undesired phenomena (when more qubits are available,quantum error correctioncan help). In 2023,Jin-Yi Caishowed that in the presence of noise, Shor's algorithm failsasymptotically almost surelyfor large semiprimes that are products of two primes inOEISsequence A073024.[5]These primesp{\displaystyle p}have the property thatp−1{\displaystyle p-1}has a prime factor larger thanp2/3{\displaystyle p^{2/3}}, and have a positive density in the set of all primes. Hence error correction will be needed to be able to factor all numbers with Shor's algorithm.
The problem that we are trying to solve is:given an oddcomposite numberN{\displaystyle N}, find itsinteger factors.
To achieve this, Shor's algorithm consists of two parts:
A complete factoring algorithm is possible if we're able to efficiently factor arbitraryN{\displaystyle N}into just two integersp{\displaystyle p}andq{\displaystyle q}greater than 1, since if eitherp{\displaystyle p}orq{\displaystyle q}are not prime, then the factoring algorithm can in turn be run on those until only primes remain.
A basic observation is that, usingEuclid's algorithm, we can always compute theGCDbetween two integers efficiently. In particular, this means we can check efficiently whetherN{\displaystyle N}is even, in which case 2 is trivially a factor. Let us thus assume thatN{\displaystyle N}is odd for the remainder of this discussion. Afterwards, we can use efficient classical algorithms to check whetherN{\displaystyle N}is aprime power.[21]For prime powers, efficient classical factorization algorithms exist,[22]hence the rest of the quantum algorithm may assume thatN{\displaystyle N}is not a prime power.
If those easy cases do not produce a nontrivial factor ofN{\displaystyle N}, the algorithm proceeds to handle the remaining case. We pick a random integer2≤a<N{\displaystyle 2\leq a<N}. A possible nontrivial divisor ofN{\displaystyle N}can be found by computinggcd(a,N){\displaystyle \gcd(a,N)}, which can be done classically and efficiently using theEuclidean algorithm. If this produces a nontrivial factor (meaninggcd(a,N)≠1{\displaystyle \gcd(a,N)\neq 1}), the algorithm is finished, and the other nontrivial factor isN/gcd(a,N){\displaystyle N/\gcd(a,N)}. If a nontrivial factor was not identified, then this means thatN{\displaystyle N}and the choice ofa{\displaystyle a}arecoprime, soa{\displaystyle a}is contained in themultiplicative group of integers moduloN{\displaystyle N}, having amultiplicative inversemoduloN{\displaystyle N}. Thus,a{\displaystyle a}has amultiplicative orderr{\displaystyle r}moduloN{\displaystyle N}, meaning
andr{\displaystyle r}is the smallest positive integer satisfying this congruence.
The quantum subroutine findsr{\displaystyle r}. It can be seen from the congruence thatN{\displaystyle N}dividesar−1{\displaystyle a^{r}-1}, writtenN∣ar−1{\displaystyle N\mid a^{r}-1}. This can be factored usingdifference of squares:N∣(ar/2−1)(ar/2+1).{\displaystyle N\mid (a^{r/2}-1)(a^{r/2}+1).}Since we have factored the expression in this way, the algorithm doesn't work for oddr{\displaystyle r}(becausear/2{\displaystyle a^{r/2}}must be an integer), meaning that the algorithm would have to restart with a newa{\displaystyle a}. Hereafter we can therefore assume thatr{\displaystyle r}is even. It cannot be the case thatN∣ar/2−1{\displaystyle N\mid a^{r/2}-1}, since this would implyar/2≡1modN{\displaystyle a^{r/2}\equiv 1{\bmod {N}}}, which would contradictorily imply thatr/2{\displaystyle r/2}would be the order ofa{\displaystyle a}, which was alreadyr{\displaystyle r}. At this point, it may or may not be the case thatN∣ar/2+1{\displaystyle N\mid a^{r/2}+1}. IfN{\displaystyle N}does not dividear/2+1{\displaystyle a^{r/2}+1}, then this means that we are able to find a nontrivial factor ofN{\displaystyle N}. We computed=gcd(N,ar/2−1).{\displaystyle d=\gcd(N,a^{r/2}-1).}Ifd=1{\displaystyle d=1}, thenN∣ar/2+1{\displaystyle N\mid a^{r/2}+1}was true, and a nontrivial factor ofN{\displaystyle N}cannot be achieved froma{\displaystyle a}, and the algorithm must restart with a newa{\displaystyle a}. Otherwise, we have found a nontrivial factor ofN{\displaystyle N}, with the other beingN/d{\displaystyle N/d}, and the algorithm is finished. For this step, it is also equivalent to computegcd(N,ar/2+1){\displaystyle \gcd(N,a^{r/2}+1)}; it will produce a nontrivial factor ifgcd(N,ar/2−1){\displaystyle \gcd(N,a^{r/2}-1)}is nontrivial, and will not if it's trivial (whereN∣ar/2+1{\displaystyle N\mid a^{r/2}+1}).
The algorithm restated shortly follows: letN{\displaystyle N}be odd, and not a prime power. We want to output two nontrivial factors ofN{\displaystyle N}.
It has been shown that this will be likely to succeed after a few runs.[2]In practice, a single call to the quantum order-finding subroutine is enough to completely factorN{\displaystyle N}with very high probability of success if one uses a more advanced reduction.[23]
The goal of the quantum subroutine of Shor's algorithm is, givencoprime integersN{\displaystyle N}and1<a<N{\displaystyle 1<a<N}, to find theorderr{\displaystyle r}ofa{\displaystyle a}moduloN{\displaystyle N}, which is the smallest positive integer such thatar≡1(modN){\displaystyle a^{r}\equiv 1{\pmod {N}}}. To achieve this, Shor's algorithm uses a quantum circuit involving two registers. The second register usesn{\displaystyle n}qubits, wheren{\displaystyle n}is the smallest integer such thatN≤2n{\displaystyle N\leq 2^{n}}, i.e.,n=⌈log2N⌉{\displaystyle n=\left\lceil {\log _{2}N}\right\rceil }. The size of the first register determines how accurate of an approximation the circuit produces. It can be shown that using2n{\displaystyle 2n}qubits gives sufficient accuracy to findr{\displaystyle r}. The exact quantum circuit depends on the parametersa{\displaystyle a}andN{\displaystyle N}, which define the problem. The following description of the algorithm usesbra–ket notationto denote quantum states, and⊗{\displaystyle \otimes }to denote thetensor product, rather thanlogical AND.
The algorithm consists of two main steps:
The connection with quantum phase estimation was not discussed in the original formulation of Shor's algorithm,[2]but was later proposed by Kitaev.[24]
In general thequantum phase estimation algorithm, for any unitaryU{\displaystyle U}and eigenstate|ψ⟩{\displaystyle |\psi \rangle }such thatU|ψ⟩=e2πiθ|ψ⟩{\displaystyle U|\psi \rangle =e^{2\pi i\theta }|\psi \rangle }, sends input states|0⟩|ψ⟩{\displaystyle |0\rangle |\psi \rangle }to output states close to|ϕ⟩|ψ⟩{\displaystyle |\phi \rangle |\psi \rangle }, whereϕ{\displaystyle \phi }is a superposition of integers close to22nθ{\displaystyle 2^{2n}\theta }. In other words, it sends each eigenstate|ψj⟩{\displaystyle |\psi _{j}\rangle }ofU{\displaystyle U}to a state containing information close to the associated eigenvalue. For the purposes of quantum order-finding, we employ this strategy using the unitary defined by the actionU|k⟩={|ak(modN)⟩0≤k<N,|k⟩N≤k<2n.{\displaystyle U|k\rangle ={\begin{cases}|ak{\pmod {N}}\rangle &0\leq k<N,\\|k\rangle &N\leq k<2^{n}.\end{cases}}}The action ofU{\displaystyle U}on states|k⟩{\displaystyle |k\rangle }withN≤k<2n{\displaystyle N\leq k<2^{n}}is not crucial to the functioning of the algorithm, but needs to be included to ensure that the overall transformation is a well-defined quantum gate. Implementing the circuit for quantum phase estimation withU{\displaystyle U}requires being able to efficiently implement the gatesU2j{\displaystyle U^{2^{j}}}. This can be accomplished viamodular exponentiation, which is the slowest part of the algorithm.
The gate thus defined satisfiesUr=I{\displaystyle U^{r}=I}, which immediately implies that its eigenvalues are ther{\displaystyle r}-throots of unityωrk=e2πik/r{\displaystyle \omega _{r}^{k}=e^{2\pi ik/r}}. Furthermore, each eigenvalueωrj{\displaystyle \omega _{r}^{j}}has an eigenvector of the form|ψj⟩=r−1/2∑k=0r−1ωr−kj|ak⟩{\textstyle |\psi _{j}\rangle =r^{-1/2}\sum _{k=0}^{r-1}\omega _{r}^{-kj}|a^{k}\rangle }, and these eigenvectors are such that1r∑j=0r−1|ψj⟩=1r∑j=0r−1∑k=0r−1ωrjk|ak⟩=|1⟩+1r∑k=1r−1(∑j=0r−1ωrjk)|ak⟩=|1⟩,{\displaystyle {\begin{aligned}{\frac {1}{\sqrt {r}}}\sum _{j=0}^{r-1}|\psi _{j}\rangle &={\frac {1}{r}}\sum _{j=0}^{r-1}\sum _{k=0}^{r-1}\omega _{r}^{jk}|a^{k}\rangle \\&=|1\rangle +{\frac {1}{r}}\sum _{k=1}^{r-1}\left(\sum _{j=0}^{r-1}\omega _{r}^{jk}\right)|a^{k}\rangle =|1\rangle ,\end{aligned}}}where the last identity follows from thegeometric seriesformula, which implies∑j=0r−1ωrjk=0{\textstyle \sum _{j=0}^{r-1}\omega _{r}^{jk}=0}.
Usingquantum phase estimationon an input state|0⟩⊗2n|ψj⟩{\displaystyle |0\rangle ^{\otimes 2n}|\psi _{j}\rangle }would then return the integer22nj/r{\displaystyle 2^{2n}j/r}with high probability. More precisely, the quantum phase estimation circuit sends|0⟩⊗2n|ψj⟩{\displaystyle |0\rangle ^{\otimes 2n}|\psi _{j}\rangle }to|ϕj⟩|ψj⟩{\displaystyle |\phi _{j}\rangle |\psi _{j}\rangle }such that the resulting probability distributionpk≡|⟨k|ϕj⟩|2{\displaystyle p_{k}\equiv |\langle k|\phi _{j}\rangle |^{2}}is peaked aroundk=22nj/r{\displaystyle k=2^{2n}j/r}, withp22nj/r≥4/π2≈0.4053{\displaystyle p_{2^{2n}j/r}\geq 4/\pi ^{2}\approx 0.4053}. This probability can be made arbitrarily close to 1 using extra qubits.
Applying the above reasoning to the input|0⟩⊗2n|1⟩{\displaystyle |0\rangle ^{\otimes 2n}|1\rangle }, quantum phase estimation thus results in the evolution|0⟩⊗2n|1⟩=1r∑j=0r−1|0⟩⊗2n|ψj⟩→1r∑j=0r−1|ϕj⟩|ψj⟩.{\displaystyle |0\rangle ^{\otimes 2n}|1\rangle ={\frac {1}{\sqrt {r}}}\sum _{j=0}^{r-1}|0\rangle ^{\otimes 2n}|\psi _{j}\rangle \to {\frac {1}{\sqrt {r}}}\sum _{j=0}^{r-1}|\phi _{j}\rangle |\psi _{j}\rangle .}Measuring the first register, we now have a balanced probability1/r{\displaystyle 1/r}to find each|ϕj⟩{\displaystyle |\phi _{j}\rangle }, each one giving an integer approximation to22nj/r{\displaystyle 2^{2n}j/r}, which can be divided by22n{\displaystyle 2^{2n}}to get a decimal approximation forj/r{\displaystyle j/r}.
Then, we apply thecontinued-fractionalgorithm to find integersb{\displaystyle b}andc{\displaystyle c}, whereb/c{\displaystyle b/c}gives the best fraction approximation for the approximation measured from the circuit, forb,c<N{\displaystyle b,c<N}andcoprimeb{\displaystyle b}andc{\displaystyle c}. The number of qubits in the first register,2n{\displaystyle 2n}, which determines the accuracy of the approximation, guarantees thatbc=jr,{\displaystyle {\frac {b}{c}}={\frac {j}{r}},}given the best approximation from the superposition of|ϕj⟩{\displaystyle |\phi _{j}\rangle }was measured[2](which can be made arbitrarily likely by using extra bits and truncating the output). However, whileb{\displaystyle b}andc{\displaystyle c}are coprime, it may be the case thatj{\displaystyle j}andr{\displaystyle r}are not coprime. Because of that,b{\displaystyle b}andc{\displaystyle c}may have lost some factors that were inj{\displaystyle j}andr{\displaystyle r}. This can be remedied by rerunning the quantum order-finding subroutine an arbitrary number of times, to produce a list of fraction approximationsb1c1,b2c2,…,bscs,{\displaystyle {\frac {b_{1}}{c_{1}}},{\frac {b_{2}}{c_{2}}},\ldots ,{\frac {b_{s}}{c_{s}}},}wheres{\displaystyle s}is the number of times the subroutine was run. Eachck{\displaystyle c_{k}}will have different factors taken out of it because the circuit will (likely) have measured multiple different possible values ofj{\displaystyle j}. To recover the actualr{\displaystyle r}value, we can take theleast common multipleof eachck{\displaystyle c_{k}}:lcm(c1,c2,…,cs).{\displaystyle \operatorname {lcm} (c_{1},c_{2},\ldots ,c_{s}).}The least common multiple will be the orderr{\displaystyle r}of the original integera{\displaystyle a}with high probability. In practice, a single run of the quantum order-finding subroutine is in general enough if more advanced post-processing is used.[25]
Phase estimation requires choosing the size of the first register to determine the accuracy of the algorithm, and for the quantum subroutine of Shor's algorithm,2n{\displaystyle 2n}qubits is sufficient to guarantee that the optimal bitstring measured from phase estimation (meaning the|k⟩{\displaystyle |k\rangle }wherek/22n{\textstyle k/2^{2n}}is the most accurate approximation of the phase from phase estimation) will allow the actual value ofr{\displaystyle r}to be recovered.
Each|ϕj⟩{\displaystyle |\phi _{j}\rangle }before measurement in Shor's algorithm represents a superposition of integers approximating22nj/r{\displaystyle 2^{2n}j/r}. Let|k⟩{\displaystyle |k\rangle }represent the most optimal integer in|ϕj⟩{\displaystyle |\phi _{j}\rangle }. The following theorem guarantees that the continued fractions algorithm will recoverj/r{\displaystyle j/r}fromk/22n{\displaystyle k/2^{2{n}}}:
Theorem—Ifj{\displaystyle j}andr{\displaystyle r}aren{\displaystyle n}bit integers, and|jr−ϕ|≤12r2{\displaystyle \left\vert {\frac {j}{r}}-\phi \right\vert \leq {\frac {1}{2r^{2}}}}then the continued fractions algorithm run onϕ{\displaystyle \phi }will recover bothjgcd(j,r){\textstyle {\frac {j}{\gcd(j,\;r)}}}andrgcd(j,r){\textstyle {\frac {r}{\gcd(j,\;r)}}}.
[3]Ask{\displaystyle k}is the optimal bitstring from phase estimation,k/22n{\displaystyle k/2^{2{n}}}is accurate toj/r{\displaystyle j/r}by2n{\displaystyle 2n}bits. Thus,|jr−k22n|≤122n+1≤12N2≤12r2{\displaystyle \left\vert {\frac {j}{r}}-{\frac {k}{2^{2n}}}\right\vert \leq {\frac {1}{2^{2{n}+1}}}\leq {\frac {1}{2N^{2}}}\leq {\frac {1}{2r^{2}}}}which implies that the continued fractions algorithm will recoverj{\displaystyle j}andr{\displaystyle r}(or with their greatest common divisor taken out).
The runtime bottleneck of Shor's algorithm is quantummodular exponentiation, which is by far slower than thequantum Fourier transformand classical pre-/post-processing. There are several approaches to constructing and optimizing circuits for modular exponentiation. The simplest and (currently) most practical approach is to mimic conventional arithmetic circuits withreversible gates, starting withripple-carry adders. Knowing the base and the modulus of exponentiation facilitates further optimizations.[26][27]Reversible circuits typically use on the order ofn3{\displaystyle n^{3}}gates forn{\displaystyle n}qubits. Alternative techniques asymptotically improve gate counts by usingquantum Fourier transforms, but are not competitive with fewer than 600 qubits owing to high constants.
Shor's algorithms for thediscrete logand the order finding problems are instances of an algorithm solving the period finding problem.[citation needed]All three are instances of thehidden subgroup problem.
Given agroupG{\displaystyle G}with orderp{\displaystyle p}andgeneratorg∈G{\displaystyle g\in G}, suppose we know thatx=gr∈G{\displaystyle x=g^{r}\in G}, for somer∈Zp{\displaystyle r\in \mathbb {Z} _{p}}, and we wish to computer{\displaystyle r}, which is thediscrete logarithm:r=logg(x){\displaystyle r={\log _{g}}(x)}. Consider theabelian groupZp×Zp{\displaystyle \mathbb {Z} _{p}\times \mathbb {Z} _{p}}, where each factor corresponds to modular addition of values. Now, consider the function
This gives us an abelianhidden subgroup problem, wheref{\displaystyle f}corresponds to agroup homomorphism. Thekernelcorresponds to the multiples of(r,1){\displaystyle (r,1)}. So, if we can find the kernel, we can findr{\displaystyle r}. A quantum algorithm for solving this problem exists. This algorithm is, like the factor-finding algorithm, due to Peter Shor and both are implemented by creating a superposition through using Hadamard gates, followed by implementingf{\displaystyle f}as a quantum transform, followed finally by a quantum Fourier transform.[3]Due to this, the quantum algorithm for computing the discrete logarithm is also occasionally referred to as "Shor's Algorithm."
The order-finding problem can also be viewed as a hidden subgroup problem.[3]To see this, consider the group of integers under addition, and for a givena∈Z{\displaystyle a\in \mathbb {Z} }such that:ar=1{\displaystyle a^{r}=1}, the function
For any finite abelian groupG{\displaystyle G}, a quantum algorithm exists for solving the hidden subgroup forG{\displaystyle G}in polynomial time.[3]
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TheSignal Protocol(formerly known as theTextSecure Protocol) is a non-federatedcryptographic protocolthat providesend-to-end encryptionfor voice andinstant messagingconversations.[2]The protocol was developed byOpen Whisper Systemsin 2013[2]and was introduced in theopen-sourceTextSecureapp, which later becameSignal. Severalclosed-sourceapplications have implemented the protocol, such asWhatsApp, which is said to encrypt the conversations of "more than a billion people worldwide"[3]orGooglewho provides end-to-end encryption by default to allRCS-based conversations between users of theirGoogle Messagesapp for one-to-one conversations.[4]Facebook Messengeralso say they offer the protocol for optional "Secret Conversations", as doesSkypefor its "Private Conversations".
The protocol combines theDouble Ratchet Algorithm, prekeys (i.e., one-time ephemeral public keys that have been uploaded in advance to a central server), and a tripleelliptic-curve Diffie–Hellman(3-DH) handshake,[5]and usesCurve25519,AES-256, andHMAC-SHA256asprimitives.[6]
The development of the Signal Protocol was started by Trevor Perrin andMoxie Marlinspike(Open Whisper Systems) in 2013. The first version of the protocol, TextSecure v1, was based onOff-the-record messaging(OTR).[7][8]
On 24 February 2014, Open Whisper Systems introduced TextSecure v2,[9]which migrated to the Axolotl Ratchet.[7][10]The design of the Axolotl Ratchet is based on the ephemeral key exchange that was introduced by OTR and combines it with a symmetric-key ratchet modeled after theSilent Circle Instant Message Protocol(SCIMP).[1]It brought about support forasynchronous communication("offline messages") as its major new feature, as well as better resilience with distorted order of messages and simpler support for conversations with multiple participants.[11]The Axolotl Ratchet was named after the critically endangered aquatic salamanderAxolotl, which has extraordinary self-healing capabilities. The developers refer to the algorithm as self-healing because it automatically disables an attacker from accessing thecleartextof later messages after having compromised asession key.[1]
The third version of the protocol, TextSecure v3, made some changes to the cryptographic primitives and the wire protocol.[7]In October 2014, researchers fromRuhr University Bochumpublished an analysis of TextSecure v3.[6][7]Among other findings, they presented anunknown key-share attackon the protocol, but in general, they found that it was secure.[12]
In March 2016, the developers renamed the protocol as the Signal Protocol. They also renamed the Axolotl Ratchet as the Double Ratchet algorithm to better differentiate between the ratchet and the full protocol[13]because some had used the name Axolotl when referring to the full protocol.[14][13]
As of October 2016[update], the Signal Protocol is based on TextSecure v3, but with additional cryptographic changes.[7]In October 2016, researchers from the UK'sUniversity of Oxford, Australia'sQueensland University of Technology, and Canada'sMcMaster Universitypublished a formal analysis of the protocol, concluding that the protocol was cryptographically sound.[15][16]
Another audit of the protocol was published in 2017.[17]
The protocol provides confidentiality, integrity,authentication, participant consistency, destination validation,forward secrecy, post-compromise security (aka future secrecy), causality preservation, message unlinkability,message repudiation, participation repudiation, and asynchronicity.[18]It does not provide anonymity preservation and requires servers for the relaying of messages and storing of public key material.[18]
The Signal Protocol also supports end-to-end encrypted group chats. The group chat protocol is a combination of a pairwisedouble ratchetandmulticast encryption.[18]In addition to the properties provided by the one-to-one protocol, the group chat protocol provides speaker consistency, out-of-order resilience, dropped message resilience, computational equality, trust equality, subgroup messaging, as well as contractible and expandable membership.[18]
For authentication, users can manually comparepublic key fingerprintsthrough an outside channel.[19]This makes it possible for users to verify each other's identities and avoid aman-in-the-middle attack.[19]An implementation can also choose to employ atrust on first usemechanism in order to notify users if a correspondent's key changes.[19]
The Signal Protocol does not prevent a company from retaining information about when and with whom users communicate.[20][21]There can therefore be differences in how messaging service providers choose to handle this information. Signal'sprivacy policystates that recipients' identifiers are only kept on the Signal servers as long as necessary in order to transmit each message.[22]In June 2016, Moxie Marlinspike toldThe Intercept: "the closest piece of information to metadata that the Signal server stores is the last time each user connected to the server, and the precision of this information is reduced to the day, rather than the hour, minute, and second."[21]
In October 2018, Signal Messenger announced that they had implemented a "sealed sender" feature into Signal, which reduces the amount of metadata that the Signal servers have access to by concealing the sender's identifier.[23][24]The sender's identity is conveyed to the recipient in each message, but is encrypted with a key that the server does not have.[24]This is done automatically if the sender is in the recipient's contacts or has access to their Signal Profile.[24]Users can also enable an option to receive "sealed sender" messages from non-contacts and people who do not have access to their Signal Profile.[24]A contemporaneous wiretap of the user's device and/or the Signal servers may still reveal that the device's IP address accessed a Signal server to send or receive messages at certain times.[23]
Open Whisper Systems first introduced the protocol in applicationTextSecure. They later merged an encrypted voice call application namedRedPhoneinto TextSecure and renamed itSignal.
In November 2014, Open Whisper Systems announced a partnership withWhatsAppto provide end-to-end encryption by incorporating the Signal Protocol into each WhatsApp client platform.[25]Open Whisper Systems said that they had already incorporated the protocol into the latest WhatsApp client forAndroidand that support for other clients, group/media messages, and key verification would be coming soon after.[26]On April 5, 2016, WhatsApp and Open Whisper Systems announced that they had finished adding end-to-end encryption to "every form of communication" on WhatsApp, and that users could now verify each other's keys.[27][28]In February 2017, WhatsApp announced a new feature, WhatsApp Status, which uses the Signal Protocol to secure its contents.[29]In October 2016, WhatsApp's parent companyFacebookalso deployed an optional mode called Secret Conversations inFacebook Messengerwhich provides end-to-end encryption using an implementation of the Signal Protocol.[30][31][32][33]
In September 2015,G Data Softwarelaunched a new messaging app called Secure Chat which used the Signal Protocol.[34][35]G Data discontinued the service in May 2018.[36]
In September 2016,Googlelaunched a new messaging app calledAllo, which featured an optional "incognito mode" that used the Signal Protocol for end-to-end encryption.[37][38]In March 2019, Google discontinued Allo in favor of theirGoogle Messagesapp on Android.[39][40]In November 2020, Google announced that they would be using the Signal Protocol to provide end-to-end encryption by default to allRCS-based conversations between users of theirGoogle Messagesapp, starting with one-to-one conversations.[4][41]
In January 2018, Open Whisper Systems andMicrosoftannounced the addition of Signal Protocol support to an optionalSkypemode called Private Conversations.[42][43]
The Signal Protocol has had an influence on other cryptographic protocols. In May 2016,Vibersaid that their encryption protocol is a custom implementation that "uses the same concepts" as the Signal Protocol.[44][45]Forsta's developers have said that their app uses a custom implementation of the Signal Protocol.[46][47][independent source needed]
TheDouble Ratchet Algorithmthat was introduced as part of the Signal Protocol has also been adopted by other protocols.OMEMOis an XMPP Extension Protocol (XEP) that was introduced in theConversationsmessaging app and approved by theXMPP Standards Foundation(XSF) in December 2016 as XEP-0384.[48][2]Matrixis an open communications protocol that includes Olm, a library that provides optional end-to-end encryption on a room-by-room basis via a Double Ratchet Algorithm implementation.[2]The developers ofWirehave said that their app uses a custom implementation of the Double Ratchet Algorithm.[49][50][51]
Messaging Layer Security, anIETFproposal, usesAsynchronous ratcheting treesto efficiently improve upon security guarantees over Signal'sDouble Ratchet.[52]
Signal Messenger maintains areference implementationof the Signal Protocollibrarywritten inRustunder theAGPLv3license onGitHub. There are bindings to Swift, Java, TypeScript, C, and other languages that use the reference Rust implementation.
Signal maintained the following deprecated libraries:
There also exist alternative libraries written by third-parties in other languages, such asTypeScript.[53]
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Signalis an Americanopen-source,encryptedmessaging service forinstant messaging,voice calls, andvideo calls.[14][15]The instant messaging function includes sending text, voice notes, images, videos, and other files.[16]Communication may be one-to-one between users or may involve group messaging.
The application uses acentralized computingarchitecture and iscross-platform software. It is developed by the non-profitSignal Foundationand its subsidiarySignal Messenger LLC. Signal's software isfree and open-source. Its mobile clients, desktop client, andserverare all published under theAGPL-3.0-onlylicense.[a][b][11][10][12][13]The official Android app generally uses the proprietaryGoogle Play Services, although it is designed to be able to work without them. Signal is also distributed foriOSand desktop programs forWindows,macOS, andLinux. Registration for desktop use requires an iOS or Android device.[20][21]
Signal uses mobiletelephone numbersto register and manage user accounts, though configurable usernames were added in March 2024 to allow users to hide their phone numbers from other users.[22]After removing support for SMS on Android in 2023,[23][24]the app now secures all communications withend-to-end encryption. The client software includes mechanisms by which users can independently verify the identity of their contacts and the integrity of the data channel.[23][25]
Thenon-profitSignal Foundation was launched in February 2018 with initial funding of $50 million fromWhatsAppco-founderBrian Acton.[26]As of January 2025[update], the platform had approximately 70 million monthly active users. As of January 2025[update], it had been downloaded more than 220 million times.[27][28]
Signal is the successor of the RedPhone encrypted voice calling app and theTextSecureencrypted texting program. Thebeta versionsof RedPhone and TextSecure were first launched in May 2010 byWhisper Systems,[54]a startup company co-founded by security researcherMoxie Marlinspikeand roboticist Stuart Anderson.[55][56]Whisper Systems also produced a firewall and tools for encrypting other forms of data.[55][57]All of these wereproprietaryenterprise mobile security software and were only available for Android.
In November 2011, Whisper Systems announced that it had been acquired byTwitter. Neither company disclosed the financial terms of the deal.[58]The acquisition was done "primarily so that Mr. Marlinspike could help the then-startup improve its security".[59]Shortly after the acquisition, Whisper Systems' RedPhone service was made unavailable.[60]Some criticized the removal, arguing that the software was "specifically targeted [to help] people under repressive regimes" and that it left people like the Egyptians in "a dangerous position" during the events of theEgyptian revolution of 2011.[61]
Twitter released TextSecure asfree and open-source softwareunder theGPLv3license in December 2011.[55][62][63][64]RedPhone was also released under the same license in July 2012.[65]Marlinspike later left Twitter and founded Open Whisper Systems as a collaborative Open Source project for the continued development of TextSecure and RedPhone.[1][66]
Open Whisper Systems' website was launched in January 2013.[66]
In February 2014, Open Whisper Systems introduced the second version of their TextSecure Protocol (nowSignal Protocol), which added end-to-end encrypted group chat and instant messaging capabilities to TextSecure.[67]Toward the end of July 2014, they announced plans to merge the RedPhone and TextSecure applications as Signal.[68]This announcement coincided with the initial release of Signal as a RedPhone counterpart foriOS. The developers said that their next steps would be to provide TextSecure instant messaging capabilities for iOS, unify the RedPhone and TextSecure applications on Android, and launch aweb client.[68]Signal was the first iOS app to enable end-to-end encrypted voice calls for free.[1][69]TextSecure compatibility was added to the iOS application in March 2015.[70][71]
From its launch in May 2010[54]until March 2015, the Android version of Signal (then called TextSecure) included support for encrypted SMS/MMSmessaging.[72]From version 2.7.0 onward, the Android application only supported sending and receiving encrypted messages via the data channel.[73]Reasons for this included security flaws of SMS/MMS and problems with thekey exchange.[73]Open Whisper Systems' abandonment of SMS/MMS encryption prompted some users to create aforknamed Silence (initially called SMSSecure[74]) that is meant solely for the exchange of encrypted SMS and MMS messages.[75][76]
In November 2015, the TextSecure and RedPhone applications on Android were merged to become Signal for Android.[77]A month later, Open Whisper Systems announced Signal Desktop, aChrome appthat could link with a Signal mobile client.[78]At launch, the app could only be linked with the Android version of Signal.[79]On 26 September 2016, Open Whisper Systems announced that Signal Desktop could now be linked with the iOS version of Signal as well.[80]On 31 October 2017, Open Whisper Systems announced that the Chrome app wasdeprecated.[9]At the same time, they announced the release of a standalone desktop client (based on theElectronframework[12]) forWindows,macOSand certainLinux distributions.[9][81]
On 4 October 2016, theAmerican Civil Liberties Union(ACLU) and Open Whisper Systems published a series of documents revealing that OWS had received asubpoenarequiring them to provide information associated with two phone numbers for a federalgrand juryinvestigation in the first half of 2016.[82][83][84]Only one of the two phone numbers was registered on Signal, and because of how the service is designed, OWS was only able to provide "the time the user's account had been created and the last time it had connected to the service".[83][82]Along with the subpoena, OWS received agag orderrequiring OWS not to tell anyone about the subpoena for one year.[82]OWS approached the ACLU, and they were able to lift part of the gag order after challenging it in court.[82]OWS said it was the first time they had received a subpoena, and that they were "committed to treating any future requests the same way".[84]
In March 2017, Open Whisper Systems transitioned Signal's calling system from RedPhone toWebRTC, also adding the ability to make video calls with the mobile apps.[85][86][14]
On 21 February 2018,Moxie MarlinspikeandWhatsAppco-founderBrian Actonannounced the formation of theSignal Technology Foundation, a501(c)(3) nonprofit organizationwhose mission is "to support, accelerate, and broaden Signal's mission of making private communication accessible and ubiquitous".[87][26]Acton started the foundation with $50 million in funding and became the foundation's executive chairman after leaving WhatsApp's parent company Facebook in September 2017.[26]Marlinspike continued as Signal Messenger's firstCEO.[87]As of 2020[update], Signal ran entirely on donations, as anonprofit.[88]
Between November 2019 and February 2020, Signal addediPadsupport, view-once images and videos,stickers, and reactions.[89]They also announced plans for a new group messaging system and an "experimental method for storing encrypted contacts in the cloud."[89]
Signal was reportedly popularized in the United States during theGeorge Floyd protests. Heightened awareness of police monitoring led protesters to use the platform to communicate.Black Lives Matterorganizershad used the platform "for several years".[90][88]During the first week of June, the encrypted messaging app was downloaded over five times more than it had been during the week prior to themurder of George Floyd.[90]In June 2020, Signal Foundation announced a new feature that enables users to blur faces in photos, in response to increased federal efforts to monitor protesters.[88][91]
On 7 January 2021, Signal saw a surge in new user registrations, which temporarily overwhelmed Signal's capacity to deliver account verification messages.[92]CNNandMacRumorslinked the surge with aWhatsAppprivacy policy change and a Signal endorsement byElon MuskandEdward Snowdenvia Twitter.[92][93]The surge was also tied to the attack on theUnited States Capitol.[94]International newspapers reported similar trends in theUnited Arab Emirates.[95]Reutersreported that more than 100,000 people had installed Signal between 7 and 8 January.[96]
Between 12 and 14 January 2021, the number of Signal installations listed on Google Play increased from over 10 million to over 50 million.[97][98][99][100]On 15 January 2021, due to the surge of new users, Signal was overwhelmed with the new traffic and was down for all users.[101][102]On the afternoon of 16 January, Signal announced via Twitter that service had been restored.[103]
On 10 January 2022, Moxie Marlinspike announced that he was stepping down from his role as CEO of Signal Messenger.[104]He continues to remain on the Signal Foundation'sboard of directorsand Brian Acton has volunteered to serve asinterim CEOduring the search for a new CEO.[104]
In August 2022, Signal notified 1900 users that their data had been affected by theTwiliobreach including user phone numbers and SMS verification codes.[105]At least one journalist had his account re-registered to a device he did not control as a result of the attack.[106]
In September 2022 Signal Messaging LLC announced that AI researcher and noted critic of big techMeredith Whittakerwould fill the newly created position of President.[107]
Signal's userbase started in May 2010, when its predecessorTextSecurewas launched byWhisper Systems.[54]According to App Annie, Signal had approximately 20 million monthly active users at the end of December 2020.[108]In January 2022, the BBC reported that Signal was used by over 40 million people.[109]In February 2025, Signal had over 7 millionmonthly active usersin the USA according toSimilarweb.[110]
According toJohn Ratcliffe, as of 2025 Signal is installed by default on the devices of mostCIAemployees and its usage is covered by standard onboarding training.[111]
The development of Signal and its predecessors atOpen Whisper Systemswas funded by a combination of consulting contracts, donations andgrants.[112]TheFreedom of the Press Foundationacted as Signal'sfiscal sponsor.[87][113][114]Between 2013 and 2016, the project received grants from theKnight Foundation,[115]theShuttleworth Foundation,[116]and almost $3 million from the US government–sponsoredOpen Technology Fund.[117]Signal is now developed by Signal Messenger LLC, a software company founded byMoxie MarlinspikeandBrian Actonin 2018, which is wholly owned by a tax-exempt nonprofit corporation called theSignal Technology Foundation, also created by them in 2018. The Foundation was funded with an initial loan of $50 million from Acton, "to support, accelerate, and broaden Signal's mission of making private communication accessible and ubiquitous".[87][26][118]All of the organization's products are published asfree and open-source software.
In November 2023, Meredith Whittaker revealed that she expected the annual cost of running Signal to reach $50 million in 2025, with the current cost estimated around $40 million.[119]
Signal provides one-to-one and group[120]voice and video[14]calls with up to forty participants on iOS, Android, and desktop platforms.[121][122]The calls are carried via the devices' wired or wireless (carrier orWiFi) data connections.[69]The application can send text messages, documents files,[16]voice notes, pictures,stickers,GIFs,[123]and video messages. The platform also supports group messaging.
All communication sessions between Signal users are automaticallyend-to-end encrypted(the encryptionkeysare generated and stored on the devices, and not on servers).[124]To verify that a correspondent is really the person that they claim to be, Signal users can compare key fingerprints (or scan QR codes)out-of-band.[125]The platform employs atrust-on-first-usemechanism to notify the user if a correspondent's key changes.[125]
Until 2023, Android users could opt into making Signal the default SMS/MMS application, allowing them to send and receive unencrypted SMS messages in addition to the standard end-to-end encrypted Signal messages.[67]Users could then use the same application to communicate with contacts who do not have Signal.[67]As of October 2022, this feature has been deprecated due to safety and security concerns, and was removed in 2023.[126][24]
TextSecure allowed the user to set a passphrase that encrypted the local message database and the user's encryption keys.[127]This did not encrypt the user's contact database or message timestamps.[127]The Signal applications on Android and iOS can be locked with the phone's pin, passphrase, or biometric authentication.[128]The user can define a "screen lock timeout" interval, where Signal will re-encrypt the messages after a certain amount of time, providing an additional protection mechanism in case the phone is lost or stolen.[125][128]
Signal has a feature for scheduling messages.[129]In addition, timers may be attached to messages[130]to automatically delete the messages from both the sender's and the receivers' devices.[130]The time period for keeping the message may be between five seconds and one week,[130]and begins for each recipient once they have read their copy of the message.[131]The developers stressed that this is meant to be "a collaborative feature for conversations where all participants want to automate minimal data hygiene, not for situations where the recipient is an adversary".[130][131]
Signal's app icon may be changed with a variety of colour themes for customization and to hide the app. The application name can also be customized.[132]Messages can have effects like spoilers and italics, and users can add each other via QR code.[133]
Signal excludes users' messages from non-encrypted cloud backups by default.[134]
Signal allows users to automatically blur faces of people in photos to protect identities.[135][136]
Signal includes acryptocurrency walletfunctionality for storing, sending and receiving in-app payments.[137]Apart from certain regions and countries,[137]the feature was enabled globally in November 2021.[138]As of January 2022[update], the only supported payment method isMobileCoin.[137]
In February 2024, Signal added a username feature to the beta version of the app. This is a privacy feature that allows users to communicate with others without having to share their telephone number.[139][140]
Signal requires that the user provide a telephone number for verification,[141]eliminating the need for user names or passwords and facilitating contact discovery (see below).[142]The number does not have to be the same as on the device's SIM card; it can also be a VoIP number[141]or a landline as long as the user can receive the verification code and have a separate device to set up the software. A number can only be registered on one mobile device at a time.[143]Account registration requires an iOS or Android device.[20][21]
This mandatory connection to a telephone number (a feature Signal shares withWhatsApp,KakaoTalk, and others) has been criticized as a "major issue" for privacy-conscious users who are not comfortable with giving out their private number.[142]A workaround is to use a secondary phone number.[142]The ability to choose a public, changeable username instead of sharing one's phone number was a widely-requested feature.[142][144][145]This feature was added to the beta version of Signal in February 2024.[146]
Using phone numbers as identifiers may also create security risks that arise from the possibility of an attacker taking over a phone number.[142]A similar vulnerability was used to attack at least one user in August 2022, though the attack was performed via the provider of Signal's SMS services, not any user's provider.[105]The threat of this attack can be mitigated by enabling Signal's Registration Lock feature, a form oftwo-factor authenticationthat requires the user to enter a PIN to register the phone number on a new device.[147]
In July 2016, theInternet Societypublished auser studythat assessed the ability of Signal users to detect and determan-in-the-middle attacks.[25]The study concluded that 21 out of 28 participants failed to correctly comparepublic key fingerprintsin order to verify the identity of other Signal users, and that most of these users believed they had succeeded, while they had actually failed.[25]Four months later, Signal's user interface was updated to make verifying the identity of other Signal users simpler.[148]
Signal messages are encrypted with the Signal Protocol (formerly known as the TextSecure Protocol). The protocol combines theDouble Ratchet Algorithm, prekeys, and an Extended TripleDiffie–Hellman(X3DH) handshake.[149][150]It usesCurve25519,AES-256, andHMAC-SHA256asprimitives.[23]The protocol provides confidentiality, integrity,authentication, participant consistency, destination validation,forward secrecy, backward secrecy (a.k.a.future secrecy), causality preservation, message unlinkability,message repudiation, participation repudiation, andasynchronicity.[151]It does not provide anonymity preservation, and requires servers for the relaying of messages and storing of public key material.[151]
The Signal Protocol also supports end-to-end encrypted group chats. The group chat protocol is a combination of a pairwise double ratchet andmulticast encryption.[151]In addition to the properties provided by the one-to-one protocol, the group chat protocol provides speaker consistency, out-of-order resilience, dropped message resilience, computational equality, trust equality, subgroup messaging, as well as contractible and expandable membership.[151]
In October 2014, researchers fromRuhr University Bochum(RUB) published an analysis of the Signal Protocol.[23]Among other findings, they presented anunknown key-share attackon the protocol, but in general, they found that it was secure.[152]In October 2016, researchers from UK'sUniversity of Oxford,Queensland University of Technologyin Australia, and Canada'sMcMaster Universitypublished a formal analysis of the protocol.[153][154]They concluded that the protocol was cryptographically sound.[153][154]In July 2017, researchers from RUB found during another analysis of group messengers a purely theoretic attack against the group protocol of Signal: A user who knows the secret group ID of a group (due to having been a group member previously or stealing it from a member's device) can become a member of the group. Since the group ID cannot be guessed and such member changes are displayed to the remaining members, this attack is likely to be difficult to carry out without being detected.[155]
As of August 2018[update], the Signal Protocol has been implemented intoWhatsApp,Facebook Messenger,Skype,[156]andGoogle Allo,[157]making it possible for the conversations of "more than a billion people worldwide" to be end-to-end encrypted.[158]In Google Allo, Skype and Facebook Messenger, conversations are not encrypted with the Signal Protocol by default; they only offer end-to-end encryption in an optional mode.[134][159][156][160]
Up until March 2017, Signal's voice calls were encrypted withSRTPand theZRTPkey-agreement protocol, which was developed byPhil Zimmermann.[1][161]In March 2017, Signal transitioned to a newWebRTC-based calling system that introduced the ability to make video calls.[86]Signal's voice and video calling functionalities use the Signal Protocol channel for authentication instead of ZRTP.[162][85][14]
To verify that a correspondent is really the person that they claim to be, Signal users can compare key fingerprints (or scan QR codes)out-of-band.[125]The platform employs atrust on first usemechanism in order to notify the user if a correspondent's key changes.[125]
After receiving and decrypting messages, the application stored them locally on each device in aSQLitedatabase that is encrypted with SQLCipher.[163]The cryptographic key for this database is also stored locally and can be accessed if the device is unlocked.[163][164]In December 2020,Cellebritepublished a blog post announcing that one of their products could now access this key and use it to "decrypt the Signal app".[163][165]Technology reporters later published articles about how Cellebrite had claimed to have the ability to "break into the Signal app" and "crack Signal's encryption".[166][167]This latter interpretation was rejected by several experts,[168]as well as representatives from Signal, who said the original post by Cellebrite had been about accessing data on "an unlocked Android phone in their physical possession" and that they "could have just opened the app to look at the messages".[169][170]Similar extraction tools also exist for iOS devices and Signal Desktop.[171][172]
Signal uses infrastructure from large providers likeAmazon Web Services,Google Compute EngineandMicrosoft Azurearound the world.[173]In addition to routing Signal's messages, the servers also facilitate the discovery of contacts who are also registered Signal users and the automaticexchangeof users'public keys. By default, Signal's voice and video calls arepeer-to-peer.[14]If the caller is not in the receiver's address book, the call is routed through a server in order to hide the users'IP addresses.[14]
The servers store registered users' phone numbers, public key material and push tokens which are necessary for setting up calls and transmitting messages.[174]In order to determine which contacts are also Signal users,cryptographic hashesof the user's contact numbers are periodically transmitted to the server.[175]The server then checks to see if those match any of theSHA-256hashes of registered users and tells the client if any matches are found.[175]The hashed numbers are thereafter discarded from the server.[174]In 2014, Moxie Marlinspike wrote that it is easy to calculate a map of all possible hash inputs to hash outputs and reverse the mapping because of the limitedpreimagespace (the set of all possible hash inputs) of phone numbers, and that a "practical privacy preserving contact discovery remains an unsolved problem."[176][175]In September 2017, Signal's developers announced that they were working on a way for the Signal client applications to "efficiently and scalably determine whether the contacts in their address book are Signal users without revealing the contacts in their address book to the Signal service."[177][178]
All client–server communications are protected byTLS.[161][179]Signal's developers have asserted that their servers do not keep logs about who called whom and when.[180]In June 2016, Marlinspike toldThe Interceptthat "the closest piece of information to metadata that the Signal server stores is the last time each user connected to the server, and the precision of this information is reduced to the day, rather than the hour, minute, and second".[134]
The group messaging mechanism is designed so that the servers do not have access to the membership list, group title, or group icon.[73]Instead, the creation, updating, joining, and leaving of groups is done by the clients, which deliver pairwise messages to the participants in the same way that one-to-one messages are delivered.[181][182]
Signal's server architecture wasfederatedbetween December 2013 and February 2016. In December 2013, it was announced that the messaging protocol Signal uses had successfully been integrated into the Android-based open-source operating systemCyanogenMod.[183][184][185]Since CyanogenMod 11.0, the client logic was contained in a system app called WhisperPush. According to Signal's developers, the Cyanogen team ran their own Signal messaging server for WhisperPush clients, which federated with the main server, so that both clients could exchange messages with each other.[185]The WhisperPush source code was available under the GPLv3 license.[186]In February 2016, the CyanogenMod team discontinued WhisperPush and recommended that its users switch to Signal.[187]In May 2016, Moxie Marlinspike wrote that federation with the CyanogenMod servers had degraded the user experience and held back development, and that their servers will probably not federate with other servers again.[188]
In May 2016, Moxie Marlinspike requested that a third-party client called LibreSignal not use the Signal service or the Signal name.[188]As a result, on 24 May 2016 the LibreSignal project posted that the project was "abandoned".[189]The functionality provided by LibreSignal was subsequently incorporated into Signal by Marlinspike.[190]
The completesource codeof the Signal clients for Android, iOS and desktop is available onGitHubunder afree software license.[11][10][12]This enables interested parties to examine the code and help the developers verify that everything is behaving as expected. It also allows advanced users to compile their own copies of the applications and compare them with the versions that are distributed by Signal Messenger. In March 2016, Moxie Marlinspike wrote that, apart from some shared libraries that are not compiled with the project build due to a lack of Gradle NDK support, Signal for Android isreproducible.[191]Signal's servers are partially open source, but the server software's anti-spam component is proprietary and closed source due to security concerns.[13][192]
In October 2014, theElectronic Frontier Foundation(EFF) included Signal in their updated surveillance self-defense guide.[193]In November 2014, Signal received a perfect score on the EFF's secure messaging scorecard;[124]it received points for having communications encrypted in transit, having communications encrypted with keys the provider does not have access to (end-to-end encryption), making it possible for users to independently verify their correspondents' identities, having past communications secure if the keys are stolen (forward secrecy), having the code open to independent review (open source), having the security designs well-documented, and having a recent independent security audit.[124]At the time, "ChatSecure+Orbot",Pidgin(withOTR),Silent Phone, andTelegram's optional "secret chats" also received seven out of seven points on the scorecard.[124]
FormerNSAcontractorEdward Snowdenhas endorsed Signal on multiple occasions.[78]In his keynote speech atSXSWin March 2014, he praised Signal's predecessors (TextSecure and RedPhone) for their ease of use.[194][195]In December 2014,Der Spiegelleaked slides from an internal NSA presentation dating to June 2012 in which the NSA deemed Signal's encrypted voice calling component (RedPhone) on its own as a "major threat" to its mission of accessing users' private data, and when used in conjunction with other privacy tools such as Cspace,Tor,Tails, andTrueCryptwas ranked as "catastrophic" and led to a "near-total loss/lack of insight to target communications [and] presence".[196][197]
Following the2016 Democratic National Committee email leak, it was reported byVanity FairthatMarc Elias(the general counsel forHillary Clinton's presidential campaign) had instructedDNCstaffers to exclusively use Signal when saying anything negative about Republican presidential nomineeDonald Trump.[198][199]
In March 2017, Signal was approved by the sergeant at arms of theU.S. Senatefor use by senators and their staff.[200][201]
On 27 September 2019, Natalie Silvanovich, a security engineer working inGoogle's vulnerability research team atProject Zero, disclosed how a bug in theAndroidSignal client could let an attacker spy on a user without their knowledge.[202]The bug allowed an attacker to phone a target device, mute the call, and the call would complete – keeping the audio open but without the owner being aware of that (however they would still be aware of a ring and / or a vibration from the initial call).[203]The bug was fixed the same day that it was reported and patched in release 4.47.7 of the app for Android.[204]
In February 2020, theEuropean Commissionrecommended that its staff use Signal.[205]Following theGeorge Floyd protests, which began in May 2020, Signal was downloaded 121,000 times in the U.S. between 25 May and 4 June.[206]In July 2020, Signal became the most downloaded app inHong Kongon both the Apple App Store and the Google Play Store after the passage of theHong Kong national security law.[207]
As of January 2021[update], Signal is a contact method for securely providing tips to major news outlets such asThe Washington Post,[208]The Guardian,[209]The New York Times,[210]andThe Wall Street Journal.[211]
The spyware companiesCandiruandFinFisherclaim the ability to extract messages from Signal when installed on a phone using their spyware.[212][213]Some forks of Signal attempt to combat this by encrypting data at rest.
On 9 August 2022,Ismail Sabri Yaakob, thePrime Minister of Malaysia, reported that his Signal account was "hacked" and infiltrated by a third party, sending out messages and impersonating the politician. No details were disclosed regarding the method used to gain access to the account.[214]
In April 2021, Signal announced the addition of acryptocurrency walletfeature that would allow users to send and receive payments inMobileCoin.[215]This received criticism from security expertBruce Schneier, who had previously praised the software. Schneier stated that this would bloat the client and attract unwanted attention from the authorities.[216]The wallet functionality was initially only available in certain countries, but was later enabled globally in November 2021.[138]
In December 2016,Egyptblocked access to Signal.[217]In response, Signal's developers addeddomain frontingto their service.[218]This allows Signal users in a specific country to circumvent censorship by making it look like they are connecting to a different internet-based service.[218][219]As of May 2022[update], Signal's domain fronting is enabled by default in Egypt,UAE,Oman,Qatar,Iran,Cuba,UzbekistanandUkraine.[220]
As of January 2018[update], Signal was blocked in Iran.[221][222]Signal's domain fronting feature relies on theGoogle App Engine(GAE) service.[222][221]This does not work in Iran because Google has blocked Iranian access to GAE in order to comply with U.S. sanctions.[221][223]
In early 2018,Google App Enginemade an internal change to stop domain fronting for all countries. Due to this issue, Signal made a public change to useAmazon CloudFrontfor domain fronting. However,AWSalso announced that they would be making changes to their service to prevent domain fronting. As a result, Signal said that they would start investigating new methods/approaches.[224][225]Signal switched from AWS back to Google in April 2019.[226]
In January 2021, Iran removed the app from app stores,[227][228]and blocked Signal.[229]Signal was later blocked by China in March 2021, followed by its removal from the App Store in China on 19 April 2024.[230][231]
On August 9, 2024, Signal was blocked in Russia.Roskomnadzorclaimed that this was due to "violations of the law on combating terrorism and extremism".[232][233]Around the same, Signal was also blocked in Venezuela following the contested2024 presidential electionand subsequent protests.[232]
In 2020, the app was used for coordination and communication by protesters during theGeorge Floyd protestsas they relied on the app's end-to-end encryption to share information securely.[234]
In March 2021, theUnited Nationsrecommended Myanmar residents use Signal andProton Mailto pass and preserve evidence of human rights violations committed during the2021 coup.[235]
Signal'sterms of servicestates that the product may not be used to violate the law.[236]According to a former employee, Signal's leadership at the time told him they would say something "if and when people start abusing Signal or doing things that we think are terrible".[236]In January 2021, the position of Signal's leadership was to take a "hands-off approach to moderation" as the company's employees are not able to read user messages and the Signal Foundation does not "want to be a media company".[236][154]
In 2016, authorities inIndiaarrested members of a suspectedISIS-affiliated terrorist cell that communicated via Signal.[237]
Radical right-wingmilitias and white nationalists use Signal for organizing their actions, including theUnite the Right IIrally in 2018.[238][239][240][241]
The claim that Signal is used to fund terrorist or criminal activities is the justification forTurkeyto criminalize the app for the general population, whichAbdullah Bozkurtclaims is a way the "government abuses its counterterrorism laws to punish critics, opponents and dissidents."[242][243]
In March 2025, it was revealed that senior members of theTrump administration—including vice presidentJD Vance, secretary of stateMarco Rubio, and defense secretaryPete Hegseth—were using Signal to discuss sensitive information, including details ofmilitary attack plans in Yemen. The existence of the chat was disclosed byThe Atlanticeditor-in-chiefJeffrey Goldberg, who was accidentally added to the chat. Brian Hughes of theNational Security Councillater confirmed the authenticity of Goldberg's account.[244][245][246]
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End-to-end encryption(E2EE) is a method of implementing a secure communication system where only communicating users can participate. No one else, including the system provider,telecom providers,Internet providersormalicious actors, can access thecryptographic keysneeded to read or send messages.[1]
End-to-endencryptionprevents data from being read orsecretly modified, except by the true sender and intended recipients. Frequently, the messages are relayed from the sender to the recipients by a service provider. However, messages are encrypted by the sender and no third party, including the service provider, has the means to decrypt them. The recipients retrieve the encrypted messages and decrypt them independently. Since third parties cannot decrypt the data being communicated or stored, services that provide end-to-end encryption are better at protecting user data when they are affected bydata breaches.[2]Such services are also unable to share user data with government authorities, domestic or international.[3][4]
In 2022, the UK'sInformation Commissioner's Office, the government body responsible for enforcing online data standards, stated that opposition to E2EE was misinformed and the debate too unbalanced, with too little focus on benefits, since E2EE helped keep children safe online and law enforcement access to stored data on servers was "not the only way" to find abusers.[5]
In many non-E2EE messaging systems, includingemailand many chat networks, messages pass through intermediaries and are stored by a third party service provider,[6]from which they are retrieved by the recipient. Even if the messages are encrypted, they are onlyencrypted 'in transit', and are thus accessible by the service provider.[7]Server-sidedisk encryptionis also distinct from E2EE because it does not prevent the service provider from viewing the information, as they have the encryption keys and can simply decrypt it.
The lack of end-to-end encryption can allow service providers to easily provide search and other features, or to scan for illegal and unacceptable content. However, it also means that content can be read by anyone who has access to the data stored by the service provider, by design or via abackdoor. This can be a concern in many cases where privacy is important, such as in governmental andmilitary communications,financial transactions, and when sensitive information such ashealthandbiometric dataare sent. If this content were shared without E2EE, a malicious actor or adversarial government could obtain it throughunauthorized accessorsubpoenastargeted at the service provider.[4]
E2EE alone does not guaranteeprivacyorsecurity.[8]For example, data may be held unencryptedon the user's own device, or be accessible via their own app, if their login is compromised.
The term "end-to-end encryption" originally only meant that the communication is never decrypted during its transport from the sender to the receiver.[9]For example, around 2003, E2EE has been proposed as an additional layer of encryption forGSM[10]orTETRA,[11]in addition to the existing radio encryption protecting the communication between the mobile device and the network infrastructure. This has been standardized by SFPG for TETRA.[12]Note that in TETRA E2EE, the keys are generated by a Key Management Centre (KMC) or a Key Management Facility (KMF), not by the communicating users.[13]
Later, around 2014, the meaning of "end-to-end encryption" started to evolve when WhatsApp encrypted a portion of its network,[14]requiring that not only the communication stays encrypted during transport,[15]but also that the provider of the communication service is not able to decrypt the communications either by having access to the private key, or by having the capability to undetectably inject an adversarial public key as part of aman-in-the-middle attack.[citation needed]This new meaning is now the widely accepted one.[16]
As of 2016,[17]typicalserver-based communications systems do not include end-to-end encryption.[18]These systems can only guarantee the protection of communications betweenclientsandservers,[19]meaning that users have to trust the third parties who are running the servers with the sensitive content. End-to-end encryption is regarded as safer[20]because it reduces the number of parties who might be able to interfere or break the encryption.[21]In the case of instant messaging, users may use a third-party client or plugin to implement an end-to-end encryption scheme over an otherwise non-E2EE protocol.[22]
Some non-E2EE systems, such asLavabitandHushmail, have described themselves as offering "end-to-end" encryption when they did not.[23]Other systems, such asTelegramandGoogle Allo, have been criticized for not enabling end-to-end encryption by default. Telegram did not enable end-to-end encryption by default on VoIP calls while users were using desktop software version, but that problem was fixed quickly.[24][25]However, as of 2020, Telegram still features no end-to-end encryption by default, no end-to-end encryption for group chats, and no end-to-end encryption for its desktop clients. In 2022,Facebook Messengercame under scrutiny because the messages between a mother and daughter inNebraskawere used to seek criminal charges in anabortion-related case against both of them. The daughter told the police that she had a miscarriage and tried to search for the date of her miscarriage in her Messenger app. Police suspected there could be more information within the messages and obtained and served a warrant against Facebook to gain access. The messages allegedly mentioned the mother obtainingabortion pillsfor her daughter and then burning the evidence. Facebook expanded default end-to-end encryption in the Messenger app just days later.[26][27]Writing forWired, Albert Fox Cahn criticized Messenger's approach to end-to-end encryption, which was not enabled by default, required opt-in for each conversation, and split the message thread into two chats which were easy for the user to confuse.[28]
Some encryptedbackupandfile sharingservices provideclient-side encryption. This type of encryption is not referred to as end-to-end encryption because only one end has the ability to decrypt the data. However, the term "end-to-end encryption" is sometimes incorrectly used to describe client-side encryption.[29]
End-to-end encryption ensures that data is transferred securely between endpoints. But, rather than try to break the encryption, an eavesdropper may impersonate a message recipient (duringkey exchangeor by substituting theirpublic keyfor the recipient's), so that messages are encrypted with a key known to the attacker. After decrypting the message, the snoop can then encrypt it with a key that they share with the actual recipient, or their public key in case of asymmetric systems, and send the message on again to avoid detection. This is known as aman-in-the-middle attack(MITM).[1][30]
Most end-to-end encryption protocols include some form of endpointauthenticationspecifically to prevent MITM attacks. For example, one could rely oncertification authoritiesor aweb of trust.[31]An alternative technique is to generate cryptographic hashes (fingerprints) based on the communicating users’ public keys or shared secret keys. The parties compare theirfingerprintsusing an outside (out-of-band) communication channel that guarantees integrity and authenticity of communication (but not necessarily secrecy[citation needed]), before starting their conversation. If the fingerprints match, there is, in theory, no man in the middle.[1]
When displayed for human inspection, fingerprints usually use some form ofbinary-to-text encoding[citation needed].[32]These strings are then formatted into groups of characters for readability. Some clients instead display anatural languagerepresentation of the fingerprint.[33]As the approach consists of aone-to-one mappingbetween fingerprint blocks and words, there is no loss inentropy. The protocol may choose to display words in the user's native (system) language.[33]This can, however, make cross-language comparisons prone to errors.[34]
In order to improvelocalization, some protocols have chosen to display fingerprints as base 10 strings instead of more error prone hexadecimal or natural language strings.[35][34]An example of the base 10 fingerprint (calledsafety numberin Signal andsecurity codein WhatsApp) would be:
Other applications such as Telegram, instead, encode fingerprints using emojis.
Modern messaging applications can also display fingerprints asQR codesthat users can scan off each other's devices.[35]
The end-to-end encryption paradigm does not directly address risks at the communications endpoints themselves. Each user's computer can still be hacked to steal their cryptographic key (to create a MITM attack) or simply read the recipients’ decrypted messages both in real time and from log files. Even the most perfectly encrypted communication pipe is only as secure as the mailbox on the other end.[1]Major attempts to increase endpoint security have been to isolate key generation, storage and cryptographic operations to a smart card such as Google's Project Vault.[36]However, since plaintext input and output are still visible to the host system, malware can monitor conversations in real time. A more robust approach is to isolate all sensitive data to a fullyair gappedcomputer.[37]PGPhas been recommended by experts for this purpose.[38]However, asBruce Schneierpoints out,Stuxnetdeveloped by US and Israel successfully jumped air gap and reached Natanz nuclear plant's network in Iran.[39]To deal with key exfiltration with malware, one approach is to split theTrusted Computing Basebehind twounidirectionally connectedcomputers that prevent either insertion of malware, or exfiltration of sensitive data with inserted malware.[40]
A backdoor is usually a secret method of bypassing normal authentication or encryption in a computer system, a product, an embedded device, etc.[41]Companies may also willingly or unwillingly introducebackdoorsto their software that help subvert key negotiation or bypass encryption altogether. In 2013, information leaked byEdward Snowdenshowed thatSkypehad a backdoor which allowed Microsoft to hand over their users' messages to theNSAdespite the fact that those messages were officially end-to-end encrypted.[42][43]
Following terrorist attacks inSan Bernardino in 2015andPensacola in 2019, theFBIrequested backdoors toApple'siPhonesoftware. The company, however, refused to create a backdoor for the government, citing concern that such a tool could pose risk for its consumers’ privacy.[44]
While E2EE can offer privacy benefits that make it desirable in consumer-grade services, many businesses have to balance these benefits with their regulatory requirements. For example, many organizations are subject to mandates that require them to be able to decrypt any communication between their employees or between their employees and third parties.[45]This might be needed for archival purposes, for inspection byData Loss Prevention (DLP)systems, for litigation-relatedeDiscoveryor for detection ofmalwareand other threats in the data streams. For this reason, some enterprise-focused communications and information protection systems might implement encryption in a way that ensures all transmissions are encrypted with the encryption being terminated at their internal systems (on-premises or cloud-based) so they can have access to the information for inspection and processing.
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Computer science(also called computing science) is the study of the theoretical foundations ofinformationandcomputationand their implementation and application incomputersystems. One well known subject classification system for computer science is theACM Computing Classification Systemdevised by theAssociation for Computing Machinery.
Computer science can be described as all of the following:
Outline of artificial intelligence
Outline of databases
Outline of software engineering
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Arc routingproblems(ARP)are a category of general routing problems (GRP), which also includes node routing problems (NRP). The objective in ARPs and NRPs is to traverse the edges and nodes of a graph, respectively.[1]The objective of arc routing problems involves minimizing the total distance and time, which often involves minimizingdeadheadingtime, the time it takes to reach a destination. Arc routing problems can be applied togarbage collection,school busroute planning, package and newspaper delivery,deicingandsnow removalwithwinter service vehiclesthat sprinklesalton the road,[2]mail delivery, network maintenance,street sweeping, police and security guard patrolling,[1]andsnow ploughing.[3][4]Arc routings problems areNP hard, as opposed toroute inspection problemsthat can be solved inpolynomial-time.
For a real-world example of arc routing problem solving, Cristina R. Delgado Serna & Joaquín Pacheco Bonrostro applied approximation algorithms to find the best school bus routes in the Spanish province ofBurgossecondary school system. The researchers minimized the number of routes that took longer than 60 minutes to traverse first. They also minimized the duration of the longest route with a fixed maximum number of vehicles.[5]
There are generalizations of arc routing problems that introduce multiple mailmen, for example the k Chinese Postman Problem (KCPP).
The efficient scheduling and routing of vehicles can save industry and government millions of dollars every year.[2][6]Arc routing problems have applications in school bus planning, garbage and waste and refuse collection in cities, mail and package delivery by mailmen and postal services, winter gritting and laying down salt to keep roads safe in the winter, snow plowing and removal, meter reading including remote radio frequency identification meter reading technology, street maintenance and sweeping, police patrol car route planning, and more.
The basic routing problem is: given a set of nodes and/or arcs to be serviced by a fleet of vehicles, find routes for each vehicle starting and ending at a depot. A vehicle route is a sequence of points or nodes, which the vehicle must traverse in order, starting and ending at a depot.[2]
The Chinese Postman Problem (CPP) is aimed at finding the minimum length cycle for a single postman. The CPP requires all edges be traversed once, the rural postman problem (RPP) requires a subset of the edges to be traversed with the minimum length cycle.[1]
Arc routing problems impact strategic, tactical, and operational planning decisions. The strategic role of where a depot is placed depends on the most efficient arc route available. The decision of the vehicle fleet size and vehicle types with varying specifications relate to the tactical aspect of arc routing problems in operations research. Routing and scheduling decisions are operational planning decisions in arc routing problems. The operational planning decisions also includes the time that the vehicles are used by workers with staff decisions.[2]Vehicle routing decisions for the location of a depot depend on the cost of transporting materials over a geographical region. Bodin et. al applied vehicle routing to the dial a ride problem.[7]
In some situations, the set of edges that are required is different from the edges in the graph. This is modeled by the Rural Postman Problem (RPP),[1]where the required edges are a subset of the system of edges.
Finding an efficient solution with large amounts data to the Chinese Postman Problem (CPP), the Windy Postman Problem (WPP), the Rural Postman Problem (RPP), thek-Chinese postman problem (KCPP), themixed Chinese postman problem(MCPP), the Directed Chinese Postman Problem (DCPP),[8]the Downhill Plowing Problem (DPP), the Plowing with Precedence Problem (PPP), the Windy Rural Postman Problem (WRPP) and the Windy General Routing Problem (WGRP) requires using thoughtful mathematical concepts, includingheuristic optimization methods,branch-and-bound methods,integer linear programming, and applications oftraveling salesman problemalgorithms such as theHeld–Karp algorithmmakes an improvement fromO(n!){\displaystyle O(n!)}toO(2nn2){\displaystyle O(2^{n}n^{2})}.[9]In addition to these algorithms, these classes of problems can also be solved with thecutting plane algorithm,convex optimization,convex hulls,Lagrange multipliersand otherdynamic programming methods. In cases where it is not feasible to run the Held–Karp algorithm because of its high computational complexity, algorithms like this can be used to approximate the solution in a reasonable amount of time.[10]
The earliest documented reference to the area of arc routing problems is the classicbridges of Königsbergchallenge, whichEulerproved to be impossible.[4]The resident ofKonigsberg, now part ofKaliningrad, wanted to find a way to cross all seven bridges over the riverPregelwithout backtracking or retracing their steps, that is crossing each bridge once and only once. In 1736, Euler reduced the problem to a question of nodes and edges and showed that the problem was impossible. In 1873, Hierholzer did more work on the question of closed circuits.[4]
The work on the Eulerian circuits was popularized with Scientific American on July 1, 1953.[11]This work was extended by Meigu Guan, also known as Kwan Mei-Ko at Shangtun Normal College. Meigu Guan was interested in a different question instead of determining a closed circuit. Guan worked to find out a minimum length walk that traversed every edge of the graph at least once. Guan described his goal in 1962: "A mailman has to cover his assigned segment before returning to the post office. The problem is to find the shortest walking distance for the mailman."[4]
Arc routing problems (ARPs) differ in their goal and heuristics. However, all of them are known to beNP-hard.
This problem is named after the postman and his challenge to deliver mail in any order he may choose, but minimizing his costs such as time or travel distance. It is also sometimes called theundirected chinese postman problem. The undirected rural postman problem (URPP) aims to minimize the total cost of a route that maps the entire network, or in more specific cases, a route that maps every edge that requires a service. If the whole network must be mapped, the route that maps the entire network is called acovering tour. In the case where only certain edges need to be mapped, the problem aims to solve the route that optimizes the demands, crossing over into non-required routes a minimal number of times.[12]
The undirected capacitated arc routing problem consists of demands placed on the edges, and each edge must meet the demand. An example is garbage collection, where each route might require both a garbage collection and a recyclable collection. Problems in real life applications might arise if there are timing issues, such as the case in which certain routes cannot be serviced due to timing or scheduling conflicts, or constraints, such as a limited period of time. The heuristics described in this article ignore any such problems that arise due to application constraints.[12]
The URPP was first introduced in 1974 and was proven to be an NP-hard problem byLenstraandKan. The UCARP can be derived from the URPP, and thus is NP-hard as well. In 1981, another pair of computer scientists, Golden and Wong, managed to prove that even deriving a .5 approximation to the URPP was NP-hard. In 2000, Dror published a book describing different arc routing problems.
Thewindy postman problemproposed by Minieka is a variant of the route inspection problem in which the input is an undirected graph, but where each edge may have a different cost for traversing it in one direction than for traversing it in the other direction.[13]In contrast to the solutions for directed and undirected graphs, it isNP-complete.[14][15]The cost of traveling in one direction is greater when the wind is blowing in your face than when the wind is at your back, and this is the origin of the name Windy Postman problem. The work that it takes to traverse the street in one direction is different than the work it takes to traverse the street in another direction on a windy day.[8]
The windy postman problem is an arc routing problem (ARP) that contains the Mixed Chinese Postman Problem MCPP as a special case.[16]
The problem can be defined in the following manner: "Given an undirected and connected graph G=(V,E) with two non-negative costsci,j{\displaystyle c_{i,j}}andcj,i{\displaystyle c_{j,i}}associated with each edge{i,j}∈E{\displaystyle \{i,j\}\in E}corresponding to the cost of traversing it from i to j and from j to i, respectively, the WPP is to find a minimum cost tour on G traversing each edge at least once."[16]This problem was introduced by Minieka. The WPP is NP-complete in general and can be solved in polynomial time if G is Eulerian, if the cost of two opposite orientations of every cycle in G in same or if G is a series-parallel graph. The Windy Rural Postman Problem (WRPP) is a generalization of the WPP in which not all the edges in the graph have to be traversed but only those in a given subset of required edges. For example, some rural roads are not required for the postman to cross and some roads on steep hills take longer to go up than down.[10]
The Windy Rural Postman Problem (WRPP) is a generalization of the WPP in which not all the edges in the graph have to be traversed but only those in a given subset of required edges. For example, some rural roads are not required for the postman to cross and some roads on steep hills take longer to go up than down.[10]Consider an undirected graphG={E,V}{\displaystyle G=\{E,V\}}with two costscij{\displaystyle c_{ij}}andcji{\displaystyle c_{ji}}associated with the cost to traverse the edge(i,j){\displaystyle (i,j)}starting from i and j, respectively. G is the windy graph and we are interested in the subset of edges, or in mathematical symbols,ER⊆E{\displaystyle E_{R}\subseteq E}.
If the WRPP includes the additional constraint that a certain set of vertices must be visited—VR⊆V{\displaystyle V_{R}\subseteq V}, the problem turns into the Windy General Routing Problem (WGRP). Benavent proposed an integer linear programming formulation and different heuristics and lower bounds for the WRPP.[9]
Benavent et al published an evaluation of several heuristic methods used for solving the WRPP in a few seconds with a deviation no greater than 1% from the lower bound on medium sized graphs. They improved on this with a Scatter Search algorithm that reduced the difference to 0.5%. Scatter Search found solutions that deviated by less than 2% when implemented on networks with hundreds of nodes and thousands of edges.[9]
In real world applications, there are multiple vehicles that can move, which leads to the generalization named the Min-Max K-vehicles Windy Rural Postman Problem (MM K-WRPP). The min–max K-vehicles Windy Rural Postman Problem (MM K-WRPP) is defined as follows: Given a windy graphG={V,E}{\displaystyle G=\{V,E\}}, a distinguished vertex,1∈V{\displaystyle 1\in V}, representing the depot, a subset of required edgesER⊆E{\displaystyle E_{R}\subseteq E}, and a fixed number K of vehicles, the MM K-WRPP consists of finding a set of K tours for the vehicles in such a way that each tour starts and ends at the depot and each required edge is serviced by exactly one vehicle. The objective is to minimize the length of the longest tour in order to find a set of balanced routes for the vehicles. Some real-life applications of routing problems with min–max objectives are school bus routing (Delgado and Pacheco 2001), the delivery of newspapers to customers (Applegate et al. 2002) and waste collection (Lacomme et al. 2004).[10]
The best MM K_WRPP algorithm was very close to the minimum solution with 2 and 3 vehicles, less than 0.4% on average. The gap increases to about 1.00% and 1.60% at 4 and 5 vehicles.
According to Dussault et al and Benavent et al, a metaheuristics multi-objective simulating annealing algorithm (MOSA) can solve the different contraints imposed on the WRPP. The WRPP is an important Arc Routing Problem which generalizes many of the single-vehicles Arc Routing problems. In real applications of math, a solution that minimizes the total costs of all vehicles route and the length of the longest tour is preferable. It's hard to be in a location where your package is always hours late.[8]We should start with the assumption that several vehicles with a specific measurable capacity to serve customers is more realistic than one vehicle with unmeasurable infinite capacity. Rabbani et. al measured the performance of MOSA algorithms and models using a multi-objective development of Cuckoo search—developed by Yang et al,[17]also referred to as Multi-objective Cuckoo Search and abbreviated by MOCS.[8]They concluded that MOSA methods were more efficient than MOCS methods. In the future comparisons with other meta-heuristic methods could be researched, including Non-dominated Sorting Genetic Algorithm (NSGA- ), multi-objective particle swarm optimization algorithm (MOPSO) and multi-objective Imperialist Competitive Algorithm.
In the Windy Postman Problem (WPP) model, the cost of going in one direction is different than the cost it takes to go in the other direction. For example, if the wind is blowing down the street it takes more time and energy to go against the wind than with the wind. Another example of the WPP is the cost of plowing uphill is greater than the cost of plowing downhill.[3]This is modeled by a variant studied by Dussault et al, the Downhill Plowing Problem (DPP).[3]
A branch and cut algorithm was published by Angel Corberan for the windy postman problem. The algorithm is based on heuristic and exact methods for manipulating odd-cut inequality violations.[16]
Various combinatorial problems have been reduced to the Chinese Postman Problem, including finding a maximum cut in a planar graph and a minimum-mean length circuit in an undirected graph.[18]
In winter a common question is what set of routes has the smallest (minimum) maximum route length? Typically, this is assessed as an arc routing problem with a graph. The time it takes to travel a street, known as deadhead time, is faster than the time it takes to plow the snow from the streets (or deliver mail or drop off packages). Another aspect that must be considered when applying arc routing to snow plowing is the fact that on steep streets it is either difficult or impossible to plow uphill. The objective is a route that avoids plowing uphill on steep streets that completes the job faster by maximizing the deadhead time to get the location. This was modeled with a heuristic algorithm that approximates a lower bound by Dussault, Golden and Wasil.[3]This is the Downhill Plow Problem (DPP). Snow teams prefer to plow downhill and deadhill uphill. This problem assumes that the conditions are severe enough that the streets are closed and there is no traffic.
The Downhill Plowing Problem ignores the Plowing with Precedence Problem (PPP), which is built on the reasonable assumption that if the snow is too deep the snow plow cannot deadhead an unplowed street. The DPP makes the assumption that the snow level is low enough that the streets that are not plowed can be deadheaded, but that the snow is deep enough that there is no traffic. If there is traffic on the roads, the assumption that it is impossible to plow uphill can no longer be held. The simulation for the DPP deadheaded unplowed street about 5% of the time, which is a topic for future graph theory and arc routing research.
Considering an undirected graphG={V,A}{\displaystyle G=\{V,A\}}whereV{\displaystyle V}is the set of vertices and nodes andA{\displaystyle A}is the set of arcs. Each arc represented by(vi,vj){\displaystyle (v_{i},v_{j})}has four costs:cij+{\displaystyle c_{ij}^{+}}, defined as the cost of plowing fromvi{\displaystyle v_{i}}tovj{\displaystyle v_{j}},cji+{\displaystyle c_{ji}^{+}}, the cost of plowing fromvj{\displaystyle v_{j}}tovi{\displaystyle v_{i}},cij−{\displaystyle c_{ij}^{-}}, the cost of deadheading fromvi{\displaystyle v_{i}}tovj{\displaystyle v_{j}}, andcji−{\displaystyle c_{ji}^{-}}, the cost of deadheading fromvj{\displaystyle v_{j}}tovi{\displaystyle v_{i}}. The setup assumes thatvj{\displaystyle v_{j}}has a higher elevationvi{\displaystyle v_{i}}, which leads to the statement:cij+≫cji+≫cij−≥cji−{\displaystyle c_{ij}^{+}\gg c_{ji}^{+}\gg c_{ij}^{-}\geq c_{ji}^{-}}. In practice, downhill plowing time is two times as efficient as uphill plowing and deadheading is twice as efficient as plowing. The algorithm findsk{\displaystyle k}routes will each begin and end at the depotv0{\displaystyle v_{0}}, plow the arc two times because the left side and right side of the street take two passes to plow.
The best solution will minimize the maximum route length. Dussault, Golden, and Wasil found an algorithm that did not exceed the lower bound by 5.5% in over 80 test runs. The deviation increased as the complexity of the model increased because there are more unoptimized approximations than optimized approximation as the model grows. An improvement on Dussault et. al's DPP algorithm might have penalties for making U-turns and left hand turns, or going straight across an intersection, which take additional time and pushes snow into the middle of the intersection, respectively. (see The Directed Rural Postman Problem with Turn Penalties problem, often referred to as the DRPP-TP below).
Thek-Chinese Postman can be stated as follows: "given a connected edge-weighted graphGand integerspandk, decide whether there are at leastkclosed walks such that every edge ofGis contained in at least one of them and the total weight of the edges in the walks is at mostp?" The process of obtaining the solution to thek-CPP is NP complete. Gutin, Muciaccia, and Yeo proved in 2013 that thek-CPP is fixed-parameter tractable.[19]The authors prove thek-CPP admits a kernel withO(k2log(k)){\displaystyle O(k^{2}\log(k))}vertices and the directed version of thek-CPP is NP complete.
The rural postman problem (RPP) makes some routes mandatory and absolute but the person traversing the graph does not have to go in one particular direction. The RPP is NP hard and complete, in the same way that the kCPP, the DPP, the PPP, are NP hard. Benevant studied a generalization of this named Directed Rural Postman Problem with Turn Penalties (DRPP-TP).[20]Benevant's algorithm approximated the solution by transforming the DRPP-TP into an asymmetrical traveling salesman problem (ATSP).
Most algorithms require a pre-processing of the graph, which simplifies the initial graph by removing all edges that are not in the shortest path between two required edges. Another simplification that the pre-processing adds is that it transforms the shortest path between 2 required edges into a single, non-required edge, regardless of the number of edges in the path, provided that there were no required edges in the path.
Once the pre-processing is done, the problem can be generalized into aconvex hullproblem, with the edges being the points of the hull. The convex hull problem can be solved through linear programming or through convex hull algorithms, but the process of finding the convex hull is an exponential problem.
Methods of solving the URPP after the pre-processing is done consist of thecutting plane algorithmand thebranch & cut methodology.[21]
This is a list of computational complexities for different arc routing problems.
O((|E|−|V|)|V|2){\displaystyle O((|E|-|V|)|V|^{2})}-time algorithm[23]
O(|V|3){\displaystyle O(|V|^{3})}-time solvable if each vertex has even degree[22]
In FPT with respect to |A|[26]
In XP with respect to treewidth[27]
P in some special cases[28][29]
O(k|V|4){\displaystyle O(k|V|^{4})}-time solvable if precedence relation linear
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The game ofGois one of the most popular games in the world. As a result of its elegant and simple rules, the game has long been an inspiration formathematicalresearch.Shen Kuo, an 11th century Chinese scholar, estimated in hisDream Pool Essaysthat the number of possible board positions is around 10172. In more recent years, research of the game byJohn H. Conwayled to the development of thesurreal numbersand contributed to development ofcombinatorial game theory(with Go Infinitesimals[1]being a specific example of its use in Go).
Generalized Go is played onn×nboards, and thecomputational complexityof determining the winner in a given position of generalized Go depends crucially on theko rules.
Go is “almost” inPSPACE, since in normal play, moves are not reversible, and it is only through capture that there is the possibility of the repeating patterns necessary for a harder complexity.
Without ko, Go isPSPACE-hard.[2]This is proved by reducingTrue Quantified Boolean Formula, which is known to be PSPACE-complete, togeneralized geography, to planar generalized geography, toplanar generalized geography with maximum degree 3, finally to Go positions.
Go with superko is not known to be in PSPACE. Though actual games seem never to last longer thann2{\displaystyle n^{2}}moves, in general it is not known if there were a polynomial bound on the length of Go games. If there were, Go would be PSPACE-complete. As it currently stands, it might be PSPACE-complete, EXPTIME-complete, or even EXPSPACE-complete.
Japanese ko rules state that only the basic ko, that is, a move that reverts the board to the situation one move previously, is forbidden. Longer repetitive situations are allowed, thus potentially allowing a game to loop forever, such as the triple ko, where there are three kos at the same time, allowing a cycle of 12 moves.
With Japanese ko rules, Go isEXPTIME-complete.[3]
Thesuperko rule(also called the positional superko rule) states that a repetition of any board position that has previously occurred is forbidden. This is the ko rule used in most Chinese and US rulesets.
It is an open problem what the complexity class of Go is under superko rule. Though Go with Japanese ko rule is EXPTIME-complete, both the lower and the upper bounds of Robson’s EXPTIME-completeness proof[3]break when the superko rule is added.
It is known that it is at least PSPACE-hard, since the proof in[2]of the PSPACE-hardness of Go does not rely on the ko rule, or lack of the ko rule. It is also known that Go is in EXPSPACE.[4]
Robson[4]showed that if the superko rule, that is, “no previous position may ever be recreated”, is added to certain two-player games that are EXPTIME-complete, then the new games would be EXPSPACE-complete. Intuitively, this is because an exponential amount of space is required even to determine the legal moves from a position, because the game history leading up to a position could be exponentially long.
As a result, superko variants (moves that repeat a previous board position are not allowed) of generalizedchessandcheckersare EXPSPACE-complete, since generalized chess[5]and checkers[6]are EXPTIME-complete. However, this result does not apply to Go.[4]
A Go endgame begins when the board is divided into areas that are isolated from all other local areas by living stones, such that each local area has a polynomial size canonical game tree. In the language ofcombinatorial game theory, it happens when a Go game decomposes into a sum of subgames with polynomial size canonical game trees.
With that definition, Go endgames are PSPACE-hard.[7]
This is proven by converting theQuantified Boolean Formulaproblem, which is PSPACE-complete, into a sum of small (with polynomial size canonical game trees) Go subgames. Note that the paper does not prove that Go endgames are in PSPACE, so they might not be PSPACE-complete.
Determining which side wins aladdercapturing race is PSPACE-complete, whether Japanese ko rule or superko rule is in place.[8]This is proven by simulating QBF, known to be PSPACE-complete, with ladders that bounce around the board like light beams.
Since each location on the board can be either empty, black, or white, there are a total of 3n2possible board positions on a square board with lengthn; however not all of them are legal.Trompand Farnebäck derived a recursive formula for legal positionsL(m,n){\displaystyle L(m,n)}of a rectangle board with lengthmandn.[9]The exact number ofL(19,19){\displaystyle L(19,19)}was obtained in 2016.[10]They also find an asymptotic formulaL≈ABm+nCmn{\displaystyle L\approx AB^{m+n}C^{mn}}, whereA≈0.8506399258457145{\displaystyle A\approx 0.8506399258457145},B≈0.96553505933837387{\displaystyle B\approx 0.96553505933837387}andC≈2.975734192043357249381{\displaystyle C\approx 2.975734192043357249381}. It has been estimated that the observable universe contains around 1080atoms, far fewer than the number of possible legal positions of regular board size (m=n=19). As the board gets larger, the percentage of the positions that are legal decreases.
Thecomputer scientistVictor Allisnotes that typical games between experts last about 150 moves, with an average of about 250 choices per move, suggesting agame-tree complexityof 10360.[12]For the number oftheoretically possiblegames, including games impossible to play in practice, Tromp and Farnebäck give lower and upper bounds of 101048and 1010171respectively.[9]The lower bound was improved to agoogolplexby Walraet and Tromp.[13]The most commonly quoted number for the number of possible games, 10700[14]is derived from a simple permutation of 361 moves or361! = 10768. Another common derivation is to assumeNintersections andLlongest game forNLtotal games. For example, 400 moves, as seen in some professional games, would be one out of 361400or 1 × 101023possible games.
The total number of possible games is a function both of the size of the board and the number of moves played. While most games last less than 400 or even 200 moves, many more are possible.
The total number of possible games can be estimated from the board size in a number of ways, some more rigorous than others. The simplest, a permutation of the board size, (N)L, fails to include illegal captures and positions. TakingNas the board size (19 × 19 = 361) andLas the longest game,NLforms an upper limit. A more accurate limit is presented in the Tromp/Farnebäck paper.
10700is thus an overestimate of the number of possible games that can be played in 200 moves and an underestimate of the number of games that can be played in 361 moves. Since there are about 31 million seconds in a year, it would take about2+1⁄4years, playing 16 hours a day at one move per second, to play 47 million moves.
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Asolved gameis agamewhose outcome (win, lose ordraw) can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied toabstract strategy games, and especially to games with full information and no element of chance; solving such a game may usecombinatorial game theoryor computer assistance.
Atwo-player gamecan be solved on several levels:[1][2]
Despite their name, many game theorists believe that "ultra-weak" proofs are the deepest, most interesting and valuable. "Ultra-weak" proofs require a scholar to reason about the abstract properties of the game, and show how these properties lead to certain outcomes if perfect play is realized.[citation needed]
By contrast, "strong" proofs often proceed bybrute force—using a computerto exhaustively search agame treeto figure out what would happen if perfect play were realized. The resulting proof gives an optimal strategy for every possible position on the board. However, these proofs are not as helpful in understanding deeper reasons why some games are solvable as a draw, and other, seemingly very similar games are solvable as a win.
Given the rules of any two-person game with a finite number of positions, one can always trivially construct aminimaxalgorithm that would exhaustively traverse the game tree. However, since for many non-trivial games such an algorithm would require an infeasible amount of time to generate a move in a given position, a game is not considered to be solved weakly or strongly unless the algorithm can be run by existing hardware in a reasonable time. Many algorithms rely on a huge pre-generated database and are effectively nothing more.
As a simple example of a strong solution, the game oftic-tac-toeis easily solvable as a draw for both players with perfect play (a result manually determinable). Games likenimalso admit a rigorous analysis usingcombinatorial game theory.
Whether a game is solved is not necessarily the same as whether it remains interesting for humans to play. Even a strongly solved game can still be interesting if its solution is too complex to be memorized; conversely, a weakly solved game may lose its attraction if the winning strategy is simple enough to remember (e.g.,Maharajah and the Sepoys). An ultra-weak solution (e.g.,ChomporHexon a sufficiently large board) generally does not affect playability.
Ingame theory,perfect playis the behavior or strategy of a player that leads to the best possible outcome for that player regardless of the response by the opponent. Perfect play for a game is known when the game is solved.[1]Based on the rules of a game, every possible final position can be evaluated (as a win, loss or draw). Bybackward reasoning, one can recursively evaluate a non-final position as identical to the position that is one move away and best valued for the player whose move it is. Thus a transition between positions can never result in a better evaluation for the moving player, and a perfect move in a position would be a transition between positions that are equally evaluated. As an example, a perfect player in a drawn position would always get a draw or win, never a loss. If there are multiple options with the same outcome, perfect play is sometimes considered the fastest method leading to a good result, or the slowest method leading to a bad result.
Perfect play can be generalized to non-perfect informationgames, as the strategy that would guarantee the highest minimalexpected outcomeregardless of the strategy of the opponent. As an example, the perfect strategy forrock paper scissorswould be to randomly choose each of the options with equal (1/3) probability. The disadvantage in this example is that this strategy will never exploit non-optimal strategies of the opponent, so the expected outcome of this strategy versus any strategy will always be equal to the minimal expected outcome.
Although the optimal strategy of a game may not (yet) be known, a game-playing computer might still benefit from solutions of the game from certainendgamepositions (in the form ofendgame tablebases), which will allow it to play perfectly after some point in the game.Computer chessprograms are well known for doing this.
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Solving chessconsists of finding an optimal strategy for the game ofchess; that is, one by which one of the players (White or Black) can always force either a victory or a draw (seesolved game). It is also related to more generally solvingchess-likegames (i.e.combinatorial gamesofperfect information) such asCapablanca chessandinfinite chess. In a weaker sense,solving chessmay refer to proving which one of the three possible outcomes (White wins; Black wins; draw) is the result of two perfect players, without necessarily revealing the optimal strategy itself (seeindirect proof).[1]
No complete solution for chess in either of the two sensesis known, nor is it expected that chess will be solved in the near future (if ever). Progress to date is extremely limited; there aretablebasesof perfect endgame play with a small number of pieces (up to seven), and somechess variantshave been solved at least weakly. Calculated estimates ofgame-tree complexityand state-space complexity of chess exist which provide a bird's eye view of the computational effort that might be required to solve the game.
Endgame tablebasesare computerized databases that contain precalculated exhaustive analyses of positions with small numbers of pieces remaining on the board. Tablebases have solved chess to a limited degree, determining perfect play in a number ofendgames, including all non-trivial endgames with no more than seven pieces or pawns (including the two kings).[2]
One consequence of developing the seven-piece endgame tablebase is that many interesting theoretical chess endings have been found. The longest seven-piece example is a mate-in-549 position discovered in the Lomonosov tablebase by Guy Haworth, ignoring the50-move rule.[3][4]Such a position is beyond the ability of any human to solve, and no chess engine plays it correctly, either, without access to the tablebase, which initially (in 2014) required 140 TB of storage space and the use of a supercomputer but was later reduced down to 18.4 TB through the Syzygy tablebase. As of January 2023, the longest known forced mating sequence for the eight-piece tablebase (also ignoring the 50-move rule) was 584 moves. This was discovered in mid-2022 by Marc Bourzutschky.[5]The eight-piece tablebase is currently incomplete, though, so it is not guaranteed that this is the absolute limit for the eight-piece tablebase.
A variant first described byClaude Shannonprovides an argument about the game-theoretic value of chess: he proposes allowing the move of “pass”. In this variant, it is provable with astrategy stealing argumentthat the first player has at least a draw thus: if the first player has a winning move in the initial position, let him play it, else pass. The second player now faces the same situation owing to the mirror symmetry of the initial position: if the first player had no winning move in the first instance, the second player has none now. Therefore, the second player can at best draw, and the first player can at least draw, so a perfect game results in the first player winning or drawing.[6]
Somechess variantswhich are simpler than chess have been solved. A winning strategy for Black inMaharajah and the Sepoyscan be easily memorised. The 5×5Gardner's Minichessvariant has beenweakly solvedas a draw.[7]Althoughlosing chessis played on an 8×8 board, its forced capture rule greatly limits its complexity, and a computational analysis managed to weakly solve this variant as a win for White.[8]
The prospect of solving individual, specific, chess-like games becomes more difficult as the board-size is increased, such as in large chess variants, andinfinite chess.[9]
Information theoristClaude Shannonin 1950 outlined a theoretical procedure for playing a perfect game (i.e. solving chess):
"With chess it is possible, in principle, to play a perfect game or construct a machine to do so as follows: One considers in a given position all possible moves, then all moves for the opponent, etc., to the end of the game (in each variation). The end must occur, by the rules of the games after a finite number of moves (remembering the50 move drawing rule). Each of these variations ends in win, loss or draw. By working backward from the end one can determine whether there is a forced win, the position is a draw or is lost."
Shannon then went on to estimate that solving chess according to that procedure would require comparing some 10120(Shannon number) possible game variations, or having a "dictionary" denoting an optimal move for each of the approximately 1043possible board positions (currently known to be about 5x1044).[6][10]The number of mathematical operations required to solve chess, however, may be significantly different than the number of operations required to produce the entiregame-treeof chess. In particular, if White has a forced win, only a subset of the game-tree would require evaluation to confirm that a forced-win exists (i.e. with no refutations from Black). Furthermore, Shannon's calculation for the complexity of chess assumes an average game length of 40 moves, but there is no mathematical basis to say that a forced win by either side would have any relation to this game length. Indeed, some expertly played games (grandmaster-level play) have been as short as 16 moves. For these reasons, mathematicians and game theorists have been reluctant to categorically state that solving chess is an intractable problem.[6][11]
In 1950, Shannon calculated, based on a game tree complexity of 10120and a computer operating at one megahertz (a big stretch at that time: the UNIVAC 1 introduced in 1951 could perform ~2000 operations per second or 2 kilohertz) that could evaluate a terminal node in 1 microsecond would take 1090years to make its first move. Even allowing for technological advances, solving chess within a practical time frame would therefore seem beyond any conceivable technology.
Hans-Joachim Bremermann, a professor ofmathematicsandbiophysicsat theUniversity of California at Berkeley, further argued in a 1965 paper that the "speed, memory, and processing capacity of any possible future computer equipment are limited by specific physical barriers: thelight barrier, thequantum barrier, and thethermodynamical barrier. These limitations imply, for example, that no computer, however constructed, will ever be able to examine the entire tree of possible move sequences of the game of chess." Nonetheless, Bremermann did not foreclose the possibility that a computer would someday be able to solve chess. He wrote, "In order to have a computer play a perfect or nearly perfect game, it will be necessary either to analyze the game completely ... or to analyze the game in an approximate way and combine this with a limited amount of tree searching. ... A theoretical understanding of such heuristic programming, however, is still very much wanting."[12]
Recent scientific advances have not significantly changed these assessments. The game ofcheckerswas (weakly) solved in 2007,[13]but it has roughly the square root of the number of positions in chess.Jonathan Schaeffer, the scientist who led the effort, said a breakthrough such asquantum computingwould be needed before solving chess could even be attempted, but he does not rule out the possibility, saying that the one thing he learned from his 16-year effort of solving checkers "is to never underestimate the advances in technology".[14]
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TheShannon number, named after the American mathematicianClaude Shannon, is a conservative lower bound of thegame-tree complexityofchessof 10120, based on an average of about 103possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves.
Shannon showed a calculation for the lower bound of the game-tree complexity of chess, resulting in about 10120possible games, to demonstrate the impracticality ofsolving chessbybrute force, in his 1950 paper "Programming a Computer for Playing Chess".[1](This influential paper introduced the field ofcomputer chess.)
Shannon also estimated the number of possible positions, of the general order of63!32!8!2{\displaystyle {\frac {63!}{32!{8!}^{2}}}}, or roughly3.7∗1043{\displaystyle 3.7*10^{43}}. This includes some illegal positions (e.g., pawns on the first rank, both kings in check) and excludes legal positions following captures and promotions.
After each player has moved a piece 5 times each (10ply) there are 69,352,859,712,417 possible games that could have been played.
Taking Shannon's numbers into account,Victor Alliscalculated anupper boundof 5×1052for the number of positions, and estimated the true number to be about 1050.[4]Later work proved an upper bound of 8.7×1045,[5]and showed an upper bound 4×1037in the absence of promotions.[6][7]
Allis also estimated the game-tree complexity to be at least 10123, "based on an average branching factor of 35 and an average game length of 80". As a comparison, thenumber of atoms in the observable universe, to which it is often compared, is roughly estimated to be 1080.
John Trompand Peter Österlund estimated the number of legal chess positions with a 95% confidence level at(4.822±0.028)×1044{\displaystyle (4.822\pm 0.028)\times 10^{44}}, based on an efficiently computable bijection between integers and chess positions.[5]
As a comparison to the Shannon number, if chess is analyzed for the number of "sensible" games that can be played (not counting ridiculous or obvious game-losing moves such as moving a queen to be immediately captured by a pawn without compensation), then the result is closer to around 1040games. This is based on having a choice of about three sensible moves at each ply (half-move), and a game length of 80 plies (or, equivalently, 40 moves).[8]
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This is a list of some of the more commonly known problems that areNP-completewhen expressed asdecision problems. As there are thousands of such problems known, this list is in no way comprehensive. Many problems of this type can be found inGarey & Johnson (1979).
Graphsoccur frequently in everyday applications. Examples include biological or social networks, which contain hundreds, thousands and even billions of nodes in some cases (e.g.FacebookorLinkedIn).
General
Specific problems
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Here are some of the more commonly known problems that arePSPACE-completewhen expressed asdecision problems. This list is in no way comprehensive.
Generalizedversions of:
Type inhabitation problemfor simply typed lambda calculus
Integer circuitevaluation[24]
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Digital physicsis a speculative idea suggesting that theuniversecan be conceived of as a vast, digital computation device, or as the output of adeterministicorprobabilisticcomputer program.[1]The hypothesis that the universe is adigital computerwas proposed byKonrad Zusein his 1969 bookRechnender Raum[2](Calculating-space).[3]The term "digital physics" was coined in 1978 byEdward Fredkin,[4]who later came to prefer the term "digital philosophy".[5]Fredkin taught a graduate course called "digital physics" at MIT in 1978, and collaborated withTommaso Toffolion "conservative logic" whileNorman Margolusserved as a graduate student in his research group.[6]
Digital physicsposits that there exists, at least in principle, aprogramfor auniversal computerthat computes theevolutionof theuniverse. The computer could be, for example, a hugecellular automaton.[1][7]It is deeply connected to the concept ofinformation theory, particularly the idea that the universe's fundamental building blocks might be bits of information rather than traditional particles or fields.
However, extant models of digital physics face challenges, particularly in reconciling with several continuoussymmetries[8]in physical laws, e.g.,rotational symmetry,translational symmetry,Lorentz symmetry, and theLie groupgauge invariance ofYang–Mills theories, all of which are central to current physical theories. Moreover, existing models of digital physics violate various well-established features ofquantum physics, as they belong to a class of theories involving localhidden variables. These models have so far been disqualified experimentally by physicists usingBell's theorem.[9][10]
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Hypercomputationorsuper-Turing computationis a set of hypotheticalmodels of computationthat can provide outputs that are notTuring-computable. For example, a machine that could solve thehalting problemwould be a hypercomputer; so too would one that couldcorrectly evaluate every statementinPeano arithmetic.
TheChurch–Turing thesisstates that any "computable" function that can be computed by a mathematician with a pen and paper using a finite set of simple algorithms, can be computed by a Turing machine. Hypercomputers compute functions that aTuring machinecannot and which are, hence, not computable in the Church–Turing sense.
Technically, the output of arandom Turing machineis uncomputable; however, most hypercomputing literature focuses instead on thecomputationof deterministic, rather than random, uncomputable functions.
A computational model going beyond Turing machines was introduced byAlan Turingin his 1938 PhD dissertationSystems of Logic Based on Ordinals.[1]This paper investigated mathematical systems in which anoraclewas available, which could compute a single arbitrary (non-recursive) function fromnaturalsto naturals. He used this device to prove that even in those more powerful systems,undecidabilityis still present. Turing's oracle machines are mathematical abstractions, and are not physically realizable.[2]
In a sense, most functions are uncomputable: there areℵ0{\displaystyle \aleph _{0}}computable functions, but there are anuncountablenumber (2ℵ0{\displaystyle 2^{\aleph _{0}}}) of possible super-Turing functions.[3]
Hypercomputer models range from useful but probably unrealizable (such as Turing's original oracle machines), to less-useful random-function generators that are more plausibly "realizable" (such as arandom Turing machine).
A system granted knowledge of the uncomputable, oracularChaitin's constant(a number with an infinite sequence of digits that encode the solution to the halting problem) as an input can solve a large number of useful undecidable problems; a system granted an uncomputable random-number generator as an input can create random uncomputable functions, but is generally not believed to be able to meaningfully solve "useful" uncomputable functions such as the halting problem. There are an unlimited number of different types of conceivable hypercomputers, including:
In order to work correctly, certain computations by the machines below literally require infinite, rather than merely unlimited but finite, physical space and resources; in contrast, with a Turing machine, any given computation that halts will require only finite physical space and resources.
A Turing machine that cancompleteinfinitely many steps in finite time, a feat known as asupertask. Simply being able to run for an unbounded number of steps does not suffice. One mathematical model is theZeno machine(inspired byZeno's paradox). The Zeno machine performs its first computation step in (say) 1 minute, the second step in ½ minute, the third step in ¼ minute, etc. By summing1 + ½ + ¼ + ...(ageometric series) we see that the machine performs infinitely many steps in a total of 2 minutes. According toOron Shagrir, Zeno machines introduce physical paradoxes and its state is logically undefined outside of one-side open period of [0, 2), thus undefined exactly at 2 minutes after beginning of the computation.[13]
It seems natural that the possibility of time travel (existence ofclosed timelike curves(CTCs)) makes hypercomputation possible by itself. However, this is not so since a CTC does not provide (by itself) the unbounded amount of storage that an infinite computation would require. Nevertheless, there are spacetimes in which the CTC region can be used for relativistic hypercomputation.[14]According to a 1992 paper,[15]a computer operating in aMalament–Hogarth spacetimeor in orbit around a rotatingblack hole[16]could theoretically perform non-Turing computations for an observer inside the black hole.[17][18]Access to a CTC may allow the rapid solution toPSPACE-completeproblems, a complexity class which, while Turing-decidable, is generally considered computationally intractable.[19][20]
Some scholars conjecture that aquantum mechanicalsystem which somehow uses an infinite superposition of states could compute a non-computable function.[21]This is not possible using the standardqubit-modelquantum computer, because it is proven that a regular quantum computer isPSPACE-reducible(a quantum computer running inpolynomial timecan be simulated by a classical computer running inpolynomial space).[22]
Some physically realizable systems will always eventually converge to the correct answer, but have the defect that they will often output an incorrect answer and stick with the incorrect answer for an uncomputably large period of time before eventually going back and correcting the mistake.
In mid 1960s,E Mark GoldandHilary Putnamindependently proposed models ofinductive inference(the "limiting recursive functionals"[23]and "trial-and-error predicates",[24]respectively). These models enable some nonrecursive sets of numbers or languages (including allrecursively enumerablesets of languages) to be "learned in the limit"; whereas, by definition, only recursive sets of numbers or languages could be identified by a Turing machine. While the machine will stabilize to the correct answer on any learnable set in some finite time, it can only identify it as correct if it is recursive; otherwise, the correctness is established only by running the machine forever and noting that it never revises its answer. Putnam identified this new interpretation as the class of "empirical" predicates, stating: "if we always 'posit' that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are neversurethat we have the correct answer.)"[24]L. K. Schubert's 1974 paper "Iterated Limiting Recursion and the Program Minimization Problem"[25]studied the effects of iterating the limiting procedure; this allows anyarithmeticpredicate to be computed. Schubert wrote, "Intuitively, iterated limiting identification might be regarded as higher-order inductive inference performed collectively by an ever-growing community of lower order inductive inference machines."
A symbol sequence iscomputable in the limitif there is a finite, possibly non-halting program on auniversal Turing machinethat incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π and of every othercomputable real, but still excludes all noncomputable reals. The 'Monotone Turing machines' traditionally used indescription sizetheory cannot edit their previous outputs; generalized Turing machines, as defined byJürgen Schmidhuber, can. He defines the constructively describable symbol sequences as those that have a finite, non-halting program running on a generalized Turing machine, such that any output symbol eventually converges; that is, it does not change any more after some finite initial time interval. Due to limitations first exhibited byKurt Gödel(1931), it may be impossible to predict the convergence time itself by a halting program, otherwise thehalting problemcould be solved. Schmidhuber ([26][27]) uses this approach to define the set of formally describable or constructively computable universes or constructivetheories of everything. Generalized Turing machines can eventually converge to a correct solution of the halting problem by evaluating aSpecker sequence.
Many hypercomputation proposals amount to alternative ways to read anoracleoradvice functionembedded into an otherwise classical machine. Others allow access to some higher level of thearithmetic hierarchy. For example, supertasking Turing machines, under the usual assumptions, would be able to compute any predicate in thetruth-table degreecontainingΣ10{\displaystyle \Sigma _{1}^{0}}orΠ10{\displaystyle \Pi _{1}^{0}}. Limiting-recursion, by contrast, can compute any predicate or function in the correspondingTuring degree, which is known to beΔ20{\displaystyle \Delta _{2}^{0}}. Gold further showed that limiting partial recursion would allow the computation of precisely theΣ20{\displaystyle \Sigma _{2}^{0}}predicates.
Martin Davis, in his writings on hypercomputation,[35][36]refers to this subject as "a myth" and offers counter-arguments to the
physical realizability of hypercomputation. As for its theory, he argues against
the claims that this is a new field founded in the 1990s. This point of view relies
on the history ofcomputability theory(degrees of unsolvability, computability over
functions, real numbers and ordinals), as also mentioned above.
In his argument, he makes a remark that all of hypercomputation is little more than: "if non-computable inputs are permitted, then non-computable outputs are attainable."[37]
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Amatrioshka brain[1][2]is a hypotheticalmegastructureof immense computational capacity powered by aDyson sphere. It was proposed in 1997 byRobert J. Bradbury(1956–2011[3]). It is an example of a class-Bstellar engine, employing the entire energy output of a star to drivecomputersystems.[4]This concept derives its name from the nesting Russianmatryoshka dolls.[5]The concept was deployed by Bradbury in the anthologyYear Million: Science at the Far Edge of Knowledge.[6][7]
The concept of a matrioshka brain comes from the idea of using Dyson spheres to power an enormous, star-sized computer. The term "matrioshka brain" originates frommatryoshka dolls, which are wooden Russian nesting dolls. Matrioshka brains are composed of several Dyson spheres nested inside one another, the same way that matryoshka dolls are composed of multiple nested doll components.
The innermost Dyson sphere of the matrioshka brain would draw energy directly from the star it surrounds and give off large amounts ofwaste heatwhile computing at a high temperature. The next surrounding Dyson sphere would absorb this waste heat and use it for its computational purposes, all while giving off waste heat of its own. This heat would be absorbed by the next sphere, and so on, with each sphere radiating at a lower temperature than the one before it. For this reason, Matrioshka brains with more nested Dyson spheres would tend to be more efficient, as they would waste less heat energy. The inner shells could run at nearly the same temperature as the star itself, while the outer ones would be close to the temperature of interstellar space. The engineering requirements and resources needed for this would be enormous.
The term "matrioshka brain" was invented by Robert Bradbury as an alternative to theJupiter brain[8]—a concept similar to the matrioshka brain, but on a smaller planetary scale and optimized for minimal signalpropagation delay. A matrioshka brain design is concentrated on sheer capacity and the maximum amount of energy extracted from its source star, while a Jupiter brain is optimized for computational speed.[9]Jupiter brains are related to the idea of the hypothetical materialcomputronium, which could be enmassed to sizes of entire planets and even stars.[10]
Some possible uses of such an immense computational resource have been proposed.
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The study of thephysics ofcomputationrelates to understanding the fundamentalphysical limits of computers. This field has led to the investigation of howthermodynamicslimits information processing, the understanding ofchaosanddynamical systems, and a rapidly growing effort to invent newquantum computers.
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Programmable matterismatterwhich has the ability to change its physical properties (shape, density,moduli, conductivity, optical properties, etc.) in a programmable fashion, based upon user input or autonomous sensing. Programmable matter is thus linked to the concept of a material which inherently has the ability to perform information processing.
Programmable matter is a term originally coined in 1991 byToffoliandMargolusto refer to an ensemble of fine-grained computing elements arranged in space.[1]Their paper describes a computingsubstratethat is composed of fine-grained compute nodes distributed throughout space which communicate using only nearest neighbor interactions. In this context, programmable matter refers to compute models similar tocellular automataandlattice gas automata.[2]The CAM-8 architecture is an example hardware realization of this model.[3]This function is also known as "digital referenced areas" (DRA) in some forms ofself-replicating machinescience.[4]
In the early 1990s, there was a significant amount of work in reconfigurable modular robotics with a philosophy similar to programmable matter.[4]
Assemiconductortechnology,nanotechnology, and self-replicating machine technology have advanced, the use of the term programmable matter has changed to reflect the fact that
it is possible to build an ensemble of elements which can be "programmed" to change their physical properties in reality, not just insimulation. Thus, programmable matter has come to mean "any bulk substance which can be programmed to change its physical properties."
In the summer of 1998, in a discussion on artificial atoms and programmable matter,Wil McCarthyand G. Snyder coined the term "quantum wellstone" (or simply "wellstone") to describe this hypothetical but plausible form of programmable matter. McCarthy has used the term in his fiction.
In 2002, Seth Goldstein and Todd Mowry started the claytronics project atCarnegie Mellon Universityto investigate the underlying hardware and software mechanisms necessary to realize programmable matter.
In 2004, theDARPAInformation Science and Technology group (ISAT) examined the potential of programmable matter. This resulted in the 2005–2006 study "Realizing Programmable Matter", which laid out a multi-year program for the research and development of programmable matter.
In 2007, programmable matter was the subject of a DARPA research solicitation and subsequent program.[5][6]
From 2016 to 2022, theANRhas funded several research programs coordinated by Julien Bourgeois and Benoit Piranda at theFEMTO-ST Institute, which is taking the lead in the Claytronics project initiated by Intel and Carnegie Mellon University.[7]
In one school of thought, the programming could be external to the material and might be achieved by the "application of light, voltage, electric or magnetic fields, etc." (McCarthy 2006). For example, aliquid crystal displayis a form of programmable matter. A second school of thought is that the individual units of the ensemble can compute and the result of their computation is a change in the ensemble's physical properties. An example of this more ambitious form of programmable matter is claytronics.
There are many proposed implementations of programmable matter. Scale is one key differentiator between different forms of programmable matter. At one end of the spectrum, reconfigurable modular robotics pursues a form of programmable matter where the individual units are in the centimeter size range.[4][8][9]At the nanoscale end of the spectrum, there are a tremendous number of different bases for programmable matter, ranging from shape changing molecules[10]toquantum dots. Quantum dots are in fact often referred to as artificial atoms. In the micrometer to sub-millimeter range examples includeMEMS-based units, cells created usingsynthetic biology, and theutility fogconcept.
An important sub-group of programmable matter arerobotic materials, which combine the structural aspects of a composite with the affordances offered by tight integration of sensors, actuators, computation, and communication,[11]while foregoing reconfiguration by particle motion.
There are many conceptions of programmable matter, and thus many discrete avenues of research using the name. Below are some specific examples of programmable matter.
Shape-changing and locomotion of solid objects are possible with solid-liquid phase change pumping.[12]This approach allows deforming objects into any intended shape with sub-millimetre resolution and freely changing their topology.
These include materials that can change their properties based on some input, but do not have the ability to do complex computation by themselves.
The physical properties of several complex fluids can be modified by applying a current or voltage, as is the case withliquid crystals.
Metamaterials are artificialcompositesthat can be controlled to react in ways that do not occur in nature. One example developed by David Smith and then by John Pendry and David Schuri is of a material that can have itsindex of refractiontuned so that it can have a different index of refraction at different points in the material. If tuned properly, this could result in aninvisibility cloak.
A further example of programmable -mechanical- metamaterial is presented by Bergamini et al.[13]Here, a pass band within the phononic bandgap is introduced, by exploiting variable stiffness of piezoelectric elements linking aluminum stubs to the aluminum plate to create a phononic crystal as in the work of Wu et al.[14]The piezoelectric elements are shunted to ground over synthetic inductors. Around the resonance frequency of the LC circuit formed by the piezoelectric and the inductors, the piezoelectric elements exhibit near zero stiffness, thus effectively disconnecting the stubs from the plate. This is considered an example of programmable mechanical metamaterial.[13]
In 2021, Chen et al. demonstrated a mechanical metamaterial whose unit cells can each store a binary digit analogous to a bit inside a hard disk drive.[15]Similarly, these mechanical unit cells are programmed through the interaction between two electromagnetic coils in the Maxwell configuration, and an embedded magnetorheological elastomer. Different binary states are associated with different stress-strain response of the material.
An active area of research is in molecules that can change their shape, as well as other properties, in response to external stimuli. These molecules can be used individually or en masse to form new kinds of materials. For example,J Fraser Stoddart's group at UCLA has been developing molecules that can change their electrical properties.[10]
An electropermanent magnet is a type ofmagnetwhich consists of both anelectromagnetand a dual materialpermanent magnet, in which themagnetic fieldproduced by the electromagnet is used to change the magnetization of the permanent magnet. The permanent magnet consists of magnetically hard and soft materials, of which only the soft material can have its magnetization changed. When the magnetically soft and hard materials have opposite magnetizations the magnet has no net field, and when they are aligned the magnet displays magnetic behaviour.[16]
They allow creating controllable permanent magnets where the magnetic effect can be maintained without requiring a continuous supply of electrical energy. For these reasons, electropermanent magnets are essential components of the research studies aiming to build programmable magnets that can give rise to self-building structures.[16][17]
Self-reconfiguring modular robotics involves a group of basic robot modules working together to dynamically form shapes and create behaviours suitable for many tasks, similar to programmable matter. SRCMR aims to offer significant improvement to many kinds of objects or systems by introducing many new possibilities. For example: 1. Most important is the incredible flexibility that comes from the ability to change the physical structure and behavior of a solution by changing the software that controls modules. 2. The ability to self-repair by automatically replacing a broken module will make SRCMR solution incredibly resilient. 3. Reducing the environmental footprint by reusing the same modules in many different solutions. Self-reconfiguring modular robotics enjoys a vibrant and active research community.[18]
Claytronics is an emerging field ofengineeringconcerning reconfigurablenanoscalerobots('claytronicatoms', orcatoms) designed to form much larger scalemachinesor mechanisms. The catoms will be sub-millimeter computers that will eventually have the ability to move around, communicate with other computers, change color, andelectrostaticallyconnect to other catoms to form different shapes.
Cellular automata are a useful concept to abstract some of the concepts of discrete units interacting to give a desired overall behavior.
Quantum wells can hold one or more electrons. Those electrons behave likeartificial atomswhich, like real atoms, can formcovalent bonds, but these are extremely weak. Because of their larger sizes, other properties are also widely different.
Synthetic biology is a field that aims to engineer cells with "novel biological functions."[citation needed]Suchcellsare usually used to create larger systems (e.g.,biofilms) which can be "programmed" utilizing syntheticgene networkssuch asgenetic toggle switches, to change their color, shape, etc. Such bioinspired approaches to materials production has been demonstrated, using self-assembling bacterial biofilm materials that can be programmed for specific functions, such as substrate adhesion,nanoparticletemplating, and protein immobilization.[19]
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Inphilosophy, asupertaskis acountably infinitesequence of operations that occur sequentially within a finite interval of time.[1]Supertasks are calledhypertaskswhen the number of operations becomesuncountably infinite. A hypertask that includes one task for eachordinal numberis called anultratask.[2]The term "supertask" was coined by the philosopherJames F. Thomson, who devisedThomson's lamp. The term "hypertask" derives from Clark and Read in their paper of that name.[3]
The origin of the interest in supertasks is normally attributed toZeno of Elea. Zeno claimed thatmotion was impossible. He argued as follows: suppose our burgeoning "mover", Achilles say, wishes to move from A to B. To achieve this he must traverse half the distance from A to B. To get from the midpoint of AB to B, Achilles must traverse halfthisdistance, and so on and so forth. However many times he performs one of these "traversing" tasks, there is another one left for him to do before he arrives at B. Thus it follows, according to Zeno, that motion (travelling a non-zero distance in finite time) is a supertask. Zeno further argues that supertasks are not possible (how can this sequence be completed if for each traversing there is another one to come?). It follows that motion is impossible.
Zeno's argument takes the following form:
Most subsequent philosophers reject Zeno's bold conclusion in favor of common sense. Instead, they reverse the argument and take it as aproof by contradictionwhere the possibility of motion is taken for granted. They accept the possibility of motion and applymodus tollens(contrapositive) to Zeno's argument to reach the conclusion that either motion is not a supertask or not all supertasks are impossible.[citation needed]
Zeno himself also discusses the notion of what he calls "Achillesand the tortoise". Suppose that Achilles is the fastest runner, and moves at a speed of 1 m/s. Achilles chases a tortoise, an animal renowned for being slow, that moves at 0.1 m/s. However, the tortoise starts 0.9 metres ahead. Common sense seems to decree that Achilles will catch up with the tortoise after exactly 1 second, but Zeno argues that this is not the case. He instead suggests that Achilles must inevitably come up to the point where the tortoise has started from, but by the time he has accomplished this, the tortoise will already have moved on to another point. This continues, and every time Achilles reaches the mark where the tortoise was, the tortoise will have reached a new point that Achilles will have to catch up with; while it begins with 0.9 metres, it becomes an additional 0.09 metres, then 0.009 metres, and so on, infinitely. While these distances will grow very small, they will remain finite, while Achilles' chasing of the tortoise will become an unending supertask. Much commentary has been made on this particular paradox; many assert that it finds a loophole in common sense.[4]
James F. Thomsonbelieved that motion was not a supertask, and he emphatically denied that supertasks are possible. He considered a lamp that may either be on or off. At timet= 0the lamp is off, and the switch is flipped on att= 1/2; after that, the switch is flipped after waiting for half the time as before. Thomson asks what is the state att= 1, when the switch has been flipped infinitely many times. He reasons that it cannot be on because there was never a time when it was not subsequently turned off, and vice versa, and reaches a contradiction. He concludes that supertasks are impossible.[5]
Paul Benacerrafbelieves that supertasks are at least logically possible despite Thomson's apparent contradiction. Benacerraf agrees with Thomson insofar as that the experiment he outlined does not determine the state of the lamp at t = 1. However he disagrees with Thomson that he can derive a contradiction from this, since the state of the lamp at t = 1 cannot be logically determined by the preceding states.[citation needed]
Most of the modern literature comes from the descendants of Benacerraf, those who tacitly accept the possibility of supertasks. Philosophers who reject their possibility tend not to reject them on grounds such as Thomson's but because they have qualms with the notion of infinity itself. Of course there are exceptions. For example, McLaughlin claims that Thomson's lamp is inconsistent if it is analyzed withinternal set theory, a variant ofreal analysis.
If supertasks are possible, then the truth or falsehood of unknown propositions of number theory, such asGoldbach's conjecture, or evenundecidablepropositions could be determined in a finite amount of time by a brute-force search of the set of all natural numbers. This would, however, be in contradiction with theChurch–Turing thesis. Some have argued this poses a problem forintuitionism, since the intuitionist must distinguish between things that cannot in fact be proven (because they are too long or complicated; for exampleBoolos's "Curious Inference"[6]) but nonetheless are considered "provable", and those whichareprovable by infinite brute force in the above sense.
Some have claimed, Thomson's lamp is physically impossible since it must have parts moving at speeds faster than thespeed of light(e.g., the lamp switch).Adolf Grünbaumsuggests that the lamp could have a strip of wire which, when lifted, disrupts the circuit and turns off the lamp; this strip could then be lifted by a smaller distance each time the lamp is to be turned off, maintaining a constant velocity.
However, such a design would ultimately fail, as eventually the distance between the contacts would be so small as to allow electrons to jump the gap, preventing the circuit from being broken at all. Still, for either a human or any device, to perceive or act upon the state of the lamp some measurement has to be done, for example the light from the lamp would have to reach an eye or a sensor.
Any such measurement will take a fixed frame of time, no matter how small and, therefore, at some point measurement of the state will be impossible. Since the state at t=1 cannot be determined even in principle, it is not meaningful to speak of the lamp being either on or off.
Other physically possible supertasks have been suggested. In one proposal, one person (or entity) counts upward from 1, taking an infinite amount of time, while another person observes this from a frame of reference where this occurs in a finite space of time. For the counter, this is not a supertask, but for the observer, it is. (This could theoretically occur due totime dilation, for example if the observer were falling into ablack holewhile observing a counter whose position is fixed relative to the singularity.)
Gustavo E. Romeroin the paper 'The collapse of supertasks'[7]maintains that any attempt to carry out a supertask will result in the formation of ablack hole, making supertasks physically impossible.
The impact of supertasks on theoretical computer science has triggered some new and interesting work, for example Hamkins and Lewis – "Infinite Time Turing Machine".[8]
Suppose there is a jar capable of containing infinitely many marbles and an infinite collection of marbles labelled 1, 2, 3, and so on. At timet= 0, marbles 1 through 10 are placed in the jar and marble 1 is taken out. Att= 0.5, marbles 11 through 20 are placed in the jar and marble 2 is taken out; att= 0.75, marbles 21 through 30 are put in the jar and marble 3 is taken out; and in general at timet= 1 − 0.5n, marbles 10n+ 1 through 10n+ 10 are placed in the jar and marblen+ 1 is taken out. How many marbles are in the jar at timet= 1?
One argument states that there should be infinitely many marbles in the jar, because at each step beforet= 1 the number of marbles increases from the previous step and does so unboundedly. A second argument, however, shows that the jar is empty. Consider the following argument: if the jar is non-empty, then there must be a marble in the jar. Let us say that that marble is labeled with the numbern. But at timet= 1 − 0.5n- 1, thenth marble has been taken out, so marblencannot be in the jar. This is a contradiction, so the jar must be empty. The Ross–Littlewood paradox is that here we have two seemingly perfectly good arguments with completely opposite conclusions.
There has been considerable interest inJ. A. Benardete’s “Paradox of the Gods”:[9]
A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods.[10]
Inspired byJ. A. Benardete’s paradox regarding an infinite series of assassins,[11]David Chalmersdescribes the paradox as follows:
There are countably many grim reapers, one for every positive integer. Grim reaper 1 is disposed to kill you with a scythe at 1pm, if and only if you are still alive then (otherwise his scythe remains immobile throughout), taking 30 minutes about it. Grim reaper 2 is disposed to kill you with a scythe at 12:30 pm, if and only if you are still alive then, taking 15 minutes about it. Grim reaper 3 is disposed to kill you with a scythe at 12:15 pm, and so on. You are still alive just before 12pm, you can only die through the motion of a grim reaper’s scythe, and once dead you stay dead. On the face of it, this situation seems conceivable — each reaper seems conceivable individually and intrinsically, and it seems reasonable to combine distinct individuals with distinct intrinsic properties into one situation. But a little reflection reveals that the situation as described is contradictory. I cannot survive to any moment past 12pm (a grim reaper would get me first), but I cannot be killed (for grim reapernto kill me, I must have survived grim reapern+1, which is impossible).[12]
It has gained significance in philosophy via its use in arguing for a finite past, thereby bearing relevance to theKalam cosmological argument.[13][14][15][16]
Proposed byE. Brian Davies,[17]this is a machine that can, in the space of half an hour, create an exact replica of itself that is half its size and capable of twice its replication speed. This replica will in turn create an even faster version of itself with the same specifications, resulting in a supertask that finishes after an hour. If, additionally, the machines create a communication link between parent and child machine that yields successively faster bandwidth and the machines are capable of simple arithmetic, the machines can be used to perform brute-force proofs of unknown conjectures. However, Davies also points out that – due to fundamental properties of the real universe such asquantum mechanics,thermal noiseandinformation theory– his machine cannot actually be built.
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Aparameterized approximation algorithmis a type ofalgorithmthat aims to find approximate solutions toNP-hardoptimization problemsinpolynomial timein the input size and a function of a specific parameter. These algorithms are designed to combine the best aspects of both traditionalapproximation algorithmsandfixed-parameter tractability.
In traditional approximation algorithms, the goal is to find solutions that are at most a certain factorαaway from the optimal solution, known as anα-approximation, in polynomial time. On the other hand, parameterized algorithms are designed to find exact solutions to problems, but with the constraint that the running time of the algorithm is polynomial in the input size and a function of a specific parameterk. The parameter describes some property of the input and is small in typical applications. The problem is said to be fixed-parameter tractable (FPT) if there is an algorithm that can find the optimum solution inf(k)nO(1){\displaystyle f(k)n^{O(1)}}time, wheref(k){\displaystyle f(k)}is a function independent of the input sizen.
A parameterized approximation algorithm aims to find a balance between these two approaches by finding approximate solutions in FPT time: the algorithm computes anα-approximation inf(k)nO(1){\displaystyle f(k)n^{O(1)}}time, wheref(k){\displaystyle f(k)}is a function independent of the input sizen. This approach aims to overcome the limitations of both traditional approaches by having stronger guarantees on the solution quality compared to traditional approximations while still having efficient running times as in FPT algorithms. An overview of the research area studying parameterized approximation algorithms can be found in the survey of Marx[1]and the more recent survey by Feldmann et al.[2]
The full potential of parameterized approximation algorithms is utilized when a givenoptimization problemis shown to admit anα-approximation algorithm running inf(k)nO(1){\displaystyle f(k)n^{O(1)}}time, while in contrast the problem neither has a polynomial-timeα-approximation algorithm (under somecomplexity assumption, e.g.,P≠NP{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}), nor an FPT algorithm for the given parameterk(i.e., it is at leastW[1]-hard).
For example, some problems that areAPX-hardandW[1]-hardadmit aparameterized approximation scheme (PAS), i.e., for anyε>0{\displaystyle \varepsilon >0}a(1+ε){\displaystyle (1+\varepsilon )}-approximation can be computed inf(k,ε)ng(ε){\displaystyle f(k,\varepsilon )n^{g(\varepsilon )}}time for some functionsfandg. This then circumvents the lower bounds in terms of polynomial-time approximation and fixed-parameter tractability. A PAS is similar in spirit to apolynomial-time approximation scheme (PTAS)but additionally exploits a given parameterk. Since the degree of the polynomial in the runtime of a PAS depends on a functiong(ε){\displaystyle g(\varepsilon )}, the value ofε{\displaystyle \varepsilon }is assumed to be arbitrary but constant in order for the PAS to run in FPT time. If this assumption is unsatisfying,ε{\displaystyle \varepsilon }is treated as a parameter as well to obtain anefficientparameterized approximation scheme (EPAS), which for anyε>0{\displaystyle \varepsilon >0}computes a(1+ε){\displaystyle (1+\varepsilon )}-approximation inf(k,ε)nO(1){\displaystyle f(k,\varepsilon )n^{O(1)}}time for some functionf. This is similar in spirit to anefficient polynomial-time approximation scheme (EPTAS).
Thek-cutproblem has no polynomial-time(2−ε){\displaystyle (2-\varepsilon )}-approximation algorithm for anyε>0{\displaystyle \varepsilon >0}, assumingP≠NP{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}and thesmall set expansion hypothesis.[3]It is also W[1]-hard parameterized by the numberkof required components.[4]However an EPAS exists, which computes a(1+ε){\displaystyle (1+\varepsilon )}-approximation in(k/ε)O(k)nO(1){\displaystyle (k/\varepsilon )^{O(k)}n^{O(1)}}time.[5]
TheSteiner Tree problemis FPT parameterized by the number of terminals.[6]However, for the "dual" parameter consisting of the numberkof non-terminals contained in the optimum solution, the problem isW[2]-hard(due to a folklore reduction from theDominating Set problem). Steiner Tree is also known to beAPX-hard.[7]However, there is an EPAS computing a(1+ε){\displaystyle (1+\varepsilon )}-approximation in2O(k2/ε4)nO(1){\displaystyle 2^{O(k^{2}/\varepsilon ^{4})}n^{O(1)}}time.[8]The more general Steiner Forest problem is NP-hard on graphs of treewidth 3. However, on graphs oftreewidthtan EPAS can compute a(1+ε){\displaystyle (1+\varepsilon )}-approximation in2O(t2εlogtε)nO(1){\displaystyle 2^{O({\frac {t^{2}}{\varepsilon }}\log {\frac {t}{\varepsilon }})}n^{O(1)}}time.[9]
It is known that the Strongly Connected Steiner Subgraph problem is W[1]-hard parameterized by the numberkof terminals,[10]and also does not admit anO(log2−εn){\displaystyle O(\log ^{2-\varepsilon }n)}-approximation in polynomial time (under standardcomplexity assumptions).[11]However a 2-approximation can be computed in3knO(1){\displaystyle 3^{k}n^{O(1)}}time.[12]Furthermore, this is best possible, since no(2−ε){\displaystyle (2-\varepsilon )}-approximation can be computed inf(k)nO(1){\displaystyle f(k)n^{O(1)}}time for any functionf, under Gap-ETH.[13]
For the well-studied metric clustering problems ofk-medianandk-meansparameterized by the numberkof centers, it is known that no(1+2/e−ε){\displaystyle (1+2/e-\varepsilon )}-approximation for k-Median and no(1+8/e−ε){\displaystyle (1+8/e-\varepsilon )}-approximation for k-Means can be computed inf(k)nO(1){\displaystyle f(k)n^{O(1)}}time for any functionf, under Gap-ETH.[14]Matching parameterized approximation algorithms exist,[14]but it is not known whether matching approximations can be computed in polynomial time.
Clustering is often considered in settings of low dimensional data, and thus a practically relevant parameterization is by thedimensionof the underlyingmetric. In theEuclidean space, the k-Median and k-Means problems admit an EPAS parameterized by the dimensiond,[15][16]and also an EPAS parameterized byk.[17][18]The former was generalized to an EPAS for the parameterization by thedoubling dimension.[19]For the loosely relatedhighway dimensionparameter, only an approximation scheme withXPruntime is known to date.[20]
For the metrick-center problema 2-approximation can be computed in polynomial time. However, when parameterizing by either the numberkof centers,[21]thedoubling dimension(in fact the dimension of aManhattan metric),[22]or thehighway dimension,[21]no parameterized(2−ε){\displaystyle (2-\varepsilon )}-approximation algorithm exists, under standardcomplexity assumptions. Furthermore, the k-Center problem is W[1]-hard even onplanar graphswhen simultaneously parameterizing it by the numberkof centers, thedoubling dimension, thehighway dimension, and thepathwidth.[23]However, when combiningkwith the doubling dimension an EPAS exists,[23]and the same is true when combiningkwith thehighway dimension.[24]For the more general version with vertex capacities, an EPAS exists for the parameterization by k and the doubling dimension, but not when using k and the highway dimension as the parameter.[25]Regarding the pathwidth, k-Center admits an EPAS even for the more generaltreewidthparameter, and also forcliquewidth.[26]
Anoptimizationvariant of thek-Clique problemis the Densestk-Subgraph problem (which is a 2-aryConstraint Satisfaction problem), where the task is to find a subgraph onkvertices with maximum number of edges. It is not hard to obtain a(k−1){\displaystyle (k-1)}-approximation by just picking amatchingof sizek/2{\displaystyle k/2}in the given input graph, since the maximum number of edges onkvertices is always at most(k2)=k(k−1)/2{\displaystyle {k \choose 2}=k(k-1)/2}. This is alsoasymptoticallyoptimal, since under Gap-ETHnok1−o(1){\displaystyle k^{1-o(1)}}-approximation can be computed in FPT time parameterized byk.[27]
For theDominating set problemit is W[1]-hard to compute anyg(k){\displaystyle g(k)}-approximation inf(k)nO(1){\displaystyle f(k)n^{O(1)}}time for any functionsgandf.[28]
Kernelizationis a technique used infixed-parameter tractabilityto pre-process an instance of anNP-hardproblem in order to remove "easy parts" and reveal the NP-hard core of the instance. A kernelization algorithm takes an instanceIand a parameterk, and returns a new instanceI′{\displaystyle I'}with parameterk′{\displaystyle k'}such that the size ofI′{\displaystyle I'}andk′{\displaystyle k'}is bounded as a function of the input parameterk, and the algorithm runs in polynomial time. Anα-approximate kernelization algorithmis a variation of this technique that is used in parameterized approximation algorithms. It returns a kernelI′{\displaystyle I'}such that anyβ-approximation inI′{\displaystyle I'}can be converted into anαβ-approximation to the input instanceIin polynomial time. This notion was introduced by Lokshtanov et al.,[29]but there are other related notions in the literature such as Turing kernels[30]andα-fidelity kernelization.[31]
As for regular (non-approximate) kernels, a problem admits an α-approximate kernelization algorithmif and only ifit has a parameterized α-approximation algorithm. The proof of this fact is very similar tothe one for regular kernels.[29]However the guaranteed approximate kernel might be of exponential size (or worse) in the input parameter. Hence it becomes interesting to find problems that admit polynomial sized approximate kernels. Furthermore, apolynomial-sized approximate kernelization scheme (PSAKS)is anα-approximate kernelization algorithm that computes a polynomial-sized kernel and for whichαcan be set to1+ε{\displaystyle 1+\varepsilon }for anyε>0{\displaystyle \varepsilon >0}.
For example, while the ConnectedVertex Coverproblem is FPT parameterized by the solution size, it does not admit a (regular) polynomial sized kernel (unlessNP⊆coNP/poly{\displaystyle {\textsf {NP}}\subseteq {\textsf {coNP/poly}}}), but a PSAKS exists.[29]Similarly, the Steiner Tree problem is FPT parameterized by the number of terminals, does not admit a polynomial sized kernel (unlessNP⊆coNP/poly{\displaystyle {\textsf {NP}}\subseteq {\textsf {coNP/poly}}}), but a PSAKS exists.[29]When parameterizing Steiner Tree by the number of non-terminals in the optimum solution, the problem is W[2]-hard (and thus admits no exact kernel at all, unless FPT=W[2]), but still admits a PSAKS.[8]
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Incomputer scienceandoperations research,approximation algorithmsareefficientalgorithmsthat findapproximatesolutions tooptimization problems(in particularNP-hardproblems) with provable guarantees on the distance of the returned solution to the optimal one.[1]Approximation algorithms naturally arise in the field oftheoretical computer scienceas a consequence of the widely believedP ≠ NPconjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly inpolynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that providesbothis the classic approximation algorithm ofLenstra,ShmoysandTardos[2]forschedulingon unrelated parallel machines.
The design andanalysisof approximation algorithms crucially involves amathematical proofcertifying the quality of the returned solutions in the worst case.[1]This distinguishes them fromheuristicssuch asannealingorgenetic algorithms, which find reasonably good solutions on some inputs, but provide no clear indication at the outset on when they may succeed or fail.
There is widespread interest intheoretical computer scienceto better understand the limits to which we can approximate certain famous optimization problems. For example, one of the long-standing open questions in computer science is to determine whether there is an algorithm that outperforms the 2-approximation for the Steiner Forest problem by Agrawal et al.[3]The desire to understand hard optimization problems from the perspective of approximability is motivated by the discovery of surprising mathematical connections and broadly applicable techniques to design algorithms for hard optimization problems. One well-known example of the former is theGoemans–Williamson algorithmformaximum cut, which solves a graph theoretic problem using high dimensional geometry.[4]
A simple example of an approximation algorithm is one for theminimum vertex coverproblem, where the goal is to choose the smallest set of vertices such that every edge in the input graph contains at least one chosen vertex. One way to find avertex coveris to repeat the following process: find an uncovered edge, add both its endpoints to the cover, and remove all edges incident to either vertex from the graph. As any vertex cover of the input graph must use a distinct vertex to cover each edge that was considered in the process (since it forms amatching), the vertex cover produced, therefore, is at most twice as large as the optimal one. In other words, this is aconstant-factor approximation algorithmwith an approximation factor of 2. Under the recentunique games conjecture, this factor is even the best possible one.[5]
NP-hard problems vary greatly in their approximability; some, such as theknapsack problem, can be approximated within a multiplicative factor1+ϵ{\displaystyle 1+\epsilon }, for any fixedϵ>0{\displaystyle \epsilon >0}, and therefore produce solutions arbitrarily close to the optimum (such a family of approximation algorithms is called apolynomial-time approximation schemeor PTAS). Others are impossible to approximate within any constant, or even polynomial, factor unlessP = NP, as in the case of themaximum clique problem. Therefore, an important benefit of studying approximation algorithms is a fine-grained classification of the difficulty of various NP-hard problems beyond the one afforded by thetheory of NP-completeness. In other words, although NP-complete problems may be equivalent (under polynomial-time reductions) to each other from the perspective of exact solutions, the corresponding optimization problems behave very differently from the perspective of approximate solutions.
By now there are several established techniques to design approximation algorithms. These include the following ones.
While approximation algorithms always provide an a priori worst case guarantee (be it additive or multiplicative), in some cases they also provide an a posteriori guarantee that is often much better. This is often the case for algorithms that work by solving aconvex relaxationof the optimization problem on the given input. For example, there is a different approximation algorithm for minimum vertex cover that solves alinear programming relaxationto find a vertex cover that is at most twice the value of the relaxation. Since the value of the relaxation is never larger than the size of the optimal vertex cover, this yields another 2-approximation algorithm. While this is similar to the a priori guarantee of the previous approximation algorithm, the guarantee of the latter can be much better (indeed when the value of the LP relaxation is far from the size of the optimal vertex cover).
Approximation algorithms as a research area is closely related to and informed byinapproximability theorywhere the non-existence of efficient algorithms with certain approximation ratios is proved (conditioned on widely believed hypotheses such as the P ≠ NP conjecture) by means ofreductions. In the case of the metric traveling salesman problem, the best known inapproximability result rules out algorithms with an approximation ratio less than 123/122 ≈ 1.008196 unless P = NP, Karpinski, Lampis, Schmied.[6]Coupled with the knowledge of the existence of Christofides' 1.5 approximation algorithm, this tells us that the threshold of approximability for metric traveling salesman (if it exists) is somewhere between 123/122 and 1.5.
While inapproximability results have been proved since the 1970s, such results were obtained by ad hoc means and no systematic understanding was available at the time. It is only since the 1990 result of Feige, Goldwasser, Lovász, Safra and Szegedy on the inapproximability ofIndependent Set[7]and the famousPCP theorem,[8]that modern tools for proving inapproximability results were uncovered. The PCP theorem, for example, shows thatJohnson's1974 approximation algorithms forMax SAT,set cover,independent setandcoloringall achieve the optimal approximation ratio, assuming P ≠ NP.[9]
Not all approximation algorithms are suitable for direct practical applications. Some involve solving non-triviallinear programming/semidefiniterelaxations (which may themselves invoke theellipsoid algorithm), complex data structures, or sophisticated algorithmic techniques, leading to difficult implementation issues or improved running time performance (over exact algorithms) only on impractically large inputs. Implementation and running time issues aside, the guarantees provided by approximation algorithms may themselves not be strong enough to justify their consideration in practice. Despite their inability to be used "out of the box" in practical applications, the ideas and insights behind the design of such algorithms can often be incorporated in other ways in practical algorithms. In this way, the study of even very expensive algorithms is not a completely theoretical pursuit as they can yield valuable insights.
In other cases, even if the initial results are of purely theoretical interest, over time, with an improved understanding, the algorithms may be refined to become more practical. One such example is the initial PTAS forEuclidean TSPbySanjeev Arora(and independently byJoseph Mitchell) which had a prohibitive running time ofnO(1/ϵ){\displaystyle n^{O(1/\epsilon )}}for a1+ϵ{\displaystyle 1+\epsilon }approximation.[10]Yet, within a year these ideas were incorporated into a near-linear timeO(nlogn){\displaystyle O(n\log n)}algorithm for any constantϵ>0{\displaystyle \epsilon >0}.[11]
Given an optimization problem:
Π:I×S{\displaystyle \Pi :I\times S}
whereΠ{\displaystyle \Pi }is an approximation problem,I{\displaystyle I}the set of inputs andS{\displaystyle S}the set of solutions, we can define the cost function:
c:S→R+{\displaystyle c:S\rightarrow \mathbb {R} ^{+}}
and the set of feasible solutions:
∀i∈I,S(i)=s∈S:iΠs{\displaystyle \forall i\in I,S(i)={s\in S:i\Pi _{s}}}
finding the best solutions∗{\displaystyle s^{*}}for a maximization or a minimization problem:
s∗∈S(i){\displaystyle s^{*}\in S(i)},c(s∗)=min/maxc(S(i)){\displaystyle c(s^{*})=min/max\ c(S(i))}
Given a feasible solutions∈S(i){\displaystyle s\in S(i)}, withs≠s∗{\displaystyle s\neq s^{*}}, we would want a guarantee of the quality of the solution, which is a performance to be guaranteed (approximation factor).
Specifically, havingAΠ(i)∈Si{\displaystyle A_{\Pi }(i)\in S_{i}}, the algorithm has anapproximation factor(orapproximation ratio) ofρ(n){\displaystyle \rho (n)}if∀i∈Is.t.|i|=n{\displaystyle \forall i\in I\ s.t.|i|=n}, we have:
The approximation can be proventight(tight approximation) by demonstrating that there exist instances where the algorithm performs at the approximation limit, indicating the tightness of the bound. In this case, it's enough to construct an input instance designed to force the algorithm into a worst-case scenario.
For some approximation algorithms it is possible to prove certain properties about the approximation of the optimum result. For example, aρ-approximation algorithmAis defined to be an algorithm for which it has been proven that the value/cost,f(x), of the approximate solutionA(x) to an instancexwill not be more (or less, depending on the situation) than a factorρtimes the value, OPT, of an optimum solution.
The factorρis called therelative performance guarantee. An approximation algorithm has anabsolute performance guaranteeorbounded errorc, if it has been proven for every instancexthat
Similarly, theperformance guarantee,R(x,y), of a solutionyto an instancexis defined as
wheref(y) is the value/cost of the solutionyfor the instancex. Clearly, the performance guarantee is greater than or equal to 1 and equal to 1 if and only ifyis an optimal solution. If an algorithmAguarantees to return solutions with a performance guarantee of at mostr(n), thenAis said to be anr(n)-approximation algorithm and has anapproximation ratioofr(n). Likewise, a problem with anr(n)-approximation algorithm is said to be r(n)-approximableor have an approximation ratio ofr(n).[12][13]
For minimization problems, the two different guarantees provide the same result and that for maximization problems, a relative performance guarantee of ρ is equivalent to a performance guarantee ofr=ρ−1{\displaystyle r=\rho ^{-1}}. In the literature, both definitions are common but it is clear which definition is used since, for maximization problems, as ρ ≤ 1 while r ≥ 1.
Theabsolute performance guaranteePA{\displaystyle \mathrm {P} _{A}}of some approximation algorithmA, wherexrefers to an instance of a problem, and whereRA(x){\displaystyle R_{A}(x)}is the performance guarantee ofAonx(i.e. ρ for problem instancex) is:
That is to say thatPA{\displaystyle \mathrm {P} _{A}}is the largest bound on the approximation ratio,r, that one sees over all possible instances of the problem. Likewise, theasymptotic performance ratioRA∞{\displaystyle R_{A}^{\infty }}is:
That is to say that it is the same as theabsolute performance ratio, with a lower boundnon the size of problem instances. These two types of ratios are used because there exist algorithms where the difference between these two is significant.
In the literature, an approximation ratio for a maximization (minimization) problem ofc- ϵ (min:c+ ϵ) means that the algorithm has an approximation ratio ofc∓ ϵ for arbitrary ϵ > 0 but that the ratio has not (or cannot) be shown for ϵ = 0. An example of this is the optimal inapproximability — inexistence of approximation — ratio of 7 / 8 + ϵ for satisfiableMAX-3SATinstances due toJohan Håstad.[14]As mentioned previously, whenc= 1, the problem is said to have apolynomial-time approximation scheme.
An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of sizengoes to infinity asndoes. In this case, the approximation ratio isc∓k/ OPT =c∓ o(1) for some constantscandk. Given arbitrary ϵ > 0, one can choose a large enoughNsuch that the termk/ OPT < ϵ for everyn ≥ N. For every fixed ϵ, instances of sizen < Ncan be solved by brute force, thereby showing an approximation ratio — existence of approximation algorithms with a guarantee — ofc∓ ϵ for every ϵ > 0.
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Intheoretical computer science,circuit complexityis a branch ofcomputational complexity theoryin whichBoolean functionsare classified according to the size or depth of theBoolean circuitsthat compute them. A related notion is the circuit complexity of arecursive languagethat isdecidedby auniformfamily of circuitsC1,C2,…{\displaystyle C_{1},C_{2},\ldots }(see below).
Proving lower bounds on size of Boolean circuits computing explicit Boolean functions is a popular approach to separating complexity classes. For example, aprominentcircuit classP/polyconsists of Boolean functions computable by circuits of polynomial size. Proving thatNP⊈P/poly{\displaystyle {\mathsf {NP}}\not \subseteq {\mathsf {P/poly}}}would separatePandNP(see below).
Complexity classesdefined in terms of Boolean circuits includeAC0,AC,TC0,NC1,NC, andP/poly.
A Boolean circuit withn{\displaystyle n}inputbitsis adirected acyclic graphin which every node (usually calledgatesin this context) is either an input node ofin-degree0 labelled by one of then{\displaystyle n}input bits, anAND gate, anOR gate, or aNOT gate. One of these gates is designated as the output gate. Such a circuit naturally computes a function of itsn{\displaystyle n}inputs. Thesizeof a circuit is the number of gates it contains and itsdepthis the maximal length of a path from an input gate to the output gate.
There are two major notions of circuit complexity.[1]Thecircuit-size complexityof a Boolean functionf{\displaystyle f}is the minimal size of any circuit computingf{\displaystyle f}. Thecircuit-depth complexityof a Boolean functionf{\displaystyle f}is the minimal depth of any circuit computingf{\displaystyle f}.
These notions generalize when one considers the circuit complexity of any language that contains strings with different bit lengths, especially infiniteformal languages. Boolean circuits, however, only allow a fixed number of input bits. Thus, no single Boolean circuit is capable of deciding such a language. To account for this possibility, one considers families of circuitsC1,C2,…{\displaystyle C_{1},C_{2},\ldots }where eachCn{\displaystyle C_{n}}accepts inputs of sizen{\displaystyle n}. Each circuit family will naturally generate the language by circuitCn{\displaystyle C_{n}}outputting1{\displaystyle 1}when a lengthn{\displaystyle n}string is a member of the family, and0{\displaystyle 0}otherwise. We say that a family of circuits issize minimalif there is no other family that decides on inputs of any size,n{\displaystyle n}, with a circuit of smaller size thanCn{\displaystyle C_{n}}(respectively fordepth minimalfamilies). Thus, circuit complexity is meaningful even fornon-recursive languages. The notion of auniform familyenables variants of circuit complexity to be related to algorithm based complexity measures of recursive languages. However, the non-uniform variant is helpful to find lower bounds on how complex any circuit family must be in order to decide given languages.
Hence, thecircuit-size complexityof a formal languageA{\displaystyle A}is defined as the functiont:N→N{\displaystyle t:\mathbb {N} \to \mathbb {N} }, that relates a bit length of an input,n{\displaystyle n}, to the circuit-size complexity of a minimal circuitCn{\displaystyle C_{n}}that decides whether inputs of that length are inA{\displaystyle A}. Thecircuit-depth complexityis defined similarly.
Boolean circuits are one of the prime examples of so-called non-uniformmodels of computationin the sense that inputs of different lengths are processed by different circuits, in contrast with uniform models such asTuring machineswhere the same computational device is used for all possible input lengths. An individualcomputational problemis thus associated with a particularfamilyof Boolean circuitsC1,C2,…{\displaystyle C_{1},C_{2},\dots }where eachCn{\displaystyle C_{n}}is the circuit handling inputs ofnbits. Auniformitycondition is often imposed on these families, requiring the existence of some possiblyresource-boundedTuring machine that, on inputn, produces a description of the individual circuitCn{\displaystyle C_{n}}. When this Turing machine has a running time polynomial inn, the circuit family is said to be P-uniform. The stricter requirement ofDLOGTIME-uniformity is of particular interest in the study of shallow-depth circuit-classes such as AC0or TC0. When no resource bounds are specified, a language is recursive (i.e., decidable by a Turing machine) if and only if the language is decided by a uniform family of Boolean circuits.
A family of Boolean circuits{Cn:n∈N}{\displaystyle \{C_{n}:n\in \mathbb {N} \}}ispolynomial-time uniformif there exists adeterministic Turing machineM, such that
A family of Boolean circuits{Cn:n∈N}{\displaystyle \{C_{n}:n\in \mathbb {N} \}}islogspace uniformif there exists adeterministic Turing machineM, such that
Circuit complexity goes back toShannonin 1949,[2]who proved that almost all Boolean functions onnvariables require circuits of size Θ(2n/n). Despite this fact, complexity theorists have so far been unable to prove a superlinear lower bound for any explicit function.
Superpolynomial lower bounds have been proved under certain restrictions on the family of circuits used. The first function for which superpolynomial circuit lower bounds were shown was theparity function, which computes the sum of its input bits modulo 2. The fact that parity is not contained inAC0was first established independently by Ajtai in 1983[3][4]and by Furst, Saxe and Sipser in 1984.[5]Later improvements byHåstadin 1987[6]established that any family of constant-depth circuits computing the parity function requires exponential size. Extending a result of Razborov,[7]Smolensky in 1987[8]proved that this is true even if the circuit is augmented with gates computing the sum of its input bits modulo some odd primep.
Thek-clique problemis to decide whether a given graph onnvertices has a clique of sizek. For any particular choice of the constantsnandk, the graph can be encoded in binary using(n2){\displaystyle {n \choose 2}}bits, which indicate for each possible edge whether it is present. Then thek-clique problem is formalized as a functionfk:{0,1}(n2)→{0,1}{\displaystyle f_{k}:\{0,1\}^{n \choose 2}\to \{0,1\}}such thatfk{\displaystyle f_{k}}outputs 1 if and only if the graph encoded by the string contains a clique of sizek. This family of functions is monotone and can be computed by a family of circuits, but it has been shown that it cannot be computed by a polynomial-size family of monotone circuits (that is, circuits with AND and OR gates but without negation). The original result ofRazborovin 1985[7]was later improved to an exponential-size lower bound by Alon and Boppana in 1987.[9]In 2008, Rossman[10]showed that constant-depth circuits with AND, OR, and NOT gates require sizeΩ(nk/4){\displaystyle \Omega (n^{k/4})}to solve thek-clique problem even in theaverage case. Moreover, there is a circuit of sizenk/4+O(1){\displaystyle n^{k/4+O(1)}}that computesfk{\displaystyle f_{k}}.
In 1999,RazandMcKenzielater showed that the monotone NC hierarchy is infinite.[11]
The Integer Division Problem lies in uniformTC0.[12]
Circuit lower bounds are generally difficult. Known results include
It is open whether NEXPTIME has nonuniform TC0circuits.
Proofs of circuit lower bounds are strongly connected toderandomization. A proof thatP=BPP{\displaystyle {\mathsf {P}}={\mathsf {BPP}}}would imply that eitherNEXP⊈P/poly{\displaystyle {\mathsf {NEXP}}\not \subseteq {\mathsf {P/poly}}}or that permanent cannot be computed by nonuniform arithmetic circuits (polynomials) of polynomial size and polynomial degree.[16]
In 1997, Razborov and Rudich showed that many known circuit lower bounds for explicit Boolean functions imply the existence of so callednatural propertiesuseful against the respective circuit class.[17]On the other hand, natural properties useful against P/poly would break strong pseudorandom generators. This is often interpreted as a "natural proofs" barrier for proving strong circuit lower bounds. In 2016, Carmosino, Impagliazzo, Kabanets and Kolokolova proved that natural properties can be also used to construct efficient learning algorithms.[18]
Many circuit complexity classes are defined in terms of class hierarchies. For each non-negative integeri, there is a classNCi, consisting of polynomial-size circuits of depthO(logi(n)){\displaystyle O(\log ^{i}(n))}, using boundedfan-inAND, OR, and NOT gates. The union NC of all of these classes is a subject to discussion. By considering unbounded fan-in gates, the classesACiand AC (which is equal to NC) can be constructed. Many other circuit complexity classes with the same size and depth restrictions can be constructed by allowing different sets of gates.
If a certain language,A{\displaystyle A}, belongs to thetime-complexity classTIME(t(n)){\displaystyle {\text{TIME}}(t(n))}for some functiont:N→N{\displaystyle t:\mathbb {N} \to \mathbb {N} }, thenA{\displaystyle A}has circuit complexityO(t(n)logt(n)){\displaystyle {\mathcal {O}}(t(n)\log t(n))}. If the Turing Machine that accepts the language isoblivious(meaning that it reads and writes the same memory cells regardless of input), thenA{\displaystyle A}has circuit complexityO(t(n)){\displaystyle {\mathcal {O}}(t(n))}.[19]
A monotone Boolean circuit is one that has only AND and OR gates, but no NOT gates. A monotone circuit can only compute a monotone Boolean function, which is a functionf:{0,1}n→{0,1}{\displaystyle f:\{0,1\}^{n}\to \{0,1\}}where for everyx,y∈{0,1}n{\displaystyle x,y\in \{0,1\}^{n}},x≤y⟹f(x)≤f(y){\displaystyle x\leq y\implies f(x)\leq f(y)}, wherex≤y{\displaystyle x\leq y}means thatxi≤yi{\displaystyle x_{i}\leq y_{i}}for alli∈{1,…,n}{\displaystyle i\in \{1,\ldots ,n\}}.
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https://en.wikipedia.org/wiki/Circuit_complexity
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Mathematical logicis the study offormal logicwithinmathematics. Major subareas includemodel theory,proof theory,set theory, andrecursion theory(also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establishfoundations of mathematics.
Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development ofaxiomaticframeworks forgeometry,arithmetic, andanalysis. In the early 20th century it was shaped byDavid Hilbert'sprogramto prove the consistency of foundational theories. Results ofKurt Gödel,Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as inreverse mathematics) rather than trying to find theories in which all of mathematics can be developed.
TheHandbook of Mathematical Logic[1]in 1977 makes a rough division of contemporary mathematical logic into four areas:
Additionally, sometimes the field ofcomputational complexity theoryis also included together with mathematical logic.[2][3]Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp.Gödel's incompleteness theoremmarks not only a milestone in recursion theory and proof theory, but has also led toLöb's theoremin modal logic. The method offorcingis employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics.
The mathematical field ofcategory theoryuses many formal axiomatic methods, and includes the study ofcategorical logic, but category theory is not ordinarily considered a subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians includingSaunders Mac Lanehave proposed category theory as a foundational system for mathematics, independent of set theory. These foundations usetoposes, which resemble generalized models of set theory that may employ classical or nonclassical logic.
Mathematical logic emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics.[4]Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method.[5]Before this emergence, logic was studied withrhetoric, withcalculationes,[6]through thesyllogism, and withphilosophy. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.
Theories of logic were developed in many cultures in history, including in ancientChina,India,Greece,Roman empireand theIslamic world. Greek methods, particularlyAristotelian logic(or term logic) as found in theOrganon, found wide application and acceptance in Western science and mathematics for millennia.[7]TheStoics, especiallyChrysippus, began the development ofpropositional logic. In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians includingLeibnizandLambert, but their labors remained isolated and little known.
In the middle of the nineteenth century,George Booleand thenAugustus De Morganpresented systematic mathematical treatments of logic. Their work, building on work by algebraists such asGeorge Peacock, extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study offoundations of mathematics.[8]In 1847,Vatroslav Bertićmade substantial work on algebraization of logic, independently from Boole.[9]Charles Sanders Peircelater built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885.
Gottlob Fregepresented an independent development of logic with quantifiers in hisBegriffsschrift, published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, untilBertrand Russellbegan to promote it near the turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts.
From 1890 to 1905,Ernst SchröderpublishedVorlesungen über die Algebra der Logikin three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century.
Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry.
In logic, the termarithmeticrefers to the theory of thenatural numbers.Giuseppe Peano[10]published a set of axioms for arithmetic that came to bear his name (Peano axioms), using a variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same timeRichard Dedekindshowed that the natural numbers are uniquely characterized by theirinductionproperties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms.[11]Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from thesuccessor functionand mathematical induction.
In the mid-19th century, flaws in Euclid's axioms for geometry became known.[12]In addition to the independence of theparallel postulate, established byNikolai Lobachevskyin 1826,[13]mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert[14]developed a complete set ofaxioms for geometry, building onprevious workby Pasch.[15]The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and thereal line. This would prove to be a major area of research in the first half of the 20th century.
The 19th century saw great advances in the theory ofreal analysis, including theories of convergence of functions andFourier series. Mathematicians such asKarl Weierstrassbegan to construct functions that stretched intuition, such asnowhere-differentiable continuous functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate thearithmetization of analysis, which sought to axiomatize analysis using properties of the natural numbers. The modern(ε, δ)-definition of limitandcontinuous functionswas already developed byBolzanoin 1817,[16]but remained relatively unknown.Cauchyin 1821 defined continuity in terms ofinfinitesimals(see Cours d'Analyse, page 34). In 1858, Dedekind proposed a definition of the real numbers in terms ofDedekind cutsof rational numbers, a definition still employed in contemporary texts.[17]
Georg Cantordeveloped the fundamental concepts of infinite set theory. His early results developed the theory ofcardinalityandprovedthat the reals and the natural numbers have different cardinalities.[18]Over the next twenty years, Cantor developed a theory oftransfinite numbersin a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced thediagonal argument, and used this method to proveCantor's theoremthat no set can have the same cardinality as itspowerset. Cantor believed that every set could bewell-ordered, but was unable to produce a proof for this result, leaving it as an open problem in 1895.[19]
In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency.
In 1900,Hilbertposed a famous list of23 problemsfor the next century. The first two of these were to resolve thecontinuum hypothesisand prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over theintegershas a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert'sEntscheidungsproblem, posed in 1928. This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false.
Ernst Zermelogave a proof thatevery set could be well-ordered, a resultGeorg Cantorhad been unable to obtain.[20]To achieve the proof, Zermelo introduced theaxiom of choice, which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof.[21]This paper led to the general acceptance of the axiom of choice in the mathematics community.
Skepticism about the axiom of choice was reinforced by recently discovered paradoxes innaive set theory.Cesare Burali-Forti[22]was the first to state a paradox: theBurali-Forti paradoxshows that the collection of allordinal numberscannot form a set. Very soon thereafter,Bertrand RusselldiscoveredRussell's paradoxin 1901, andJules RicharddiscoveredRichard's paradox.[23]
Zermelo provided the first set of axioms for set theory.[24]These axioms, together with the additionalaxiom of replacementproposed byAbraham Fraenkel, are now calledZermelo–Fraenkel set theory(ZF). Zermelo's axioms incorporated the principle oflimitation of sizeto avoid Russell's paradox.
In 1910, the first volume ofPrincipia Mathematicaby Russell andAlfred North Whiteheadwas published. This seminal work developed the theory of functions and cardinality in a completely formal framework oftype theory, which Russell and Whitehead developed in an effort to avoid the paradoxes.Principia Mathematicais considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics.[25]
Fraenkel[26]proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory withurelements. Later work byPaul Cohen[27]showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method offorcing, which is now an important tool for establishingindependence resultsin set theory.[28]
Leopold Löwenheim[29]andThoralf Skolem[30]obtained theLöwenheim–Skolem theorem, which says thatfirst-order logiccannot control thecardinalitiesof infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has acountablemodel. This counterintuitive fact became known asSkolem's paradox.
In his doctoral thesis,Kurt Gödelproved thecompleteness theorem, which establishes a correspondence between syntax and semantics in first-order logic.[31]Gödel used the completeness theorem to prove thecompactness theorem, demonstrating the finitary nature of first-orderlogical consequence. These results helped establish first-order logic as the dominant logic used by mathematicians.
In 1931, Gödel publishedOn Formally Undecidable Propositions of Principia Mathematica and Related Systems, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known asGödel's incompleteness theorem, establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time.[a]
Gödel's theorem shows that aconsistencyproof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using a finitistic system together with a principle oftransfinite induction.[32]Gentzen's result introduced the ideas ofcut eliminationandproof-theoretic ordinals, which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types.[33]
The first textbook on symbolic logic for the layman was written byLewis Carroll,[34]author ofAlice's Adventures in Wonderland, in 1896.[35]
Alfred Tarskideveloped the basics ofmodel theory.
Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonymNicolas Bourbakito publishÉléments de mathématique, a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as the wordsbijection,injection, andsurjection, and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics.
The study of computability came to be known as recursion theory orcomputability theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.[b]When these definitions were shown equivalent to Turing's formalization involvingTuring machines, it became clear that a new concept – thecomputable function– had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could only be implied in the original paper.
Numerous results in recursion theory were obtained in the 1940s byStephen Cole KleeneandEmil Leon Post. Kleene[36]introduced the concepts of relative computability, foreshadowed by Turing,[37]and thearithmetical hierarchy. Kleene later generalized recursion theory to higher-order functionals. Kleene andGeorg Kreiselstudied formal versions of intuitionistic mathematics, particularly in the context of proof theory.
At its core, mathematical logic deals with mathematical concepts expressed usingformal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixedformal language. The systems ofpropositional logicandfirst-order logicare the most widely studied today, because of their applicability tofoundations of mathematicsand because of their desirable proof-theoretic properties.[c]Stronger classical logics such assecond-order logicorinfinitary logicare also studied, along withNon-classical logicssuch asintuitionistic logic.
First-order logicis a particularformal system of logic. Itssyntaxinvolves only finite expressions aswell-formed formulas, while itssemanticsare characterized by the limitation of allquantifiersto a fixeddomain of discourse.
Early results from formal logic established limitations of first-order logic. TheLöwenheim–Skolem theorem(1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up toisomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark.
Gödel's completeness theoremestablished the equivalence between semantic and syntactic definitions oflogical consequencein first-order logic.[31]It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. Thecompactness theoremfirst appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development ofmodel theory, and they are a key reason for the prominence of first-order logic in mathematics.
Gödel's incompleteness theoremsestablish additional limits on first-order axiomatizations.[38]Thefirst incompleteness theoremstates that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in somenon-standard models of arithmeticwhich may be consistent with the logical system). For example, in every logical system capable of expressing thePeano axioms, the Gödel sentence holds for the natural numbers but cannot be proved.
Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are notelementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem. Thesecond incompleteness theoremstates that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show thatHilbert's programcannot be reached.
Many logics besides first-order logic are studied. These includeinfinitary logics, which allow for formulas to provide an infinite amount of information, andhigher-order logics, which include a portion of set theory directly in their semantics.
The most well studied infinitary logic isLω1,ω{\displaystyle L_{\omega _{1},\omega }}. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula ofLω1,ω{\displaystyle L_{\omega _{1},\omega }}such as
Higher-order logics allow for quantification not only of elements of thedomain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.
Another type of logics arefixed-point logicsthat allowinductive definitions, like one writes forprimitive recursive functions.
One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal orfuzzy logic.
Lindström's theoremimplies that the only extension of first-order logic satisfying both thecompactness theoremand thedownward Löwenheim–Skolem theoremis first-order logic.
Modal logicsinclude additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability[39]and set-theoretic forcing.[40]
Intuitionistic logicwas developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include thelaw of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic iscomputable; this is not true in classical theories of arithmetic such asPeano arithmetic.
Algebraic logicuses the methods ofabstract algebrato study the semantics of formal logics. A fundamental example is the use ofBoolean algebrasto representtruth valuesin classical propositional logic, and the use ofHeyting algebrasto represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such ascylindric algebras.
Set theoryis the study ofsets, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. Thefirst such axiomatization, due to Zermelo,[24]was extended slightly to becomeZermelo–Fraenkel set theory(ZF), which is now the most widely used foundational theory for mathematics.
Other formalizations of set theory have been proposed, includingvon Neumann–Bernays–Gödel set theory(NBG),Morse–Kelley set theory(MK), andNew Foundations(NF). Of these, ZF, NBG, and MK are similar in describing acumulative hierarchyof sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system ofKripke–Platek set theoryis closely related to generalized recursion theory.
Two famous statements in set theory are theaxiom of choiceand thecontinuum hypothesis. The axiom of choice, first stated by Zermelo,[20]was proved independent of ZF by Fraenkel,[26]but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single setCthat contains exactly one element from each set in the collection. The setCis said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive.Stefan BanachandAlfred Tarskishowed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size.[41]This theorem, known as theBanach–Tarski paradox, is one of many counterintuitive results of the axiom of choice.
The continuum hypothesis, first proposed as a conjecture by Cantor, was listed byDavid Hilbertas one of his 23 problems in 1900. Gödel showed that the continuum hypothesis cannot be disproven from the axioms of Zermelo–Fraenkel set theory (with or without the axiom of choice), by developing theconstructible universeof set theory in which the continuum hypothesis must hold. In 1963,Paul Cohenshowed that the continuum hypothesis cannot be proven from the axioms of Zermelo–Fraenkel set theory.[27]This independence result did not completely settle Hilbert's question, however, as it is possible that new axioms for set theory could resolve the hypothesis. Recent work along these lines has been conducted byW. Hugh Woodin, although its importance is not yet clear.[42]
Contemporary research in set theory includes the study oflarge cardinalsanddeterminacy. Large cardinals arecardinal numberswith particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, aninaccessible cardinal, already implies the consistency of ZFC. Despite the fact that large cardinals have extremely highcardinality, their existence has many ramifications for the structure of the real line.Determinacyrefers to the possible existence of winning strategies for certain two-player games (the games are said to bedetermined). The existence of these strategies implies structural properties of the real line and otherPolish spaces.
Model theorystudies the models of various formal theories. Here atheoryis a set of formulas in a particular formal logic andsignature, while amodelis a structure that gives a concrete interpretation of the theory. Model theory is closely related touniversal algebraandalgebraic geometry, although the methods of model theory focus more on logical considerations than those fields.
The set of all models of a particular theory is called anelementary class; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes.
The method ofquantifier eliminationcan be used to show that definable sets in particular theories cannot be too complicated. Tarski established quantifier elimination forreal-closed fields, a result which also shows the theory of the field of real numbers isdecidable.[43]He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic. A modern subfield developing from this is concerned witho-minimal structures.
Morley's categoricity theorem, proved byMichael D. Morley,[44]states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e. all models of this cardinality are isomorphic, then it is categorical in all uncountable cardinalities.
A trivial consequence of thecontinuum hypothesisis that a complete theory with less than continuum many nonisomorphic countable models can have only countably many.Vaught's conjecture, named afterRobert Lawson Vaught, says that this is true even independently of the continuum hypothesis. Many special cases of this conjecture have been established.
Recursion theory, also calledcomputability theory, studies the properties ofcomputable functionsand theTuring degrees, which divide the uncomputable functions into sets that have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Recursion theory grew from the work ofRózsa Péter,Alonzo ChurchandAlan Turingin the 1930s, which was greatly extended byKleeneandPostin the 1940s.[45]
Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers. The fundamental results establish a robust, canonical class of computable functions with numerous independent, equivalent characterizations usingTuring machines,λ calculus, and other systems. More advanced results concern the structure of the Turing degrees and thelatticeofrecursively enumerable sets.
Generalized recursion theory extends the ideas of recursion theory to computations that are no longer necessarily finite. It includes the study of computability in higher types as well as areas such ashyperarithmetical theoryandα-recursion theory.
Contemporary research in recursion theory includes the study of applications such asalgorithmic randomness,computable model theory, andreverse mathematics, as well as new results in pure recursion theory.
An important subfield of recursion theory studies algorithmic unsolvability; adecision problemorfunction problemisalgorithmically unsolvableif there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that theEntscheidungsproblemis algorithmically unsolvable. Turing proved this by establishing the unsolvability of thehalting problem, a result with far-ranging implications in both recursion theory and computer science.
There are many known examples of undecidable problems from ordinary mathematics. Theword problem for groupswas proved algorithmically unsolvable byPyotr Novikovin 1955 and independently by W. Boone in 1959. Thebusy beaverproblem, developed byTibor Radóin 1962, is another well-known example.
Hilbert's tenth problemasked for an algorithm to determine whether a multivariate polynomial equation with integer coefficients has a solution in the integers. Partial progress was made byJulia Robinson,Martin DavisandHilary Putnam. The algorithmic unsolvability of the problem was proved byYuri Matiyasevichin 1970.[46]
Proof theoryis the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques. Several deduction systems are commonly considered, includingHilbert-style deduction systems, systems ofnatural deduction, and thesequent calculusdeveloped by Gentzen.
The study ofconstructive mathematics, in the context of mathematical logic, includes the study of systems in non-classical logic such as intuitionistic logic, as well as the study ofpredicativesystems. An early proponent of predicativism wasHermann Weyl, who showed it is possible to develop a large part of real analysis using only predicative methods.[47]
Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical (or nonconstructive) systems and provability in intuitionistic (or constructive, respectively) systems is of particular interest. Results such as theGödel–Gentzen negative translationshow that it is possible to embed (ortranslate) classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs.
Recent developments in proof theory include the study ofproof miningbyUlrich Kohlenbachand the study ofproof-theoretic ordinalsbyMichael Rathjen.
"Mathematical logic has been successfully applied not only to mathematics and its foundations (G. Frege,B. Russell,D. Hilbert,P. Bernays,H. Scholz,R. Carnap,S. Lesniewski,T. Skolem), but also to physics (R. Carnap, A. Dittrich, B. Russell,C. E. Shannon,A. N. Whitehead,H. Reichenbach, P. Fevrier), to biology (J. H. Woodger,A. Tarski), to psychology (F. B. Fitch,C. G. Hempel), to law and morals (K. Menger, U. Klug, P. Oppenheim), to economics (J. Neumann,O. Morgenstern), to practical questions (E. C. Berkeley, E. Stamm), and even to metaphysics (J. [Jan] Salamucha, H. Scholz,J. M. Bochenski). Its applications to the history of logic have proven extremely fruitful (J. Lukasiewicz, H. Scholz,B. Mates, A. Becker,E. Moody, J. Salamucha, K. Duerr, Z. Jordan,P. Boehner, J. M. Bochenski, S. [Stanislaw] T. Schayer,D. Ingalls)."[48]"Applications have also been made to theology (F. Drewnowski, J. Salamucha, I. Thomas)."[48]
The study ofcomputability theory in computer scienceis closely related to the study of computability in mathematical logic. There is a difference of emphasis, however.Computer scientistsoften focus on concrete programming languages andfeasible computability, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability.
The theory ofsemantics of programming languagesis related tomodel theory, as isprogram verification(in particular,model checking). TheCurry–Howard correspondencebetween proofs and programs relates toproof theory, especiallyintuitionistic logic. Formal calculi such as thelambda calculusandcombinatory logicare now studied as idealizedprogramming languages.
Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such asautomated theorem provingandlogic programming.
Descriptive complexity theoryrelates logics tocomputational complexity. The first significant result in this area,Fagin's theorem(1974) established thatNPis precisely the set of languages expressible by sentences of existentialsecond-order logic.
In the 19th century, mathematicians became aware of logical gaps and inconsistencies in their field. It was shown thatEuclid's axioms for geometry, which had been taught for centuries as an example of the axiomatic method, were incomplete. The use ofinfinitesimals, and the very definition offunction, came into question in analysis, as pathological examples such as Weierstrass' nowhere-differentiablecontinuous function were discovered.
Cantor's study of arbitrary infinite sets also drew criticism.Leopold Kroneckerfamously stated "God made the integers; all else is the work of man," endorsing a return to the study of finite, concrete objects in mathematics. Although Kronecker's argument was carried forward by constructivists in the 20th century, the mathematical community as a whole rejected them.David Hilbertargued in favor of the study of the infinite, saying "No one shall expel us from the Paradise that Cantor has created."
Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics. In addition to removing ambiguity from previously naive terms such as function, it was hoped that this axiomatization would allow for consistency proofs. In the 19th century, the main method of proving the consistency of a set of axioms was to provide a model for it. Thus, for example,non-Euclidean geometrycan be proved consistent by definingpointto mean a point on a fixed sphere andlineto mean agreat circleon the sphere. The resulting structure, a model ofelliptic geometry, satisfies the axioms of plane geometry except the parallel postulate.
With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis that it is impossible to prove a contradiction. This idea led to the study ofproof theory. Moreover, Hilbert proposed that the analysis should be entirely concrete, using the termfinitaryto refer to the methods he would allow but not precisely defining them. This project, known asHilbert's program, was seriously affected by Gödel's incompleteness theorems, which show that the consistency of formal theories of arithmetic cannot be established using methods formalizable in those theories. Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms oftransfinite induction, and the techniques he developed to do so were seminal in proof theory.
A second thread in the history of foundations of mathematics involvesnonclassical logicsandconstructive mathematics. The study of constructive mathematics includes many different programs with various definitions ofconstructive. At the most accommodating end, proofs in ZF set theory that do not use the axiom of choice are called constructive by many mathematicians. More limited versions of constructivism limit themselves tonatural numbers,number-theoretic functions, and sets of natural numbers (which can be used to represent real numbers, facilitating the study ofmathematical analysis). A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist.
In the early 20th century,Luitzen Egbertus Jan Brouwerfoundedintuitionismas a part ofphilosophy of mathematics. This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able tointuitthe statement, to not only believe its truth but understand the reason for its truth. A consequence of this definition of truth was the rejection of thelaw of the excluded middle, for there are statements that, according to Brouwer, could not be claimed to be true while their negations also could not be claimed true. Brouwer's philosophy was influential, and the cause of bitter disputes among prominent mathematicians. Kleene and Kreisel would later study formalized versions of intuitionistic logic (Brouwer rejected formalization, and presented his work in unformalized natural language). With the advent of theBHK interpretationandKripke models, intuitionism became easier to reconcile with classical mathematics.
"Die Ausführung dieses Vorhabens hat eine wesentliche Verzögerung dadurch erfahren, daß in einem Stadium, in dem die Darstellung schon ihrem Abschuß nahe war, durch das Erscheinen der Arbeiten von Herbrand und von Gödel eine veränderte Situation im Gebiet der Beweistheorie entstand, welche die Berücksichtigung neuer Einsichten zur Aufgabe machte. Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien."
"Carrying out this plan [by Hilbert for an exposition on proof theory for mathematical logic] has experienced an essential delay because, at the stage at which the exposition was already near to its conclusion, there occurred an altered situation in the area of proof theory due to the appearance of works by Herbrand and Gödel, which necessitated the consideration of new insights. Thus the scope of this book has grown, so that a division into two volumes seemed advisable."
Bochenski, Jozef Maria, ed. (1959).A Precis of Mathematical Logic. Synthese Library, Vol. 1. Translated by Otto Bird.Dordrecht:Springer.doi:10.1007/978-94-017-0592-9.ISBN9789048183296.{{cite book}}:ISBN / Date incompatibility (help)
Cantor, Georg (1874)."Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"(PDF).Journal für die Reine und Angewandte Mathematik.1874(77):258–262.doi:10.1515/crll.1874.77.258.S2CID199545885.Carroll, Lewis(1896).Symbolic Logic. Kessinger Legacy Reprints.ISBN9781163444955.{{cite book}}:ISBN / Date incompatibility (help)
Soare, Robert Irving (22 December 2011)."Computability Theory and Applications: The Art of Classical Computability"(PDF).Department of Mathematics. University of Chicago. Retrieved23 August2017.Swineshead, Richard(1498).Calculationes Suiseth Anglici(in Lithuanian). Papie: Per Franciscum Gyrardengum.{{cite book}}: CS1 maint: publisher location (link)
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Proof theoryis a major branch[1]ofmathematical logicandtheoretical computer sciencewithin whichproofsare treated as formalmathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented asinductively defineddata structuressuch aslists, boxed lists, ortrees, which are constructed according to theaxiomsandrules of inferenceof a given logical system. Consequently, proof theory issyntacticin nature, in contrast tomodel theory, which issemanticin nature.
Some of the major areas of proof theory includestructural proof theory,ordinal analysis,provability logic,reverse mathematics,proof mining,automated theorem proving, andproof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy.
Although the formalisation of logic was much advanced by the work of such figures asGottlob Frege,Giuseppe Peano,Bertrand Russell, andRichard Dedekind, the story of modern proof theory is often seen as being established byDavid Hilbert, who initiated what is calledHilbert's programin theFoundations of Mathematics. The central idea of this program was that if we could givefinitaryproofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provableΠ10{\displaystyle \Pi _{1}^{0}}sentences) are finitarily true; once so grounded we do not care about the non-finitary meaning of their existential theorems, regarding these as pseudo-meaningful stipulations of the existence of ideal entities.
The failure of the program was induced byKurt Gödel'sincompleteness theorems, which showed that anyω-consistent theorythat is sufficiently strong to express certain simple arithmetic truths, cannot prove its own consistency, which on Gödel's formulation is aΠ10{\displaystyle \Pi _{1}^{0}}sentence. However, modified versions of Hilbert's program emerged and research has been carried out on related topics. This has led, in particular, to:
In parallel to the rise and fall of Hilbert's program, the foundations ofstructural proof theorywere being founded.Jan Łukasiewiczsuggested in 1926 that one could improve onHilbert systemsas a basis for the axiomatic presentation of logic if one allowed the drawing of conclusions from assumptions in the inference rules of the logic. In response to this,Stanisław Jaśkowski(1929) andGerhard Gentzen(1934) independently provided such systems, called calculi ofnatural deduction, with Gentzen's approach introducing the idea of symmetry between the grounds for asserting propositions, expressed inintroduction rules, and the consequences of accepting propositions in theelimination rules, an idea that has proved very important in proof theory.[2]Gentzen (1934) further introduced the idea of thesequent calculus, a calculus advanced in a similar spirit that better expressed the duality of the logical connectives,[3]and went on to make fundamental advances in the formalisation of intuitionistic logic, and provide the firstcombinatorial proofof the consistency ofPeano arithmetic. Together, the presentation of natural deduction and the sequent calculus introduced the fundamental idea ofanalytic proofto proof theory.
Structural proof theory is the subdiscipline of proof theory that studies the specifics ofproof calculi. The three most well-known styles of proof calculi are:
Each of these can give a complete and axiomatic formalization ofpropositionalorpredicate logicof either theclassicalorintuitionisticflavour, almost anymodal logic, and manysubstructural logics, such asrelevance logicorlinear logic. Indeed, it is unusual to find a logic that resists being represented in one of these calculi.
Proof theorists are typically interested in proof calculi that support a notion ofanalytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that arecut-free. Much of the interest in cut-free proofs comes from thesubformula property: every formula in the end sequent of a cut-free proof is a subformula of one of the premises. This allows one to show consistency of the sequent calculus easily; if the empty sequent were derivable it would have to be a subformula of some premise, which it is not. Gentzen's midsequent theorem, the Craig interpolation theorem, and Herbrand's theorem also follow as corollaries of the cut-elimination theorem.
Gentzen's natural deduction calculus also supports a notion of analytic proof, as shown byDag Prawitz. The definition is slightly more complex: we say the analytic proofs are thenormal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such asJean-Yves Girard'sproof netsalso support a notion of analytic proof.
A particular family of analytic proofs arising inreductive logicarefocused proofswhich characterise a large family of goal-directed proof-search procedures. The ability to transform a proof system into a focused form is a good indication of its syntactic quality, in a manner similar to how admissibility of cut shows that a proof system is syntactically consistent.[4]
Structural proof theory is connected totype theoryby means of theCurry–Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in thetyped lambda calculus. This provides the foundation for theintuitionistic type theorydeveloped byPer Martin-Löf, and is often extended to a three way correspondence, the third leg of which are thecartesian closed categories.
Other research topics in structural theory include analytic tableau, which apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics, and the proof theory of substructural logics.
Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for subsystems of arithmetic, analysis, and set theory.Gödel's second incompleteness theoremis often interpreted as demonstrating that finitistic consistency proofs are impossible for theories of sufficient strength. Ordinal analysis allows one to measure precisely the infinitary content of the consistency of theories. For a consistent recursively axiomatized theory T, one can prove in finitistic arithmetic that the well-foundedness of a certain transfinite ordinal implies the consistency of T. Gödel's second incompleteness theorem implies that the well-foundedness of such an ordinal cannot be proved in the theory T.
Consequences of ordinal analysis include (1) consistency of subsystems of classical second order arithmetic and set theory relative to constructive theories, (2) combinatorial independence results, and (3) classifications of provably total recursive functions and provably well-founded ordinals.
Ordinal analysis was originated by Gentzen, who proved the consistency of Peano Arithmetic usingtransfinite inductionup to ordinal ε0. Ordinal analysis has been extended to many fragments of first and second order arithmetic and set theory. One major challenge has been the ordinal analysis of impredicative theories. The first breakthrough in this direction was Takeuti's proof of the consistency of Π11-CA0using the method of ordinal diagrams.
Provability logicis amodal logic, in which the box operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably richformal theory. As basic axioms of the provability logic GL (Gödel-Löb), which captures provable inPeano Arithmetic, one takes modal analogues of the Hilbert-Bernays derivability conditions andLöb's theorem(if it is provable that the provability of A implies A, then A is provable).
Some of the basic results concerning the incompleteness of Peano Arithmetic and related theories have analogues in provability logic. For example, it is a theorem in GL that if a contradiction is not provable then it is not provable that a contradiction is not provable (Gödel's second incompleteness theorem). There are also modal analogues of the fixed-point theorem.Robert Solovayproved that the modal logic GL is complete with respect to Peano Arithmetic. That is, the propositional theory of provability in Peano Arithmetic is completely represented by the modal logic GL. This straightforwardly implies that propositional reasoning about provability in Peano Arithmetic is complete and decidable.
Other research in provability logic has focused on first-order provability logic,polymodal provability logic(with one modality representing provability in the object theory and another representing provability in the meta-theory), andinterpretability logicsintended to capture the interaction between provability and interpretability. Some very recent research has involved applications of graded provability algebras to the ordinal analysis of arithmetical theories.
Reverse mathematics is a program inmathematical logicthat seeks to determine which axioms are required to prove theorems of mathematics.[5]The field was founded byHarvey Friedman. Its defining method can be described as "going backwards from thetheoremsto theaxioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that theaxiom of choiceandZorn's lemmaare equivalent overZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.
In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems. For example, to study the theorem "Every bounded sequence ofreal numbershas asupremum" it is necessary to use a base system that can speak of real numbers and sequences of real numbers.
For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system (stronger than the base system) that is necessary to prove that theorem. To show that a systemSis required to prove a theoremT, two proofs are required. The first proof showsTis provable fromS; this is an ordinary mathematical proof along with a justification that it can be carried out in the systemS. The second proof, known as areversal, shows thatTitself impliesS; this proof is carried out in the base system. The reversal establishes that no axiom systemS′that extends the base system can be weaker thanSwhile still provingT.
One striking phenomenon in reverse mathematics is the robustness of theBig Fiveaxiom systems. In order of increasing strength, these systems are named by the initialisms RCA0, WKL0, ACA0, ATR0, and Π11-CA0. Nearly every theorem of ordinary mathematics that has been reverse mathematically analyzed has been proven equivalent to one of these five systems. Much recent research has focused on combinatorial principles that do not fit neatly into this framework, like RT22(Ramsey's theorem for pairs).
Research in reverse mathematics often incorporates methods and techniques fromrecursion theoryas well as proof theory.
Functional interpretations are interpretations of non-constructive theories in functional ones. Functional interpretations usually proceed in two stages. First, one "reduces" a classical theory C to an intuitionistic one I. That is, one provides a constructive mapping that translates the theorems of C to the theorems of I. Second, one reduces the intuitionistic theory I to a quantifier free theory of functionals F. These interpretations contribute to a form of Hilbert's program, since they prove the consistency of classical theories relative to constructive ones. Successful functional interpretations have yielded reductions of infinitary theories to finitary theories and impredicative theories to predicative ones.
Functional interpretations also provide a way to extract constructive information from proofs in the reduced theory. As a direct consequence of the interpretation one usually obtains the result that any recursive function whose totality can be proven either in I or in C is represented by a term of F. If one can provide an additional interpretation of F in I, which is sometimes possible, this characterization is in fact usually shown to be exact. It often turns out that the terms of F coincide with a natural class of functions, such as the primitive recursive or polynomial-time computable functions. Functional interpretations have also been used to provide ordinal analyses of theories and classify their provably recursive functions.
The study of functional interpretations began with Kurt Gödel's interpretation of intuitionistic arithmetic in a quantifier-free theory of functionals of finite type. This interpretation is commonly known as theDialectica interpretation. Together with the double-negation interpretation of classical logic in intuitionistic logic, it provides a reduction of classical arithmetic to intuitionistic arithmetic.
Theinformalproofs of everyday mathematical practice are unlike theformalproofs of proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience. For most mathematicians, writing a fully formal proof is too pedantic and long-winded to be in common use.
Formal proofs are constructed with the help of computers ininteractive theorem proving.
Significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, whereasfindingproofs (automated theorem proving) is generally hard. An informal proof in the mathematics literature, by contrast, requires weeks ofpeer reviewto be checked, and may still contain errors.
Inlinguistics,type-logical grammar,categorial grammarandMontague grammarapply formalisms based on structural proof theory to give a formalnatural language semantics.
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Incomputational complexity theory, acomplexity classis asetofcomputational problems"of related resource-basedcomplexity".[1]The two most commonly analyzed resources aretimeandmemory.
In general, a complexity class is defined in terms of a type of computational problem, amodel of computation, and a bounded resource liketimeormemory. In particular, most complexity classes consist ofdecision problemsthat are solvable with aTuring machine, and are differentiated by their time or space (memory) requirements. For instance, the classPis the set of decision problems solvable by a deterministic Turing machine inpolynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g.counting problemsandfunction problems) and using other models of computation (e.g.probabilistic Turing machines,interactive proof systems,Boolean circuits, andquantum computers).
The study of the relationships between complexity classes is a major area of research intheoretical computer science. There are often general hierarchies of complexity classes; for example, it is known that a number of fundamental time and space complexity classes relate to each other in the following way:L⊆NL⊆P⊆NP⊆PSPACE⊆EXPTIME⊆NEXPTIME⊆EXPSPACE(where ⊆ denotes thesubsetrelation). However, many relationships are not yet known; for example, one of the most famousopen problemsin computer science concerns whetherPequalsNP. The relationships between classes often answer questions about the fundamental nature of computation. ThePversusNPproblem, for instance, is directly related to questions of whethernondeterminismadds any computational power to computers and whether problems having solutions that can be quickly checked for correctness can also be quickly solved.
Complexity classes aresetsof relatedcomputational problems. They are defined in terms of the computational difficulty of solving the problems contained within them with respect to particular computational resources like time or memory. More formally, the definition of a complexity class consists of three things: a type of computational problem, a model of computation, and a bounded computational resource. In particular, most complexity classes consist ofdecision problemsthat can be solved by aTuring machinewith boundedtimeorspaceresources. For example, the complexity classPis defined as the set ofdecision problemsthat can be solved by adeterministic Turing machineinpolynomial time.
Intuitively, acomputational problemis just a question that can be solved by analgorithm. For example, "is thenatural numbern{\displaystyle n}prime?" is a computational problem. A computational problem is mathematically represented as thesetof answers to the problem. In the primality example, the problem (call itPRIME{\displaystyle {\texttt {PRIME}}}) is represented by the set of all natural numbers that are prime:PRIME={n∈N|nis prime}{\displaystyle {\texttt {PRIME}}=\{n\in \mathbb {N} |n{\text{ is prime}}\}}. In the theory of computation, these answers are represented asstrings; for example, in the primality example the natural numbers could be represented as strings ofbitsthat representbinary numbers. For this reason, computational problems are often synonymously referred to as languages, since strings of bits representformal languages(a concept borrowed fromlinguistics); for example, saying that thePRIME{\displaystyle {\texttt {PRIME}}}problem is in the complexity classPis equivalent to saying that the languagePRIME{\displaystyle {\texttt {PRIME}}}is inP.
The most commonly analyzed problems in theoretical computer science aredecision problems—the kinds of problems that can be posed asyes–no questions. The primality example above, for instance, is an example of a decision problem as it can be represented by the yes–no question "is thenatural numbern{\displaystyle n}prime". In terms of the theory of computation, a decision problem is represented as the set of input strings that a computer running a correctalgorithmwould answer "yes" to. In the primality example,PRIME{\displaystyle {\texttt {PRIME}}}is the set of strings representing natural numbers that, when input into a computer running an algorithm that correctlytests for primality, the algorithm answers "yes, this number is prime". This "yes-no" format is often equivalently stated as "accept-reject"; that is, an algorithm "accepts" an input string if the answer to the decision problem is "yes" and "rejects" if the answer is "no".
While some problems cannot easily be expressed as decision problems, they nonetheless encompass a broad range of computational problems.[2]Other types of problems that certain complexity classes are defined in terms of include:
To make concrete the notion of a "computer", in theoretical computer science problems are analyzed in the context of acomputational model. Computational models make exact the notions of computational resources like "time" and "memory". Incomputational complexity theory, complexity classes deal with theinherentresource requirements of problems and not the resource requirements that depend upon how a physical computer is constructed. For example, in the real world different computers may require different amounts of time and memory to solve the same problem because of the way that they have been engineered. By providing an abstract mathematical representations of computers, computational models abstract away superfluous complexities of the real world (like differences inprocessorspeed) that obstruct an understanding of fundamental principles.
The most commonly used computational model is theTuring machine. While other models exist and many complexity classes are defined in terms of them (see section"Other models of computation"), the Turing machine is used to define most basic complexity classes. With the Turing machine, instead of using standard units of time like the second (which make it impossible to disentangle running time from the speed of physical hardware) and standard units of memory likebytes, the notion of time is abstracted as the number of elementary steps that a Turing machine takes to solve a problem and the notion of memory is abstracted as the number of cells that are used on the machine's tape. These are explained in greater detail below.
It is also possible to use theBlum axiomsto define complexity classes without referring to a concretecomputational model, but this approach is less frequently used in complexity theory.
ATuring machineis a mathematical model of a general computing machine. It is the most commonly used model in complexity theory, owing in large part to the fact that it is believed to be as powerful as any other model of computation and is easy to analyze mathematically. Importantly, it is believed that if there exists an algorithm that solves a particular problem then there also exists a Turing machine that solves that same problem (this is known as theChurch–Turing thesis); this means that it is believed thateveryalgorithm can be represented as a Turing machine.
Mechanically, a Turing machine (TM) manipulates symbols (generally restricted to the bits 0 and 1 to provide an intuitive connection to real-life computers) contained on an infinitely long strip of tape. The TM can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6". The Turing machine starts with only the input string on its tape and blanks everywhere else. The TM accepts the input if it enters a designated accept state and rejects the input if it enters a reject state. The deterministic Turing machine (DTM) is the most basic type of Turing machine. It uses a fixed set of rules to determine its future actions (which is why it is called "deterministic").
A computational problem can then be defined in terms of a Turing machine as the set of input strings that a particular Turing machine accepts. For example, the primality problemPRIME{\displaystyle {\texttt {PRIME}}}from above is the set of strings (representing natural numbers) that a Turing machine running an algorithm that correctlytests for primalityaccepts. A Turing machine is said torecognizea language (recall that "problem" and "language" are largely synonymous in computability and complexity theory) if it accepts all inputs that are in the language and is said todecidea language if it additionally rejects all inputs that are not in the language (certain inputs may cause a Turing machine to run forever, sodecidabilityplaces the additional constraint overrecognizabilitythat the Turing machine must halt on all inputs). A Turing machine that "solves" a problem is generally meant to mean one that decides the language.
Turing machines enable intuitive notions of "time" and "space". Thetime complexityof a TM on a particular input is the number of elementary steps that the Turing machine takes to reach either an accept or reject state. Thespace complexityis the number of cells on its tape that it uses to reach either an accept or reject state.
The deterministic Turing machine (DTM) is a variant of the nondeterministic Turing machine (NTM). Intuitively, an NTM is just a regular Turing machine that has the added capability of being able to explore multiple possible future actions from a given state, and "choosing" a branch that accepts (if any accept). That is, while a DTM must follow only one branch of computation, an NTM can be imagined as a computation tree, branching into many possible computational pathways at each step (see image). If at least one branch of the tree halts with an "accept" condition, then the NTM accepts the input. In this way, an NTM can be thought of as simultaneously exploring all computational possibilities in parallel and selecting an accepting branch.[3]NTMs are not meant to be physically realizable models, they are simply theoretically interesting abstract machines that give rise to a number of interesting complexity classes (which often do have physically realizable equivalent definitions).
Thetime complexityof an NTM is the maximum number of steps that the NTM uses onanybranch of its computation.[4]Similarly, thespace complexityof an NTM is the maximum number of cells that the NTM uses on any branch of its computation.
DTMs can be viewed as a special case of NTMs that do not make use of the power of nondeterminism. Hence, every computation that can be carried out by a DTM can also be carried out by an equivalent NTM. It is also possible to simulate any NTM using a DTM (the DTM will simply compute every possible computational branch one-by-one). Hence, the two are equivalent in terms of computability. However, simulating an NTM with a DTM often requires greater time and/or memory resources; as will be seen, how significant this slowdown is for certain classes of computational problems is an important question in computational complexity theory.
Complexity classes group computational problems by their resource requirements. To do this, computational problems are differentiated byupper boundson the maximum amount of resources that the most efficient algorithm takes to solve them. More specifically, complexity classes are concerned with therate of growthin the resources required to solve particular computational problems as the input size increases. For example, the amount of time it takes to solve problems in the complexity classPgrows at apolynomialrate as the input size increases, which is comparatively slow compared to problems in the exponential complexity classEXPTIME(or more accurately, for problems inEXPTIMEthat are outside ofP, sinceP⊆EXPTIME{\displaystyle {\mathsf {P}}\subseteq {\mathsf {EXPTIME}}}).
Note that the study of complexity classes is intended primarily to understand theinherentcomplexity required to solve computational problems. Complexity theorists are thus generally concerned with finding the smallest complexity class that a problem falls into and are therefore concerned with identifying which class a computational problem falls into using themost efficientalgorithm. There may be an algorithm, for instance, that solves a particular problem in exponential time, but if the most efficient algorithm for solving this problem runs in polynomial time then the inherent time complexity of that problem is better described as polynomial.
Thetime complexityof an algorithm with respect to the Turing machine model is the number of steps it takes for a Turing machine to run an algorithm on a given input size. Formally, the time complexity for an algorithm implemented with a Turing machineM{\displaystyle M}is defined as the functiontM:N→N{\displaystyle t_{M}:\mathbb {N} \to \mathbb {N} }, wheretM(n){\displaystyle t_{M}(n)}is the maximum number of steps thatM{\displaystyle M}takes on any input of lengthn{\displaystyle n}.
In computational complexity theory, theoretical computer scientists are concerned less with particular runtime values and more with the general class of functions that the time complexity function falls into. For instance, is the time complexity function apolynomial? Alogarithmic function? Anexponential function? Or another kind of function?
Thespace complexityof an algorithm with respect to the Turing machine model is the number of cells on the Turing machine's tape that are required to run an algorithm on a given input size. Formally, the space complexity of an algorithm implemented with a Turing machineM{\displaystyle M}is defined as the functionsM:N→N{\displaystyle s_{M}:\mathbb {N} \to \mathbb {N} }, wheresM(n){\displaystyle s_{M}(n)}is the maximum number of cells thatM{\displaystyle M}uses on any input of lengthn{\displaystyle n}.
Complexity classes are often defined using granular sets of complexity classes calledDTIMEandNTIME(for time complexity) andDSPACEandNSPACE(for space complexity). Usingbig O notation, they are defined as follows:
Pis the class of problems that are solvable by adeterministic Turing machineinpolynomial timeandNPis the class of problems that are solvable by anondeterministic Turing machinein polynomial time. Or more formally,
Pis often said to be the class of problems that can be solved "quickly" or "efficiently" by a deterministic computer, since thetime complexityof solving a problem inPincreases relatively slowly with the input size.
An important characteristic of the classNPis that it can be equivalently defined as the class of problems whose solutions areverifiableby a deterministic Turing machine in polynomial time. That is, a language is inNPif there exists adeterministicpolynomial time Turing machine, referred to as the verifier, that takes as input a stringw{\displaystyle w}anda polynomial-sizecertificatestringc{\displaystyle c}, and acceptsw{\displaystyle w}ifw{\displaystyle w}is in the language and rejectsw{\displaystyle w}ifw{\displaystyle w}is not in the language. Intuitively, the certificate acts as aproofthat the inputw{\displaystyle w}is in the language. Formally:[5]
This equivalence between the nondeterministic definition and the verifier definition highlights a fundamental connection betweennondeterminismand solution verifiability. Furthermore, it also provides a useful method for proving that a language is inNP—simply identify a suitable certificate and show that it can be verified in polynomial time.
While there might seem to be an obvious difference between the class of problems that are efficiently solvable and the class of problems whose solutions are merely efficiently checkable,PandNPare actually at the center of one of the most famous unsolved problems in computer science: thePversusNPproblem. While it is known thatP⊆NP{\displaystyle {\mathsf {P}}\subseteq {\mathsf {NP}}}(intuitively, deterministic Turing machines are just a subclass of nondeterministic Turing machines that don't make use of their nondeterminism; or under the verifier definition,Pis the class of problems whose polynomial time verifiers need only receive the empty string as their certificate), it is not known whetherNPis strictly larger thanP. IfP=NP, then it follows that nondeterminism providesno additional computational powerover determinism with regards to the ability to quickly find a solution to a problem; that is, being able to exploreall possible branchesof computation providesat mosta polynomial speedup over being able to explore only a single branch. Furthermore, it would follow that if there exists a proof for a problem instance and that proof can be quickly be checked for correctness (that is, if the problem is inNP), then there also exists an algorithm that can quicklyconstructthat proof (that is, the problem is inP).[6]However, the overwhelming majority of computer scientists believe thatP≠NP{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}},[7]and mostcryptographic schemesemployed today rely on the assumption thatP≠NP{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}.[8]
EXPTIME(sometimes shortened toEXP) is the class of decision problems solvable by a deterministic Turing machine in exponential time andNEXPTIME(sometimes shortened toNEXP) is the class of decision problems solvable by a nondeterministic Turing machine in exponential time. Or more formally,
EXPTIMEis a strict superset ofPandNEXPTIMEis a strict superset ofNP. It is further the case thatEXPTIME⊆{\displaystyle \subseteq }NEXPTIME. It is not known whether this is proper, but ifP=NPthenEXPTIMEmust equalNEXPTIME.
While it is possible to definelogarithmictime complexity classes, these are extremely narrow classes as sublinear times do not even enable a Turing machine to read the entire input (becauselogn<n{\displaystyle \log n<n}).[a][9]However, there are a meaningful number of problems that can be solved in logarithmic space. The definitions of these classes require atwo-tape Turing machineso that it is possible for the machine to store the entire input (it can be shown that in terms ofcomputabilitythe two-tape Turing machine is equivalent to the single-tape Turing machine).[10]In the two-tape Turing machine model, one tape is the input tape, which is read-only. The other is the work tape, which allows both reading and writing and is the tape on which the Turing machine performs computations. The space complexity of the Turing machine is measured as the number of cells that are used on the work tape.
L(sometimes lengthened toLOGSPACE) is then defined as the class of problems solvable in logarithmic space on a deterministic Turing machine andNL(sometimes lengthened toNLOGSPACE) is the class of problems solvable in logarithmic space on a nondeterministic Turing machine. Or more formally,[10]
It is known thatL⊆NL⊆P{\displaystyle {\mathsf {L}}\subseteq {\mathsf {NL}}\subseteq {\mathsf {P}}}. However, it is not known whether any of these relationships is proper.
The complexity classesPSPACEandNPSPACEare the space analogues toPandNP. That is,PSPACEis the class of problems solvable in polynomial space by a deterministic Turing machine andNPSPACEis the class of problems solvable in polynomial space by a nondeterministic Turing machine. More formally,
While it is not known whetherP=NP,Savitch's theoremfamously showed thatPSPACE=NPSPACE. It is also known thatP⊆PSPACE{\displaystyle {\mathsf {P}}\subseteq {\mathsf {PSPACE}}}, which follows intuitively from the fact that, since writing to a cell on a Turing machine's tape is defined as taking one unit of time, a Turing machine operating in polynomial time can only write to polynomially many cells. It is suspected thatPis strictly smaller thanPSPACE, but this has not been proven.
The complexity classesEXPSPACEandNEXPSPACEare the space analogues toEXPTIMEandNEXPTIME. That is,EXPSPACEis the class of problems solvable in exponential space by a deterministic Turing machine andNEXPSPACEis the class of problems solvable in exponential space by a nondeterministic Turing machine. Or more formally,
Savitch's theoremshowed thatEXPSPACE=NEXPSPACE. This class is extremely broad: it is known to be a strict superset ofPSPACE,NP, andP, and is believed to be a strict superset ofEXPTIME.
Complexity classes have a variety ofclosureproperties. For example, decision classes may be closed undernegation,disjunction,conjunction, or even under allBoolean operations. Moreover, they might also be closed under a variety of quantification schemes.P, for instance, is closed under all Boolean operations, and under quantification over polynomially sized domains. Closure properties can be helpful in separating classes—one possible route to separating two complexity classes is to find some closure property possessed by one class but not by the other.
Each classXthat is not closed under negation has a complement classco-X, which consists of the complements of the languages contained inX(i.e.co-X={L|L¯∈X}{\displaystyle {\textsf {co-X}}=\{L|{\overline {L}}\in {\mathsf {X}}\}}).co-NP, for instance, is one important complement complexity class, and sits at the center of the unsolved problem over whetherco-NP=NP.
Closure properties are one of the key reasons many complexity classes are defined in the way that they are.[11]Take, for example, a problem that can be solved inO(n){\displaystyle O(n)}time (that is, in linear time) and one that can be solved in, at best,O(n1000){\displaystyle O(n^{1000})}time. Both of these problems are inP, yet the runtime of the second grows considerably faster than the runtime of the first as the input size increases. One might ask whether it would be better to define the class of "efficiently solvable" problems using some smaller polynomial bound, likeO(n3){\displaystyle O(n^{3})}, rather than all polynomials, which allows for such large discrepancies. It turns out, however, that the set of all polynomials is the smallest class of functions containing the linear functions that is also closed under addition, multiplication, and composition (for instance,O(n3)∘O(n2)=O(n6){\displaystyle O(n^{3})\circ O(n^{2})=O(n^{6})}, which is a polynomial butO(n6)>O(n3){\displaystyle O(n^{6})>O(n^{3})}).[11]Since we would like composing one efficient algorithm with another efficient algorithm to still be considered efficient, the polynomials are the smallest class that ensures composition of "efficient algorithms".[12](Note that the definition ofPis also useful because, empirically, almost all problems inPthat are practically useful do in fact have low order polynomial runtimes, and almost all problems outside ofPthat are practically useful do not have any known algorithms with small exponential runtimes, i.e. withO(cn){\displaystyle O(c^{n})}runtimes wherecis close to 1.[13])
Many complexity classes are defined using the concept of areduction. A reduction is a transformation of one problem into another problem, i.e. a reduction takes inputs from one problem and transforms them into inputs of another problem. For instance, you can reduce ordinary base-10 additionx+y{\displaystyle x+y}to base-2 addition by transformingx{\displaystyle x}andy{\displaystyle y}to their base-2 notation (e.g. 5+7 becomes 101+111). Formally, a problemX{\displaystyle X}reduces to a problemY{\displaystyle Y}if there exists a functionf{\displaystyle f}such that for everyx∈Σ∗{\displaystyle x\in \Sigma ^{*}},x∈X{\displaystyle x\in X}if and only iff(x)∈Y{\displaystyle f(x)\in Y}.
Generally, reductions are used to capture the notion of a problem being at least as difficult as another problem. Thus we are generally interested in using a polynomial-time reduction, since any problemX{\displaystyle X}that can be efficiently reduced to another problemY{\displaystyle Y}is no more difficult thanY{\displaystyle Y}. Formally, a problemX{\displaystyle X}is polynomial-time reducible to a problemY{\displaystyle Y}if there exists apolynomial-timecomputable functionp{\displaystyle p}such that for allx∈Σ∗{\displaystyle x\in \Sigma ^{*}},x∈X{\displaystyle x\in X}if and only ifp(x)∈Y{\displaystyle p(x)\in Y}.
Note that reductions can be defined in many different ways. Common reductions areCook reductions,Karp reductionsandLevin reductions, and can vary based on resource bounds, such aspolynomial-time reductionsandlog-space reductions.
Reductions motivate the concept of a problem beinghardfor a complexity class. A problemX{\displaystyle X}is hard for a class of problemsCif every problem inCcan be polynomial-time reduced toX{\displaystyle X}. Thus no problem inCis harder thanX{\displaystyle X}, since an algorithm forX{\displaystyle X}allows us to solve any problem inCwith at most polynomial slowdown. Of particular importance, the set of problems that are hard forNPis called the set ofNP-hardproblems.
If a problemX{\displaystyle X}is hard forCand is also inC, thenX{\displaystyle X}is said to becompleteforC. This means thatX{\displaystyle X}is the hardest problem inC(since there could be many problems that are equally hard, more preciselyX{\displaystyle X}is as hard as the hardest problems inC).
Of particular importance is the class ofNP-completeproblems—the most difficult problems inNP. Because all problems inNPcan be polynomial-time reduced toNP-complete problems, finding anNP-complete problem that can be solved in polynomial time would mean thatP=NP.
Savitch's theorem establishes the relationship between deterministic and nondetermistic space resources. It shows that if a nondeterministic Turing machine can solve a problem usingf(n){\displaystyle f(n)}space, then a deterministic Turing machine can solve the same problem inf(n)2{\displaystyle f(n)^{2}}space, i.e. in the square of the space. Formally, Savitch's theorem states that for anyf(n)>n{\displaystyle f(n)>n},[14]
Important corollaries of Savitch's theorem are thatPSPACE=NPSPACE(since the square of a polynomial is still a polynomial) andEXPSPACE=NEXPSPACE(since the square of an exponential is still an exponential).
These relationships answer fundamental questions about the power of nondeterminism compared to determinism. Specifically, Savitch's theorem shows that any problem that a nondeterministic Turing machine can solve in polynomial space, a deterministic Turing machine can also solve in polynomial space. Similarly, any problem that a nondeterministic Turing machine can solve in exponential space, a deterministic Turing machine can also solve in exponential space.
By definition ofDTIME, it follows thatDTIME(nk1){\displaystyle {\mathsf {DTIME}}(n^{k_{1}})}is contained inDTIME(nk2){\displaystyle {\mathsf {DTIME}}(n^{k_{2}})}ifk1≤k2{\displaystyle k_{1}\leq k_{2}}, sinceO(nk1)⊆O(nk2){\displaystyle O(n^{k_{1}})\subseteq O(n^{k_{2}})}ifk1≤k2{\displaystyle k_{1}\leq k_{2}}. However, this definition gives no indication of whether this inclusion is strict. For time and space requirements, the conditions under which the inclusion is strict are given by the time and space hierarchy theorems, respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. The hierarchy theorems enable one to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.
Thetime hierarchy theoremstates that
Thespace hierarchy theoremstates that
The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem establishes thatPis strictly contained inEXPTIME, and the space hierarchy theorem establishes thatLis strictly contained inPSPACE.
While deterministic and non-deterministicTuring machinesare the most commonly used models of computation, many complexity classes are defined in terms of other computational models. In particular,
These are explained in greater detail below.
A number of important complexity classes are defined using theprobabilistic Turing machine, a variant of theTuring machinethat can toss random coins. These classes help to better describe the complexity ofrandomized algorithms.
A probabilistic Turing machine is similar to a deterministic Turing machine, except rather than following a single transition function (a set of rules for how to proceed at each step of the computation) it probabilistically selects between multiple transition functions at each step. The standard definition of a probabilistic Turing machine specifies two transition functions, so that the selection of transition function at each step resembles a coin flip. The randomness introduced at each step of the computation introduces the potential for error; that is, strings that the Turing machine is meant to accept may on some occasions be rejected and strings that the Turing machine is meant to reject may on some occasions be accepted. As a result, the complexity classes based on the probabilistic Turing machine are defined in large part around the amount of error that is allowed. Formally, they are defined using an error probabilityϵ{\displaystyle \epsilon }. A probabilistic Turing machineM{\displaystyle M}is said to recognize a languageL{\displaystyle L}with error probabilityϵ{\displaystyle \epsilon }if:
The fundamental randomized time complexity classes areZPP,RP,co-RP,BPP, andPP.
The strictest class isZPP(zero-error probabilistic polynomial time), the class of problems solvable in polynomial time by a probabilistic Turing machine with error probability 0. Intuitively, this is the strictest class of probabilistic problems because it demandsno error whatsoever.
A slightly looser class isRP(randomized polynomial time), which maintains no error for strings not in the language but allows bounded error for strings in the language. More formally, a language is inRPif there is a probabilistic polynomial-time Turing machineM{\displaystyle M}such that if a string is not in the language thenM{\displaystyle M}always rejects and if a string is in the language thenM{\displaystyle M}accepts with a probability at least 1/2. The classco-RPis similarly defined except the roles are flipped: error is not allowed for strings in the language but is allowed for strings not in the language. Taken together, the classesRPandco-RPencompass all of the problems that can be solved by probabilistic Turing machines withone-sided error.
Loosening the error requirements further to allow fortwo-sided erroryields the classBPP(bounded-error probabilistic polynomial time), the class of problems solvable in polynomial time by a probabilistic Turing machine with error probability less than 1/3 (for both strings in the language and not in the language).BPPis the most practically relevant of the probabilistic complexity classes—problems inBPPhave efficientrandomized algorithmsthat can be run quickly on real computers.BPPis also at the center of the important unsolved problem in computer science over whetherP=BPP, which if true would mean that randomness does not increase the computational power of computers, i.e. any probabilistic Turing machine could be simulated by a deterministic Turing machine with at most polynomial slowdown.
The broadest class of efficiently-solvable probabilistic problems isPP(probabilistic polynomial time), the set of languages solvable by a probabilistic Turing machine in polynomial time with an error probability of less than 1/2 for all strings.
ZPP,RPandco-RPare all subsets ofBPP, which in turn is a subset ofPP. The reason for this is intuitive: the classes allowing zero error and only one-sided error are all contained within the class that allows two-sided error, andPPsimply relaxes the error probability ofBPP.ZPPrelates toRPandco-RPin the following way:ZPP=RP∩co-RP{\displaystyle {\textsf {ZPP}}={\textsf {RP}}\cap {\textsf {co-RP}}}. That is,ZPPconsists exactly of those problems that are in bothRPandco-RP. Intuitively, this follows from the fact thatRPandco-RPallow only one-sided error:co-RPdoes not allow error for strings in the language andRPdoes not allow error for strings not in the language. Hence, if a problem is in bothRPandco-RP, then there must be no error for strings both inandnot in the language (i.e. no error whatsoever), which is exactly the definition ofZPP.
Important randomized space complexity classes includeBPL,RL, andRLP.
A number of complexity classes are defined usinginteractive proof systems. Interactive proofs generalize the proofs definition of the complexity classNPand yield insights intocryptography,approximation algorithms, andformal verification.
Interactive proof systems areabstract machinesthat model computation as the exchange of messages between two parties: a proverP{\displaystyle P}and a verifierV{\displaystyle V}. The parties interact by exchanging messages, and an input string is accepted by the system if the verifier decides to accept the input on the basis of the messages it has received from the prover. The proverP{\displaystyle P}has unlimited computational power while the verifier has bounded computational power (the standard definition of interactive proof systems defines the verifier to be polynomially-time bounded). The prover, however, is untrustworthy (this prevents all languages from being trivially recognized by the proof system by having the computationally unbounded prover solve for whether a string is in a language and then sending a trustworthy "YES" or "NO" to the verifier), so the verifier must conduct an "interrogation" of the prover by "asking it" successive rounds of questions, accepting only if it develops a high degree of confidence that the string is in the language.[15]
The classNPis a simple proof system in which the verifier is restricted to being a deterministic polynomial-timeTuring machineand the procedure is restricted to one round (that is, the prover sends only a single, full proof—typically referred to as thecertificate—to the verifier). Put another way, in the definition of the classNP(the set of decision problems for which the problem instances, when the answer is "YES", have proofs verifiable in polynomial time by a deterministic Turing machine) is a proof system in which the proof is constructed by an unmentioned prover and the deterministic Turing machine is the verifier. For this reason,NPcan also be calleddIP(deterministic interactive proof), though it is rarely referred to as such.
It turns out thatNPcaptures the full power of interactive proof systems with deterministic (polynomial-time) verifiers because it can be shown that for any proof system with a deterministic verifier it is never necessary to need more than a single round of messaging between the prover and the verifier. Interactive proof systems that provide greater computational power over standard complexity classes thus requireprobabilisticverifiers, which means that the verifier's questions to the prover are computed usingprobabilistic algorithms. As noted in the section above onrandomized computation, probabilistic algorithms introduce error into the system, so complexity classes based on probabilistic proof systems are defined in terms of an error probabilityϵ{\displaystyle \epsilon }.
The most general complexity class arising out of this characterization is the classIP(interactive polynomial time), which is the class of all problems solvable by an interactive proof system(P,V){\displaystyle (P,V)}, whereV{\displaystyle V}is probabilistic polynomial-time and the proof system satisfies two properties: for a languageL∈IP{\displaystyle L\in {\mathsf {IP}}}
An important feature ofIPis that it equalsPSPACE. In other words, any problem that can be solved by a polynomial-time interactive proof system can also be solved by adeterministic Turing machinewith polynomial space resources, and vice versa.
A modification of the protocol forIPproduces another important complexity class:AM(Arthur–Merlin protocol). In the definition of interactive proof systems used byIP, the prover was not able to see the coins utilized by the verifier in its probabilistic computation—it was only able to see the messages that the verifier produced with these coins. For this reason, the coins are calledprivate random coins. The interactive proof system can be constrained so that the coins used by the verifier arepublic random coins; that is, the prover is able to see the coins. Formally,AMis defined as the class of languages with an interactive proof in which the verifier sends a random string to the prover, the prover responds with a message, and the verifier either accepts or rejects by applying a deterministic polynomial-time function to the message from the prover.AMcan be generalized toAM[k], wherekis the number of messages exchanged (so in the generalized form the standardAMdefined above isAM[2]). However, it is the case that for allk≥2{\displaystyle k\geq 2},AM[k]=AM[2]. It is also the case thatAM[k]⊆IP[k]{\displaystyle {\mathsf {AM}}[k]\subseteq {\mathsf {IP}}[k]}.
Other complexity classes defined using interactive proof systems includeMIP(multiprover interactive polynomial time) andQIP(quantum interactive polynomial time).
An alternative model of computation to theTuring machineis theBoolean circuit, a simplified model of thedigital circuitsused in moderncomputers. Not only does this model provide an intuitive connection between computation in theory and computation in practice, but it is also a natural model fornon-uniform computation(computation in which different input sizes within the same problem use different algorithms).
Formally, a Boolean circuitC{\displaystyle C}is adirected acyclic graphin which edges represent wires (which carry thebitvalues 0 and 1), the input bits are represented by source vertices (vertices with no incoming edges), and all non-source vertices representlogic gates(generally theAND,OR, andNOT gates). One logic gate is designated the output gate, and represents the end of the computation. The input/output behavior of a circuitC{\displaystyle C}withn{\displaystyle n}input variables is represented by theBoolean functionfC:{0,1}n→{0,1}{\displaystyle f_{C}:\{0,1\}^{n}\to \{0,1\}}; for example, on input bitsx1,x2,...,xn{\displaystyle x_{1},x_{2},...,x_{n}}, the output bitb{\displaystyle b}of the circuit is represented mathematically asb=fC(x1,x2,...,xn){\displaystyle b=f_{C}(x_{1},x_{2},...,x_{n})}. The circuitC{\displaystyle C}is said tocomputethe Boolean functionfC{\displaystyle f_{C}}.
Any particular circuit has a fixed number of input vertices, so it can only act on inputs of that size.Languages(the formal representations ofdecision problems), however, contain strings of differing lengths, so languages cannot be fully captured by a single circuit (this contrasts with the Turing machine model, in which a language is fully described by a single Turing machine that can act on any input size). A language is thus represented by acircuit family. A circuit family is an infinite list of circuits(C0,C1,C2,...){\displaystyle (C_{0},C_{1},C_{2},...)}, whereCn{\displaystyle C_{n}}is a circuit withn{\displaystyle n}input variables. A circuit family is said to decide a languageL{\displaystyle L}if, for every stringw{\displaystyle w},w{\displaystyle w}is in the languageL{\displaystyle L}if and only ifCn(w)=1{\displaystyle C_{n}(w)=1}, wheren{\displaystyle n}is the length ofw{\displaystyle w}. In other words, a stringw{\displaystyle w}of sizen{\displaystyle n}is in the language represented by the circuit family(C0,C1,C2,...){\displaystyle (C_{0},C_{1},C_{2},...)}if the circuitCn{\displaystyle C_{n}}(the circuit with the same number of input vertices as the number of bits inw{\displaystyle w}) evaluates to 1 whenw{\displaystyle w}is its input.
While complexity classes defined using Turing machines are described in terms oftime complexity, circuit complexity classes are defined in terms of circuit size — the number of vertices in the circuit. The size complexity of a circuit family(C0,C1,C2,...){\displaystyle (C_{0},C_{1},C_{2},...)}is the functionf:N→N{\displaystyle f:\mathbb {N} \to \mathbb {N} }, wheref(n){\displaystyle f(n)}is the circuit size ofCn{\displaystyle C_{n}}. The familiar function classes follow naturally from this; for example, a polynomial-size circuit family is one such that the functionf{\displaystyle f}is apolynomial.
The complexity classP/polyis the set of languages that are decidable by polynomial-size circuit families. It turns out that there is a natural connection between circuit complexity and time complexity. Intuitively, a language with small time complexity (that is, requires relatively few sequential operations on a Turing machine), also has a small circuit complexity (that is, requires relatively few Boolean operations). Formally, it can be shown that if a language is inDTIME(t(n)){\displaystyle {\mathsf {DTIME}}(t(n))}, wheret{\displaystyle t}is a functiont:N→N{\displaystyle t:\mathbb {N} \to \mathbb {N} }, then it has circuit complexityO(t2(n)){\displaystyle O(t^{2}(n))}.[16]It follows directly from this fact thatP⊂P/poly{\displaystyle {\mathsf {\color {Blue}P}}\subset {\textsf {P/poly}}}. In other words, any problem that can be solved in polynomial time by a deterministic Turing machine can also be solved by a polynomial-size circuit family. It is further the case that the inclusion is proper, i.e.P⊊P/poly{\displaystyle {\textsf {P}}\subsetneq {\textsf {P/poly}}}(for example, there are someundecidable problemsthat are inP/poly).
P/polyhas a number of properties that make it highly useful in the study of the relationships between complexity classes. In particular, it is helpful in investigating problems related toPversusNP. For example, if there is any language inNPthat is not inP/poly, thenP≠NP{\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}.[17]P/polyis also helpful in investigating properties of thepolynomial hierarchy. For example, ifNP⊆P/poly, thenPHcollapses toΣ2P{\displaystyle \Sigma _{2}^{\mathsf {P}}}. A full description of the relations betweenP/polyand other complexity classes is available at "Importance of P/poly".P/polyis also helpful in the general study of the properties ofTuring machines, as the class can be equivalently defined as the class of languages recognized by a polynomial-time Turing machine with a polynomial-boundedadvice function.
Two subclasses ofP/polythat have interesting properties in their own right areNCandAC. These classes are defined not only in terms of their circuit size but also in terms of theirdepth. The depth of a circuit is the length of the longestdirected pathfrom an input node to the output node. The classNCis the set of languages that can be solved by circuit families that are restricted not only to having polynomial-size but also to having polylogarithmic depth. The classACis defined similarly toNC, however gates are allowed to have unbounded fan-in (that is, the AND and OR gates can be applied to more than two bits).NCis a notable class because it can be equivalently defined as the class of languages that have efficientparallel algorithms.
The classesBQPandQMA, which are of key importance inquantum information science, are defined usingquantum Turing machines.
While most complexity classes studied by computer scientists are sets ofdecision problems, there are also a number of complexity classes defined in terms of other types of problems. In particular, there are complexity classes consisting ofcounting problems,function problems, andpromise problems. These are explained in greater detail below.
Acounting problemasks not onlywhethera solution exists (as with adecision problem), but askshow manysolutions exist.[18]For example, the decision problemCYCLE{\displaystyle {\texttt {CYCLE}}}askswhethera particular graphG{\displaystyle G}has asimple cycle(the answer is a simple yes/no); the corresponding counting problem#CYCLE{\displaystyle \#{\texttt {CYCLE}}}(pronounced "sharp cycle") askshow manysimple cyclesG{\displaystyle G}has.[19]The output to a counting problem is thus a number, in contrast to the output for a decision problem, which is a simple yes/no (or accept/reject, 0/1, or other equivalent scheme).[20]
Thus, whereas decision problems are represented mathematically asformal languages, counting problems are represented mathematically asfunctions: a counting problem is formalized as the functionf:{0,1}∗→N{\displaystyle f:\{0,1\}^{*}\to \mathbb {N} }such that for every inputw∈{0,1}∗{\displaystyle w\in \{0,1\}^{*}},f(w){\displaystyle f(w)}is the number of solutions. For example, in the#CYCLE{\displaystyle \#{\texttt {CYCLE}}}problem, the input is a graphG∈{0,1}∗{\displaystyle G\in \{0,1\}^{*}}(a graph represented as a string ofbits) andf(G){\displaystyle f(G)}is the number of simple cycles inG{\displaystyle G}.
Counting problems arise in a number of fields, includingstatistical estimation,statistical physics,network design, andeconomics.[21]
#P(pronounced "sharp P") is an important class of counting problems that can be thought of as the counting version ofNP.[22]The connection toNParises from the fact that the number of solutions to a problem equals the number of accepting branches in anondeterministic Turing machine's computation tree.#Pis thus formally defined as follows:
And just asNPcan be defined both in terms of nondeterminism and in terms of a verifier (i.e. as aninteractive proof system), so too can#Pbe equivalently defined in terms of a verifier. Recall that a decision problem is inNPif there exists a polynomial-time checkablecertificateto a given problem instance—that is,NPasks whether there exists a proof of membership (a certificate) for the input that can be checked for correctness in polynomial time. The class#Paskshow manysuch certificates exist.[22]In this context,#Pis defined as follows:
Counting problems are a subset of a broader class of problems calledfunction problems. A function problem is a type of problem in which the values of afunctionf:A→B{\displaystyle f:A\to B}are computed. Formally, a function problemf{\displaystyle f}is defined as a relationR{\displaystyle R}over strings of an arbitraryalphabetΣ{\displaystyle \Sigma }:
An algorithm solvesf{\displaystyle f}if for every inputx{\displaystyle x}such that there exists ay{\displaystyle y}satisfying(x,y)∈R{\displaystyle (x,y)\in R}, the algorithm produces one suchy{\displaystyle y}. This is just another way of saying thatf{\displaystyle f}is afunctionand the algorithm solvesf(x){\displaystyle f(x)}for allx∈Σ∗{\displaystyle x\in \Sigma ^{*}}.
An important function complexity class isFP, the class of efficiently solvable functions.[23]More specifically,FPis the set of function problems that can be solved by adeterministic Turing machineinpolynomial time.[23]FPcan be thought of as the function problem equivalent ofP. Importantly,FPprovides some insight into both counting problems andPversusNP. If#P=FP, then the functions that determine the number of certificates for problems inNPare efficiently solvable. And since computing the number of certificates is at least as hard as determining whether a certificate exists, it must follow that if#P=FPthenP=NP(it is not known whether this holds in the reverse, i.e. whetherP=NPimplies#P=FP).[23]
Just asFPis the function problem equivalent ofP,FNPis the function problem equivalent ofNP. Importantly,FP=FNPif and only ifP=NP.[24]
Promise problemsare a generalization of decision problems in which the input to a problem is guaranteed ("promised") to be from a particular subset of all possible inputs. Recall that with a decision problemL⊆{0,1}∗{\displaystyle L\subseteq \{0,1\}^{*}}, an algorithmM{\displaystyle M}forL{\displaystyle L}must act (correctly) oneveryw∈{0,1}∗{\displaystyle w\in \{0,1\}^{*}}. A promise problem loosens the input requirement onM{\displaystyle M}by restricting the input to some subset of{0,1}∗{\displaystyle \{0,1\}^{*}}.
Specifically, a promise problem is defined as a pair of non-intersecting sets(ΠACCEPT,ΠREJECT){\displaystyle (\Pi _{\text{ACCEPT}},\Pi _{\text{REJECT}})}, where:[25]
The input to an algorithmM{\displaystyle M}for a promise problem(ΠACCEPT,ΠREJECT){\displaystyle (\Pi _{\text{ACCEPT}},\Pi _{\text{REJECT}})}is thusΠACCEPT∪ΠREJECT{\displaystyle \Pi _{\text{ACCEPT}}\cup \Pi _{\text{REJECT}}}, which is called thepromise. Strings inΠACCEPT∪ΠREJECT{\displaystyle \Pi _{\text{ACCEPT}}\cup \Pi _{\text{REJECT}}}are said tosatisfy the promise.[25]By definition,ΠACCEPT{\displaystyle \Pi _{\text{ACCEPT}}}andΠREJECT{\displaystyle \Pi _{\text{REJECT}}}must be disjoint, i.e.ΠACCEPT∩ΠREJECT=∅{\displaystyle \Pi _{\text{ACCEPT}}\cap \Pi _{\text{REJECT}}=\emptyset }.
Within this formulation, it can be seen that decision problems are just the subset of promise problems with the trivial promiseΠACCEPT∪ΠREJECT={0,1}∗{\displaystyle \Pi _{\text{ACCEPT}}\cup \Pi _{\text{REJECT}}=\{0,1\}^{*}}. With decision problems it is thus simpler to simply define the problem as onlyΠACCEPT{\displaystyle \Pi _{\text{ACCEPT}}}(withΠREJECT{\displaystyle \Pi _{\text{REJECT}}}implicitly being{0,1}∗/ΠACCEPT{\displaystyle \{0,1\}^{*}/\Pi _{\text{ACCEPT}}}), which throughout this page is denotedL{\displaystyle L}to emphasize thatΠACCEPT=L{\displaystyle \Pi _{\text{ACCEPT}}=L}is aformal language.
Promise problems make for a more natural formulation of many computational problems. For instance, a computational problem could be something like "given aplanar graph, determine whether or not..."[26]This is often stated as a decision problem, where it is assumed that there is some translation schema that takeseverystrings∈{0,1}∗{\displaystyle s\in \{0,1\}^{*}}to a planar graph. However, it is more straightforward to define this as a promise problem in which the input is promised to be a planar graph.
Promise problems provide an alternate definition for standard complexity classes of decision problems.P, for instance, can be defined as a promise problem:[27]
Classes of decision problems—that is, classes of problems defined as formal languages—thus translate naturally to promise problems, where a languageL{\displaystyle L}in the class is simplyL=ΠACCEPT{\displaystyle L=\Pi _{\text{ACCEPT}}}andΠREJECT{\displaystyle \Pi _{\text{REJECT}}}is implicitly{0,1}∗/ΠACCEPT{\displaystyle \{0,1\}^{*}/\Pi _{\text{ACCEPT}}}.
Formulating many basic complexity classes likePas promise problems provides little additional insight into their nature. However, there are some complexity classes for which formulating them as promise problems have been useful to computer scientists. Promise problems have, for instance, played a key role in the study ofSZK(statistical zero-knowledge).[28]
The following table shows some of the classes of problems that are considered in complexity theory. If classXis a strictsubsetofY, thenXis shown belowYwith a dark line connecting them. IfXis a subset, but it is unknown whether they are equal sets, then the line is lighter and dotted. Technically, the breakdown into decidable and undecidable pertains more to the study ofcomputability theory, but is useful for putting the complexity classes in perspective.
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Incomputational complexity theory,NP(nondeterministic polynomial time) is acomplexity classused to classifydecision problems. NP is thesetof decision problems for which theproblem instances, where the answer is "yes", haveproofsverifiable inpolynomial timeby adeterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by anondeterministic Turing machine.[2][Note 1]
The first definition is the basis for the abbreviation NP; "nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess about the solution, which is generated in a nondeterministic way, while the second phase consists of a deterministic algorithm that verifies whether the guess is a solution to the problem.[3]
The complexity classP(all problems solvable, deterministically, in polynomial time) is contained in NP (problems where solutions can be verified in polynomial time), because if a problem is solvable in polynomial time, then a solution is also verifiable in polynomial time by simply solving the problem. It is widely believed, but not proven, thatP is smaller than NP, in other words, that decision problems exist that cannot be solved in polynomial time even though their solutions can be checked in polynomial time. The hardest problems in NP are calledNP-completeproblems. An algorithm solving such a problem in polynomial time is also able to solve any other NP problem in polynomial time. If P were in fact equal to NP, then a polynomial-time algorithm would exist for solving NP-complete, and by corollary, all NP problems.[4]
The complexity class NP is related to the complexity classco-NP, for which the answer "no" can be verified in polynomial time. Whether or notNP = co-NPis another outstanding question in complexity theory.[5]
The complexity class NP can be defined in terms ofNTIMEas follows:
whereNTIME(nk){\displaystyle {\mathsf {NTIME}}(n^{k})}is the set of decision problems that can be solved by anondeterministic Turing machineinO(nk){\displaystyle O(n^{k})}time.
Equivalently, NP can be defined using deterministic Turing machines as verifiers. AlanguageLis in NP if and only if there exist polynomialspandq, and a deterministic Turing machineM, such that
Manycomputer scienceproblems are contained in NP, like decision versions of manysearchand optimization problems.
In order to explain the verifier-based definition of NP, consider thesubset sum problem:
Assume that we are given someintegers, {−7, −3, −2, 5, 8}, and we wish to know whether some of these integers sum up to zero. Here the answer is "yes", since the integers {−3, −2, 5} corresponds to the sum(−3) + (−2) + 5 = 0.
To answer whether some of the integers add to zero we can create an algorithm that obtains all the possible subsets. As the number of integers that we feed into the algorithm becomes larger, both the number of subsets and the computation time grows exponentially.
But notice that if we are given a particular subset, we canefficiently verifywhether the subset sum is zero, by summing the integers of the subset. If the sum is zero, that subset is aprooforwitnessfor the answer is "yes". An algorithm that verifies whether a given subset has sum zero is averifier. Clearly, summing the integers of a subset can be done in polynomial time, and the subset sum problem is therefore in NP.
The above example can be generalized for any decision problem. Given any instance I of problemΠ{\displaystyle \Pi }and witness W, if there exists averifierV so that given the ordered pair (I, W) as input, V returns "yes" in polynomial time if the witness proves that the answer is "yes" or "no" in polynomial time otherwise, thenΠ{\displaystyle \Pi }is in NP.
The "no"-answer version of this problem is stated as: "given a finite set of integers, does every non-empty subset have a nonzero sum?". The verifier-based definition of NP doesnotrequire an efficient verifier for the "no"-answers. The class of problems with such verifiers for the "no"-answers is called co-NP. In fact, it is an open question whether all problems in NP also have verifiers for the "no"-answers and thus are in co-NP.
In some literature the verifier is called the "certifier", and the witness the "certificate".[2]
Equivalent to the verifier-based definition is the following characterization: NP is the class ofdecision problemssolvable by anondeterministic Turing machinethat runs inpolynomial time. That is to say, a decision problemΠ{\displaystyle \Pi }is in NP wheneverΠ{\displaystyle \Pi }is recognized by some polynomial-time nondeterministic Turing machineM{\displaystyle M}with anexistential acceptance condition, meaning thatw∈Π{\displaystyle w\in \Pi }if and only if some computation path ofM(w){\displaystyle M(w)}leads to an accepting state. This definition is equivalent to the verifier-based definition because a nondeterministic Turing machine could solve an NP problem in polynomial time by nondeterministically selecting a certificate and running the verifier on the certificate. Similarly, if such a machine exists, then a polynomial time verifier can naturally be constructed from it.
In this light, we can define co-NP dually as the class of decision problems recognizable by polynomial-time nondeterministic Turing machines with an existential rejection condition. Since an existential rejection condition is exactly the same thing as auniversal acceptance condition, we can understand theNP vs. co-NPquestion as asking whether the existential and universal acceptance conditions have the same expressive power for the class of polynomial-time nondeterministic Turing machines.
NP is closed underunion,intersection,concatenation,Kleene starandreversal. It is not known whether NP is closed undercomplement(this question is the so-called "NP versus co-NP" question).
Because of the many important problems in this class, there have been extensive efforts to find polynomial-time algorithms for problems in NP. However, there remain a large number of problems in NP that defy such attempts, seeming to requiresuper-polynomial time. Whether these problems are not decidable in polynomial time is one of the greatest open questions incomputer science(seePversus NP ("P = NP") problemfor an in-depth discussion).
An important notion in this context is the set ofNP-completedecision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for evenoneof them, then there is a polynomial-time algorithm forallthe problems in NP. Because of this, and because dedicated research has failed to find a polynomial algorithm for any NP-complete problem, once a problem has been proven to be NP-complete, this is widely regarded as a sign that a polynomial algorithm for this problem is unlikely to exist.
However, in practical uses, instead of spending computational resources looking for an optimal solution, a good enough (but potentially suboptimal) solution may often be found in polynomial time. Also, the real-life applications of some problems are easier than their theoretical equivalents.
The two definitions of NP as the class of problems solvable by a nondeterministicTuring machine(TM) in polynomial time and the class of problems verifiable by a deterministic Turing machine in polynomial time are equivalent. The proof is described by many textbooks, for example, Sipser'sIntroduction to the Theory of Computation, section 7.3.
To show this, first, suppose we have a deterministic verifier. A non-deterministic machine can simply nondeterministically run the verifier on all possible proof strings (this requires only polynomially many steps because it can nondeterministically choose the next character in the proof string in each step, and the length of the proof string must be polynomially bounded). If any proof is valid, some path will accept; if no proof is valid, the string is not in the language and it will reject.
Conversely, suppose we have a non-deterministic TM called A accepting a given language L. At each of its polynomially many steps, the machine'scomputation treebranches in at most a finite number of directions. There must be at least one accepting path, and the string describing this path is the proof supplied to the verifier. The verifier can then deterministically simulate A, following only the accepting path, and verifying that it accepts at the end. If A rejects the input, there is no accepting path, and the verifier will always reject.
NP contains all problems inP, since one can verify any instance of the problem by simply ignoring the proof and solving it. NP is contained inPSPACE—to show this, it suffices to construct a PSPACE machine that loops over all proof strings and feeds each one to a polynomial-time verifier. Since a polynomial-time machine can only read polynomially many bits, it cannot use more than polynomial space, nor can it read a proof string occupying more than polynomial space (so we do not have to consider proofs longer than this). NP is also contained inEXPTIME, since the same algorithm operates in exponential time.
co-NP contains those problems that have a simple proof fornoinstances, sometimes called counterexamples. For example,primality testingtrivially lies in co-NP, since one can refute the primality of an integer by merely supplying a nontrivial factor. NP and co-NP together form the first level in thepolynomial hierarchy, higher only than P.
NP is defined using only deterministic machines. If we permit the verifier to be probabilistic (this, however, is not necessarily a BPP machine[6]), we get the classMAsolvable using anArthur–Merlin protocolwith no communication from Arthur to Merlin.
The relationship betweenBPPandNPis unknown: it is not known whetherBPPis asubsetofNP,NPis a subset ofBPPor neither. IfNPis contained inBPP, which is considered unlikely since it would imply practical solutions forNP-completeproblems, thenNP=RPandPH⊆BPP.[7]
NP is a class ofdecision problems; the analogous class of function problems isFNP.
The only known strict inclusions come from thetime hierarchy theoremand thespace hierarchy theorem, and respectively they areNP⊊NEXPTIME{\displaystyle {\mathsf {NP\subsetneq NEXPTIME}}}andNP⊊EXPSPACE{\displaystyle {\mathsf {NP\subsetneq EXPSPACE}}}.
In terms ofdescriptive complexity theory, NP corresponds precisely to the set of languages definable by existentialsecond-order logic(Fagin's theorem).
NP can be seen as a very simple type ofinteractive proof system, where the prover comes up with the proof certificate and the verifier is a deterministic polynomial-time machine that checks it. It is complete because the right proof string will make it accept if there is one, and it is sound because the verifier cannot accept if there is no acceptable proof string.
A major result of complexity theory is that NP can be characterized as the problems solvable byprobabilistically checkable proofswhere the verifier uses O(logn) random bits and examines only a constant number of bits of the proof string (the classPCP(logn, 1)). More informally, this means that the NP verifier described above can be replaced with one that just "spot-checks" a few places in the proof string, and using a limited number of coin flips can determine the correct answer with high probability. This allows several results about the hardness ofapproximation algorithmsto be proven.
All problems inP, denotedP⊆NP{\displaystyle {\mathsf {P\subseteq NP}}}. Given a certificate for a problem inP, we can ignore the certificate and just solve the problem in polynomial time.
The decision problem version of theinteger factorization problem: given integersnandk, is there a factorfwith 1 <f<kandfdividingn?[8]
EveryNP-completeproblem is in NP.
TheBoolean satisfiability problem(SAT), where we want to know whether or not a certain formula inpropositional logicwithBoolean variablesis true for some value of the variables.[9]
The decision version of thetravelling salesman problemis in NP. Given an input matrix of distances betweenncities, the problem is to determine if there is a route visiting all cities with total distance less thank.
A proof can simply be a list of the cities. Then verification can clearly be done in polynomial time. It simply adds the matrix entries corresponding to the paths between the cities.
Anondeterministic Turing machinecan find such a route as follows:
One can think of each guess as "forking" a new copy of the Turing machine to follow each of the possible paths forward, and if at least one machine finds a route of distance less thank, that machine accepts the input. (Equivalently, this can be thought of as a single Turing machine that always guesses correctly)
Abinary searchon the range of possible distances can convert the decision version of Traveling Salesman to the optimization version, by calling the decision version repeatedly (a polynomial number of times).[10][8]
Thesubgraph isomorphism problemof determining whether graphGcontains a subgraph that is isomorphic to graphH.[11]
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Incomputational complexity theory,co-NPis acomplexity class. Adecision problemX is a member of co-NP if and only if itscomplementXis in the complexity classNP. The class can be defined as follows: a decision problem is in co-NP if and only if for everyno-instance we have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate.
That is,co-NPis the set of decision problems where there exists a polynomialp(n){\displaystyle p(n)}and a polynomial-time boundedTuring machineMsuch that for every instancex,xis ano-instance if and only if: for some possible certificatecof length bounded byp(n){\displaystyle p(n)}, the Turing machineMaccepts the pair(x,c).[1]
While an NP problem asks whether a given instance is ayes-instance, itscomplementasks whether an instance is ano-instance, which means the complement is in co-NP. Anyyes-instance for the original NP problem becomes ano-instance for its complement, and vice versa.
An example of anNP-completeproblem is theBoolean satisfiability problem: given a Boolean formula, is itsatisfiable(is there a possible input for which the formula outputs true)? The complementary problem asks: "given a Boolean formula, is itunsatisfiable(do all possible inputs to the formula output false)?". Since this is thecomplementof the satisfiability problem, a certificate for ano-instance is the same as for ayes-instance from the original NP problem: a set of Boolean variable assignments which make the formula true. On the other hand, a certificate of ayes-instance for the complementary problem (whatever form it might take) would be equally as complex as for theno-instance of the original NP satisfiability problem.
A problemLisco-NP-completeif and only ifLis in co-NP and for any problem in co-NP, there exists apolynomial-time reductionfrom that problem toL.
Determining if a formula inpropositional logicis atautologyis co-NP-complete: that is, if the formula evaluates to true under every possible assignment to its variables.[1]
P, the class of polynomial time solvable problems, is a subset of both NP and co-NP. P is thought to be a strict subset in both cases. Because P is closed under complementation, and NP and co-NP are complementary, it cannot be strict in one case and not strict in the other: if P equals NP, it must also equal co-NP, and vice versa.[2]
NP and co-NP are also thought to be unequal,[3]and their equality would imply the collapse of thepolynomial hierarchyPH to NP. If they are unequal, then no NP-complete problem can be in co-NP and noco-NP-completeproblem can be in NP.[4]This can be shown as follows. Suppose for the sake of contradiction there exists an NP-complete problemXthat is in co-NP. Since all problems in NP can be reduced toX, it follows that for every problem in NP, we can construct anon-deterministic Turing machinethat decides its complement in polynomial time; i.e.,NP⊆co-NP{\displaystyle {\textsf {NP}}\subseteq {\textsf {co-NP}}}. From this, it follows that the set of complements of the problems in NP is a subset of the set of complements of the problems in co-NP; i.e.,co-NP⊆NP{\displaystyle {\textsf {co-NP}}\subseteq {\textsf {NP}}}. Thusco-NP=NP{\displaystyle {\textsf {co-NP}}={\textsf {NP}}}. The proof that no co-NP-complete problem can be in NP ifNP≠co-NP{\displaystyle {\textsf {NP}}\neq {\textsf {co-NP}}}is symmetrical.
co-NP is a subset ofPH, which itself is a subset ofPSPACE.
An example of a problem that is known to belong to both NP and co-NP (but not known to be in P) isInteger factorization: given positive integersmandn, determine ifmhas a factor less thannand greater than one. Membership in NP is clear; ifmdoes have such a factor, then the factor itself is a certificate. Membership in co-NP is also straightforward: one can just list the prime factors ofm, all greater or equal ton, which the verifier can confirm to be valid by multiplication and theAKS primality test. It is presently not known whether there is a polynomial-time algorithm for factorization, equivalently that integer factorization is in P, and hence this example is interesting as one of the most natural problems known to be in NP and co-NP but not known to be in P.[5]
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Aquantum Turing machine(QTM) oruniversal quantum computeris anabstract machineused tomodelthe effects of aquantum computer. It provides a simple model that captures all of the power of quantum computation—that is, anyquantum algorithmcan be expressed formally as a particular quantum Turing machine. However, the computationally equivalentquantum circuitis a more common model.[1][2]: 2
Quantum Turing machines can be related to classical andprobabilistic Turing machinesin a framework based ontransition matrices. That is, a matrix can be specified whose product with the matrix representing a classical or probabilistic machine provides the quantumprobability matrixrepresenting the quantum machine. This was shown byLance Fortnow.[3]
A way of understanding the quantum Turing machine (QTM) is that it generalizes the classicalTuring machine(TM) in the same way that thequantum finite automaton(QFA) generalizes thedeterministic finite automaton(DFA). In essence, the internal states of a classical TM are replaced bypureormixed statesin aHilbert space; the transition function is replaced by a collection ofunitary matricesthat map the Hilbert space to itself.[4]
That is, a classical Turing machine is described by a 7-tupleM=⟨Q,Γ,b,Σ,δ,q0,F⟩{\displaystyle M=\langle Q,\Gamma ,b,\Sigma ,\delta ,q_{0},F\rangle }. Seethe formal definition of a Turing Machinefor a more in-depth understanding of each of the elements in this tuple.
For a three-tape quantum Turing machine (one tape holding the input, a second tape holding intermediate calculation results, and a third tape holding output):
The above is merely a sketch of a quantum Turing machine, rather than its formal definition, as it leaves vague several important details: for example, how often ameasurementis performed; see for example, the difference between a measure-once and a measure-many QFA. This question of measurement affects the way in which writes to the output tape are defined.
In 1980 and 1982, physicistPaul Benioffpublished articles[5][6]that first described a quantum mechanical model ofTuring machines. A 1985 article written by Oxford University physicistDavid Deutschfurther developed the idea of quantum computers by suggesting thatquantum gatescould function in a similar fashion to traditional digital computingbinarylogic gates.[4]
Iriyama,Ohya, and Volovich have developed a model of alinear quantum Turing machine(LQTM). This is a generalization of a classical QTM that has mixed states and that allows irreversible transition functions. These allow the representation of quantum measurements without classical outcomes.[7]
Aquantum Turing machine withpostselectionwas defined byScott Aaronson, who showed that the class of polynomial time on such a machine (PostBQP) is equal to the classical complexity classPP.[8]
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Incomputational complexity theory, thepolynomial hierarchy(sometimes called thepolynomial-time hierarchy) is ahierarchyofcomplexity classesthat generalize the classesNPandco-NP.[1]Each class in the hierarchy is contained withinPSPACE. The hierarchy can be defined usingoracle machinesoralternating Turing machines. It is a resource-bounded counterpart to thearithmetical hierarchyandanalytical hierarchyfrommathematical logic. The union of the classes in the hierarchy is denotedPH.
Classes within the hierarchy have complete problems (with respect topolynomial-time reductions) that ask ifquantified Boolean formulaehold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level.
There are multiple equivalent definitions of the classes of the polynomial hierarchy.
For the oracle definition of the polynomial hierarchy, define
wherePis the set ofdecision problemssolvable inpolynomial time. Then for i ≥ 0 define
wherePA{\displaystyle \mathrm {P} ^{\rm {A}}}is the set ofdecision problemssolvable in polynomial time by aTuring machineaugmented by anoraclefor some complete problem in class A; the classesNPA{\displaystyle \mathrm {NP} ^{\rm {A}}}andcoNPA{\displaystyle \mathrm {coNP} ^{\rm {A}}}are defined analogously. For example,Σ1P=NP,Π1P=coNP{\displaystyle \Sigma _{1}^{\mathrm {P} }=\mathrm {NP} ,\Pi _{1}^{\mathrm {P} }=\mathrm {coNP} }, andΔ2P=PNP{\displaystyle \Delta _{2}^{\mathrm {P} }=\mathrm {P^{NP}} }is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for some NP-complete problem.[2]
For the existential/universal definition of the polynomial hierarchy, letLbe alanguage(i.e. adecision problem, a subset of {0,1}*), letpbe apolynomial, and define
where⟨x,w⟩∈{0,1}∗{\displaystyle \langle x,w\rangle \in \{0,1\}^{*}}is some standard encoding of the pair of binary stringsxandwas a single binary string. The languageLrepresents a set of ordered pairs of strings, where the first stringxis a member of∃pL{\displaystyle \exists ^{p}L}, and the second stringwis a "short" (|w|≤p(|x|){\displaystyle |w|\leq p(|x|)}) witness testifying thatxis a member of∃pL{\displaystyle \exists ^{p}L}. In other words,x∈∃pL{\displaystyle x\in \exists ^{p}L}if and only if there exists a short witnesswsuch that⟨x,w⟩∈L{\displaystyle \langle x,w\rangle \in L}. Similarly, define
Note thatDe Morgan's lawshold:(∃pL)c=∀pLc{\displaystyle \left(\exists ^{p}L\right)^{\rm {c}}=\forall ^{p}L^{\rm {c}}}and(∀pL)c=∃pLc{\displaystyle \left(\forall ^{p}L\right)^{\rm {c}}=\exists ^{p}L^{\rm {c}}}, whereLcis the complement ofL.
LetCbe a class of languages. Extend these operators to work on whole classes of languages by the definition
Again, De Morgan's laws hold:co∃PC=∀PcoC{\displaystyle \mathrm {co} \exists ^{\mathrm {P} }{\mathcal {C}}=\forall ^{\mathrm {P} }\mathrm {co} {\mathcal {C}}}andco∀PC=∃PcoC{\displaystyle \mathrm {co} \forall ^{\mathrm {P} }{\mathcal {C}}=\exists ^{\mathrm {P} }\mathrm {co} {\mathcal {C}}}, wherecoC={Lc|L∈C}{\displaystyle \mathrm {co} {\mathcal {C}}=\left\{L^{c}|L\in {\mathcal {C}}\right\}}.
The classesNPandco-NPcan be defined asNP=∃PP{\displaystyle \mathrm {NP} =\exists ^{\mathrm {P} }\mathrm {P} }, andcoNP=∀PP{\displaystyle \mathrm {coNP} =\forall ^{\mathrm {P} }\mathrm {P} }, wherePis the class of all feasibly (polynomial-time) decidable languages. The polynomial hierarchy can be defined recursively as
Note thatNP=Σ1P{\displaystyle \mathrm {NP} =\Sigma _{1}^{\mathrm {P} }}, andcoNP=Π1P{\displaystyle \mathrm {coNP} =\Pi _{1}^{\mathrm {P} }}.
This definition reflects the close connection between the polynomial hierarchy and thearithmetical hierarchy, whereRandREplay roles analogous toPandNP, respectively. Theanalytic hierarchyis also defined in a similar way to give a hierarchy of subsets of thereal numbers.
Analternating Turing machineis a non-deterministic Turing machine with non-final states partitioned into existential and universal states. It is eventually accepting from its current configuration if: it is in an existential state and can transition into some eventually accepting configuration; or, it is in a universal state and every transition is into some eventually accepting configuration; or, it is in an accepting state.[3]
We defineΣkP{\displaystyle \Sigma _{k}^{\mathrm {P} }}to be the class of languages accepted by an alternating Turing machine in polynomial time such that the initial state is an existential state and every path the machine can take swaps at mostk– 1 times between existential and universal states. We defineΠkP{\displaystyle \Pi _{k}^{\mathrm {P} }}similarly, except that the initial state is a universal state.[4]
If we omit the requirement of at mostk– 1 swaps between the existential and universal states, so that we only require that our alternating Turing machine runs in polynomial time, then we have the definition of the classAP, which is equal toPSPACE.[5]
The union of all classes in the polynomial hierarchy is the complexity classPH.
The definitions imply the relations:
Unlike the arithmetic and analytic hierarchies, whose inclusions are known to be proper, it is an open question whether any of these inclusions are proper, though it is widely believed that they all are. If anyΣkP=Σk+1P{\displaystyle \Sigma _{k}^{\mathrm {P} }=\Sigma _{k+1}^{\mathrm {P} }}, or if anyΣkP=ΠkP{\displaystyle \Sigma _{k}^{\mathrm {P} }=\Pi _{k}^{\mathrm {P} }}, then the hierarchycollapses to level k: for alli>k{\displaystyle i>k},ΣiP=ΣkP{\displaystyle \Sigma _{i}^{\mathrm {P} }=\Sigma _{k}^{\mathrm {P} }}.[6]In particular, we have the following implications involving unsolved problems:
The case in whichNP=PHis also termed as acollapseof thePHtothe second level. The caseP=NPcorresponds to a collapse ofPHtoP.
The question of collapse to the first level is generally thought to be extremely difficult. Most researchers do not believe in a collapse, even to the second level.
The polynomial hierarchy is an analogue (at much lower complexity) of theexponential hierarchyandarithmetical hierarchy.
It is known that PH is contained withinPSPACE, but it is not known whether the two classes are equal. One useful reformulation of this problem is that PH = PSPACE if and only ifsecond-order logic over finite structuresgains no additional power from the addition of atransitive closureoperator over relations of relations (i.e., over the second-order variables).[8]
If the polynomial hierarchy has anycomplete problems, then it has only finitely many distinct levels. Since there arePSPACE-completeproblems, we know that if PSPACE = PH, then the polynomial hierarchy must collapse, since a PSPACE-complete problem would be aΣkP{\displaystyle \Sigma _{k}^{\mathrm {P} }}-complete problem for somek.[9]
Each class in the polynomial hierarchy contains≤mP{\displaystyle \leq _{\rm {m}}^{\mathrm {P} }}-complete problems (problems complete under polynomial-time many-one reductions). Furthermore, each class in the polynomial hierarchy isclosed under≤mP{\displaystyle \leq _{\rm {m}}^{\mathrm {P} }}-reductions: meaning that for a classCin the hierarchy and a languageL∈C{\displaystyle L\in {\mathcal {C}}}, ifA≤mPL{\displaystyle A\leq _{\rm {m}}^{\mathrm {P} }L}, thenA∈C{\displaystyle A\in {\mathcal {C}}}as well. These two facts together imply that ifKi{\displaystyle K_{i}}is a complete problem forΣiP{\displaystyle \Sigma _{i}^{\mathrm {P} }}, thenΣi+1P=NPKi{\displaystyle \Sigma _{i+1}^{\mathrm {P} }=\mathrm {NP} ^{K_{i}}}, andΠi+1P=coNPKi{\displaystyle \Pi _{i+1}^{\mathrm {P} }=\mathrm {coNP} ^{K_{i}}}. For instance,Σ2P=NPSAT{\displaystyle \Sigma _{2}^{\mathrm {P} }=\mathrm {NP} ^{\mathrm {SAT} }}. In other words, if a language is defined based on some oracle inC, then we can assume that it is defined based on a complete problem forC. Complete problems therefore act as "representatives" of the class for which they are complete.
There is some evidence thatBQP, the class of problems solvable in polynomial time by aquantum computer, is not contained in PH; however, it is also believed that PH is not contained in BQP.[10]=[11]
is decidable in polynomial time. The language
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Inphilosophy, asupertaskis acountably infinitesequence of operations that occur sequentially within a finite interval of time.[1]Supertasks are calledhypertaskswhen the number of operations becomesuncountably infinite. A hypertask that includes one task for eachordinal numberis called anultratask.[2]The term "supertask" was coined by the philosopherJames F. Thomson, who devisedThomson's lamp. The term "hypertask" derives from Clark and Read in their paper of that name.[3]
The origin of the interest in supertasks is normally attributed toZeno of Elea. Zeno claimed thatmotion was impossible. He argued as follows: suppose our burgeoning "mover", Achilles say, wishes to move from A to B. To achieve this he must traverse half the distance from A to B. To get from the midpoint of AB to B, Achilles must traverse halfthisdistance, and so on and so forth. However many times he performs one of these "traversing" tasks, there is another one left for him to do before he arrives at B. Thus it follows, according to Zeno, that motion (travelling a non-zero distance in finite time) is a supertask. Zeno further argues that supertasks are not possible (how can this sequence be completed if for each traversing there is another one to come?). It follows that motion is impossible.
Zeno's argument takes the following form:
Most subsequent philosophers reject Zeno's bold conclusion in favor of common sense. Instead, they reverse the argument and take it as aproof by contradictionwhere the possibility of motion is taken for granted. They accept the possibility of motion and applymodus tollens(contrapositive) to Zeno's argument to reach the conclusion that either motion is not a supertask or not all supertasks are impossible.[citation needed]
Zeno himself also discusses the notion of what he calls "Achillesand the tortoise". Suppose that Achilles is the fastest runner, and moves at a speed of 1 m/s. Achilles chases a tortoise, an animal renowned for being slow, that moves at 0.1 m/s. However, the tortoise starts 0.9 metres ahead. Common sense seems to decree that Achilles will catch up with the tortoise after exactly 1 second, but Zeno argues that this is not the case. He instead suggests that Achilles must inevitably come up to the point where the tortoise has started from, but by the time he has accomplished this, the tortoise will already have moved on to another point. This continues, and every time Achilles reaches the mark where the tortoise was, the tortoise will have reached a new point that Achilles will have to catch up with; while it begins with 0.9 metres, it becomes an additional 0.09 metres, then 0.009 metres, and so on, infinitely. While these distances will grow very small, they will remain finite, while Achilles' chasing of the tortoise will become an unending supertask. Much commentary has been made on this particular paradox; many assert that it finds a loophole in common sense.[4]
James F. Thomsonbelieved that motion was not a supertask, and he emphatically denied that supertasks are possible. He considered a lamp that may either be on or off. At timet= 0the lamp is off, and the switch is flipped on att= 1/2; after that, the switch is flipped after waiting for half the time as before. Thomson asks what is the state att= 1, when the switch has been flipped infinitely many times. He reasons that it cannot be on because there was never a time when it was not subsequently turned off, and vice versa, and reaches a contradiction. He concludes that supertasks are impossible.[5]
Paul Benacerrafbelieves that supertasks are at least logically possible despite Thomson's apparent contradiction. Benacerraf agrees with Thomson insofar as that the experiment he outlined does not determine the state of the lamp at t = 1. However he disagrees with Thomson that he can derive a contradiction from this, since the state of the lamp at t = 1 cannot be logically determined by the preceding states.[citation needed]
Most of the modern literature comes from the descendants of Benacerraf, those who tacitly accept the possibility of supertasks. Philosophers who reject their possibility tend not to reject them on grounds such as Thomson's but because they have qualms with the notion of infinity itself. Of course there are exceptions. For example, McLaughlin claims that Thomson's lamp is inconsistent if it is analyzed withinternal set theory, a variant ofreal analysis.
If supertasks are possible, then the truth or falsehood of unknown propositions of number theory, such asGoldbach's conjecture, or evenundecidablepropositions could be determined in a finite amount of time by a brute-force search of the set of all natural numbers. This would, however, be in contradiction with theChurch–Turing thesis. Some have argued this poses a problem forintuitionism, since the intuitionist must distinguish between things that cannot in fact be proven (because they are too long or complicated; for exampleBoolos's "Curious Inference"[6]) but nonetheless are considered "provable", and those whichareprovable by infinite brute force in the above sense.
Some have claimed, Thomson's lamp is physically impossible since it must have parts moving at speeds faster than thespeed of light(e.g., the lamp switch).Adolf Grünbaumsuggests that the lamp could have a strip of wire which, when lifted, disrupts the circuit and turns off the lamp; this strip could then be lifted by a smaller distance each time the lamp is to be turned off, maintaining a constant velocity.
However, such a design would ultimately fail, as eventually the distance between the contacts would be so small as to allow electrons to jump the gap, preventing the circuit from being broken at all. Still, for either a human or any device, to perceive or act upon the state of the lamp some measurement has to be done, for example the light from the lamp would have to reach an eye or a sensor.
Any such measurement will take a fixed frame of time, no matter how small and, therefore, at some point measurement of the state will be impossible. Since the state at t=1 cannot be determined even in principle, it is not meaningful to speak of the lamp being either on or off.
Other physically possible supertasks have been suggested. In one proposal, one person (or entity) counts upward from 1, taking an infinite amount of time, while another person observes this from a frame of reference where this occurs in a finite space of time. For the counter, this is not a supertask, but for the observer, it is. (This could theoretically occur due totime dilation, for example if the observer were falling into ablack holewhile observing a counter whose position is fixed relative to the singularity.)
Gustavo E. Romeroin the paper 'The collapse of supertasks'[7]maintains that any attempt to carry out a supertask will result in the formation of ablack hole, making supertasks physically impossible.
The impact of supertasks on theoretical computer science has triggered some new and interesting work, for example Hamkins and Lewis – "Infinite Time Turing Machine".[8]
Suppose there is a jar capable of containing infinitely many marbles and an infinite collection of marbles labelled 1, 2, 3, and so on. At timet= 0, marbles 1 through 10 are placed in the jar and marble 1 is taken out. Att= 0.5, marbles 11 through 20 are placed in the jar and marble 2 is taken out; att= 0.75, marbles 21 through 30 are put in the jar and marble 3 is taken out; and in general at timet= 1 − 0.5n, marbles 10n+ 1 through 10n+ 10 are placed in the jar and marblen+ 1 is taken out. How many marbles are in the jar at timet= 1?
One argument states that there should be infinitely many marbles in the jar, because at each step beforet= 1 the number of marbles increases from the previous step and does so unboundedly. A second argument, however, shows that the jar is empty. Consider the following argument: if the jar is non-empty, then there must be a marble in the jar. Let us say that that marble is labeled with the numbern. But at timet= 1 − 0.5n- 1, thenth marble has been taken out, so marblencannot be in the jar. This is a contradiction, so the jar must be empty. The Ross–Littlewood paradox is that here we have two seemingly perfectly good arguments with completely opposite conclusions.
There has been considerable interest inJ. A. Benardete’s “Paradox of the Gods”:[9]
A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods.[10]
Inspired byJ. A. Benardete’s paradox regarding an infinite series of assassins,[11]David Chalmersdescribes the paradox as follows:
There are countably many grim reapers, one for every positive integer. Grim reaper 1 is disposed to kill you with a scythe at 1pm, if and only if you are still alive then (otherwise his scythe remains immobile throughout), taking 30 minutes about it. Grim reaper 2 is disposed to kill you with a scythe at 12:30 pm, if and only if you are still alive then, taking 15 minutes about it. Grim reaper 3 is disposed to kill you with a scythe at 12:15 pm, and so on. You are still alive just before 12pm, you can only die through the motion of a grim reaper’s scythe, and once dead you stay dead. On the face of it, this situation seems conceivable — each reaper seems conceivable individually and intrinsically, and it seems reasonable to combine distinct individuals with distinct intrinsic properties into one situation. But a little reflection reveals that the situation as described is contradictory. I cannot survive to any moment past 12pm (a grim reaper would get me first), but I cannot be killed (for grim reapernto kill me, I must have survived grim reapern+1, which is impossible).[12]
It has gained significance in philosophy via its use in arguing for a finite past, thereby bearing relevance to theKalam cosmological argument.[13][14][15][16]
Proposed byE. Brian Davies,[17]this is a machine that can, in the space of half an hour, create an exact replica of itself that is half its size and capable of twice its replication speed. This replica will in turn create an even faster version of itself with the same specifications, resulting in a supertask that finishes after an hour. If, additionally, the machines create a communication link between parent and child machine that yields successively faster bandwidth and the machines are capable of simple arithmetic, the machines can be used to perform brute-force proofs of unknown conjectures. However, Davies also points out that – due to fundamental properties of the real universe such asquantum mechanics,thermal noiseandinformation theory– his machine cannot actually be built.
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Finitismis aphilosophy of mathematicsthat accepts the existence only offinitemathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g.,infinite sets) are accepted as existing.
The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While allnatural numbersare accepted as existing, thesetof all natural numbers is not considered to exist as a mathematical object. Thereforequantificationover infinite domains is not considered meaningful. The mathematical theory often associated with finitism isThoralf Skolem'sprimitive recursive arithmetic.
The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase whenGeorg Cantorin 1874 introduced what is now callednaive set theoryand used it as a base for his work ontransfinite numbers. When paradoxes such asRussell's paradox,Berry's paradoxand theBurali-Forti paradoxwere discovered in Cantor's naive set theory, the issue became a heated topic among mathematicians.
There were various positions taken by mathematicians. All agreed about finite mathematical objects such as natural numbers. However there were disagreements regarding infinite mathematical objects.
One position was theintuitionistic mathematicsthat was advocated byL. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.
Another position was endorsed byDavid Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led toHilbert's programof proving bothconsistencyandcompletenessof set theory using finitistic means as this would imply that adding ideal mathematical objects isconservativeover the finitistic part. Hilbert's views are also associated with theformalist philosophy of mathematics. Hilbert's goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due toKurt Gödel'sincompleteness theorems. However,Harvey Friedman'sgrand conjecturewould imply that most mathematical results are provable using finitistic means.
Hilbert did not give a rigorous explanation of what he considered finitistic and referred to as elementary. However, based on his work withPaul Bernayssome experts such asTait (1981)have argued thatprimitive recursive arithmeticcan be considered an upper bound on what Hilbert considered finitistic mathematics.[1]
As a result of Gödel's theorems, as it became clear that there is no hope of proving both the consistency and completeness of mathematics, and with the development of seemingly consistentaxiomatic set theoriessuch asZermelo–Fraenkel set theory, most modern mathematicians do not focus on this topic.
In her bookThe Philosophy of Set Theory,Mary Tilescharacterized those who allowpotentially infiniteobjects asclassical finitists, and those who do not allow potentially infinite objects asstrict finitists: for example, a classical finitist would allow statements such as "every natural number has asuccessor" and would accept the meaningfulness ofinfinite seriesin the sense oflimitsof finite partial sums, while a strict finitist would not. Historically, the written history of mathematics was thus classically finitist until Cantor created the hierarchy oftransfinitecardinalsat the end of the 19th century.
Leopold Kroneckerremained a strident opponent to Cantor's set theory:[2]
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.God created the integers; all else is the work of man.
Reuben Goodsteinwas another proponent of finitism. Some of his work involved building up toanalysisfrom finitist foundations.
Although he denied it, much ofLudwig Wittgenstein's writing on mathematics has a strong affinity with finitism.[4]
If finitists are contrasted withtransfinitists(proponents of e.g.Georg Cantor's hierarchy of infinities), then alsoAristotlemay be characterized as a finitist. Aristotle especially promoted thepotential infinityas a middle option between strict finitism andactual infinity(the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinitecardinalandordinalnumbers, which have nothing to do with the things in nature):
But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.
Ultrafinitism(also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large.
Towards the end of the 20th centuryJohn Penn Mayberrydeveloped a system of finitary mathematics which he called "Euclidean Arithmetic". The most striking tenet of his system is a complete and rigorous rejection of the special foundational status normally accorded to iterative processes, including in particular the construction of the natural numbers by the iteration "+1". Consequently Mayberry is in sharp dissent from those who would seek to equate finitary mathematics withPeano arithmeticor any of its fragments such asprimitive recursive arithmetic.
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Private biometricsis a form of encryptedbiometrics, also calledprivacy-preserving biometric authentication methods, in which the biometricpayloadis a one-way,homomorphically encrypted feature vectorthat is 0.05% the size of the originalbiometrictemplate and can be searched with full accuracy, speed and privacy. The feature vector'shomomorphic encryptionallows search and match to be conducted inpolynomial timeon an encrypted dataset and the search result is returned as an encrypted match. One or more computing devices may use an encrypted feature vector to verify an individualperson(1:1 verify) or identify an individual in adatastore(1:many identify) without storing, sending or receivingplaintextbiometric data within or between computing devices or any other entity. The purpose of private biometrics is to allow a person to beidentifiedorauthenticatedwhile guaranteeing individualprivacyand fundamentalhuman rightsby only operating on biometric data in the encrypted space. Some private biometrics including fingerprint authentication methods, face authentication methods, and identity-matching algorithms according to bodily features. Private biometrics are constantly evolving based on the changing nature of privacy needs, identity theft, and biotechnology.
Biometricsecuritystrengthens user authentication but, until recently, also implied important risks to personal privacy. Indeed, while compromisedpasswordscan be easily replaced and are notpersonally identifiable information(PII), biometric data is considered highly sensitive due to its personal nature, unique association with users, and the fact that compromised biometrics (biometric templates) cannot be revoked or replaced. Private biometrics have been developed to address this challenge. Private Biometrics provide the necessary biometric authentication while simultaneously minimizing user's privacy exposure through the use of one-way, fullyhomomorphic encryption.
The Biometric Open Protocol Standard,IEEE 2410-2018, was updated in 2018 to include private biometrics and stated that the one-way fully homomorphic encrypted feature vectors, “...bring a new level of consumer privacy assurance by keeping biometric data encrypted both at rest and in transit.” TheBiometric Open Protocol Standard (BOPS III)also noted a key benefit of private biometrics was the new standard allowed for simplification of theAPIsince the biometric payload was always one-way encrypted and therefore had no need forkey management.[1]
Historically, biometric matching techniques have been unable to operate in the encrypted space and have required the biometric to be visible (unencrypted) at specific points during search and match operations. This decrypt requirement made large-scale search across encrypted biometrics (“1:many identify”) infeasible due to both significant overhead issues (e.g. complex key management and significant data storage and processing requirements) and the substantial risk that the biometrics were vulnerable to loss when processed in plaintext within theapplicationoroperating system(seeFIDO Alliance, for example).
Biometric security vendors complying withdata privacy lawsand regulations (including Apple FaceID, Samsung, Google) therefore focused their efforts on the simpler 1:1 verify problem and were unable to overcome the large computational demands required forlinear scanto solve the 1:many identify problem.[2]
Today, private biometric cryptosystems overcome these limitations and risks through the use of one-way, fullyhomomorphic encryption. This form of encryption allows computations to be carried out onciphertext, allows the match to be conducted on an encrypted dataset without decrypting the reference biometric, and returns an encrypted match result. Matching in the encrypted space offers the highest levels of accuracy, speed and privacy and eliminates the risks associated with decrypting biometrics.[3]
The private biometric feature vector is much smaller (0.05% the size of the original biometric template) but yet maintains the same accuracy as the original plaintext reference biometric. In testing using Google's unified embedding forface recognitionand clusteringCNN(“Facenet”),[4]Labeled Faces in the Wild (LFW) (source), and other open source faces, private biometric feature vectors returned the same accuracy as plaintext facial recognition. Using an 8MB facial biometric, one vendor reported an accuracy rate of 98.7%. The same vendor reported accuracy increased to 99.99% when using three 8MB facial biometrics and a vote algorithm (best two out of 3) to predict.[5]
As the quality of the facial biometric image declined, accuracy degraded very slowly. For 256kB facial images (3% the quality of an 8MB picture), the same vendor reported 96.3% accuracy and that theneural networkwas able to maintain similar accuracy through boundary conditions including extreme cases of light or background.[6]
The private biometric feature vector is 4kB and contains 128floating point numbers. In contrast, plaintext biometric security instances (including Apple Face ID[7]) currently use 7MB to 8MB reference facial biometrics (templates). By using the much smaller feature vector, the resulting search performance is less than one second per prediction using a datastore of 100 million open source faces (“polynomial search”).[8]The private biometric test model used for these results was Google's unified embedding for face recognition and clusteringCNN(“Facenet”),[4]Labeled Faces in the Wild (LFW) (source), and other open source faces.
As with all ideal one-waycryptographic hashfunctions, decrypt keys do not exist for private biometrics so it isinfeasibleto generate the original biometric message from the private biometric feature vector (its hash value) except by trying all possible messages. Unlike passwords, however, no two instances of a biometric are exactly the same or, stated in another way, there is no constant biometric value, so a brute force attack using all possible faces would only produce an approximate (fuzzy) match. Privacy and fundamental human rights are therefore guaranteed.
Specifically, the private biometric feature vector is produced by a one-way cryptographic hash algorithm that maps plaintext biometric data of arbitrary size to a small feature vector of a fixed size (4kB) that is mathematically impossible to invert. The one-way encryption algorithm is typically achieved using a pre-trained convolutional neural network (CNN), which takes a vector of arbitrary real-valued scores and squashes it to a 4kB vector of values between zero and one that sum to one.[9]It is mathematically impossible to reconstruct the original plaintext image from a private biometric feature vector of 128 floating point numbers.[10]
One-way encryptions offer unlimited privacy by containing no mechanism to reverse the encryption and disclose the original data. Once a value is processed through a one-way hash, it is not possible to discover to the original value (hence the name “one-way”).[11]
The first one-way encryptions were likely developed by James H. Ellis, Clifford Cocks, and Malcolm Williamson at the UK intelligence agency GCHQ during the 1960s and 1970s and were published independently by Diffie and Hellman in 1976 (History of cryptography). Common modern one-way encryption algorithms, includingMD5(message digest) andSHA-512(secure hash algorithm) are similar to the first such algorithms in that they also contain no mechanism to disclose the original data. The output of these modern one-way encryptions offer high privacy but are not homomorphic, meaning that the results of the one-way encryptions do not allow high order math operations (such as match). For example, we cannot use twoSHA-512sums to compare the closeness of two encrypted documents. This limitation makes it impossible for these one-way encryptions to be used to support classifying models in machine learning—or nearly anything else.[citation needed]
The first one-way, homomorphically encrypted,Euclidean-measurablefeature vector for biometric processing was proposed in a paper by Streit, Streit and Suffian in 2017.[12]In this paper, the authors theorized and also demonstrated using a small sample size (n=256 faces) that (1) it was possible to use neural networks to build a cryptosystem for biometrics that produced one-way, fully homomorphic feature vectors composed of normalized floating-point values; (2) the sameneural networkwould also be useful for 1:1 verification (matching); and (3) the sameneural networkwould not be useful in 1:many identification tasks since search would occur inlinear time(i.e.non polynomial). The paper's first point was (in theory) later shown to be true, and the papers first, second and third points were later shown to be true only for small samples but not for larger samples.
A later tutorial (blog posting) by Mandel in 2018 demonstrated a similar approach to Streit, Streit and Suffian and confirmed using aFrobenius2 distance function to determine the closeness of two feature vectors. In this posting, Mandel used a Frobenius 2 distance function to determine the closeness of two feature vectors and also demonstrated successful 1:1 verification. Mandel did not offer a scheme for 1:many identification as this method would have required a non polynomial full linear scan of the entire database. The Streit, Streit and Suffian paper attempted a novel “banding” approach for 1:many identification in order to mitigate the full linear scan requirement, but it is now understood that this approach produced too much overlap to help in identification.[13]
The first claimed commercial implementation of private biometrics,Private.id, was published by Private Identity, LLC in May 2018 by using the same method to provide 1:many identification in polynomial time across a large biometrics database (100 million faces).
On the client device, Private.id transforms each reference biometric (template) into a one-way, fully homomorphic, Euclidean-measurablefeature vectorusing matrix multiplication from the neural network that may then be stored locally or transmitted. The original biometric is deleted immediately after the feature vector is computed or, if the solution isembeddedin firmware, the biometric is transient and never stored. Once the biometric is deleted, it is no longer possible to lose or compromise the biometric.[5]
The Private.idfeature vectorcan be used in one of two ways. If the feature vector is stored locally, it may be used to compute 1:1 verification with high accuracy (99% or greater) usinglinear mathematics. If the feature vector is also stored in aCloud, thefeature vectormay also be used as input for a neural network to perform 1:many identification with the same accuracy, speed and privacy as the original plaintext reference biometric (template).[5]
Private biometrics use the following two properties in deriving compliance with biometric data privacy laws and regulations worldwide. First, the private biometrics encryption is a one-way encryption, so loss of privacy by decryption is mathematically impossible and privacy is therefore guaranteed. Second, since no two instances of a biometric are exactly the same or, stated in another way, there is no constant biometric value, the private biometrics one-way encryptedfeature vectoris Euclidean Measureable in order to provide a mechanism to determine a fuzzy match in which two instances of the same identity are “closer” than two instances of a different identity.
The IEEE 2410-2018Biometric Open Protocol Standardwas updated in 2018 to include private biometrics. The specification stated that one-way fully homomorphic encrypted feature vectors, “bring a new level of consumer privacy assurance by keeping biometric data encrypted both at rest and in transit.”IEEE 2410-2018also noted a key benefit of private biometrics is that the new standard allows for simplification of theAPIsince the biometricpayloadis always one-way encrypted and there is no need for key management.[1]
Private biometrics enables passive encryption (encryption at rest), the most difficult requirement of the US Department of Defense Trusted Computer System Evaluation Criteria (TCSEC). No other cryptosystem or method provides operations on rested encrypted data, so passive encryption—an unfulfilled requirement of theTCSECsince 1983, is no longer an issue.
Private biometrics technology is an enabling technology for applications and operating systems—but itself does not directly address—the auditing and constant protection concepts introduced in theTCSEC.
Private biometrics, as implemented in a system that conforms toIEEE 2410-2018 BOPS III,[1]satisfies the privacy requirements of the US Department of Defense Standard Trusted Computer System Evaluation Criteria (TCSEC). TheTCSECsets the basic requirements for assessing the effectiveness of computer security controls built into a computer system (“Orange Book, section B1”). Today, the applications and operating systems contain features that comply withTCSEClevels C2 and B1 except they lackhomomorphic encryptionand so do not process dataencryptedat rest. We typically, if not always, obtained waivers, because there was not a known work around. Adding private biometrics to these operating systems and applications resolves this issue.
For example, consider the case of a typicalMySQLdatabase. To queryMySQLin a reasonable period of time, we need data that maps to indexes that maps to queries that maps to end user data. To do this, we work withplaintext. The only way to encrypt this is to encrypt the entire data store, and to decrypt the entire data store, prior to use. Since data use is constant, the data is never encrypted. Thus, in the past we would apply for waivers because there was no known work around. Now using private biometrics, we can match and do operations on data that is alwaysencrypted.
Private biometrics, as implemented in a system that conforms to IEEE 2410-2018BOPS III, comply with the standards of the Multiple Independent Levels of Security/Safety (MILS) architecture.MILSbuilds on the Bell and La Padula theories on secure systems that represent the foundational theories of the US DoD Standard Trusted Computer System Evaluation Criteria (TCSEC), or the DoD “Orange Book.” (See paragraphs above.)
Private biometrics’ high-assurancesecurityarchitecture is based on the concepts of separation and controlled information flow and implemented using only mechanisms that support trustworthy components, thus the security solution is non-bypassable, evaluable, always invoked and tamper proof. This is achieved using the one-way encryptedfeature vector, which elegantly allows only encrypted data (and never stores or processes plaintext) between security domains and through trustworthy security monitors.
Specifically, private biometrics systems are:
Unsecure biometric data are sensitive due to their nature and how they can be used.Implicit authenticationis a common practice when usingpasswords, as a user may prove knowledge of a password without actually revealing it. However, two biometric measurements of the samepersonmay differ, and this fuzziness of biometric measurements renders implicit authentication protocols useless in the biometrics domain.
Similarly, private equality testing, where two devices or entities want to check whether the values that they hold are the same without presenting them to each other or to any other device or entity, is well practiced and detailed solutions have been published. However, since two biometrics of the same person may not be equal, these protocols are also ineffective in the biometrics domain. For instance, if the two values differ in τ bits, then one of the parties may need to present 2τ candidate values for checking.[14]
Prior to the introduction of private biometrics, biometric techniques required the use ofplaintextsearch for matching so each biometric was required to be visible (unencrypted) at some point in the search process. It was recognized that it would be beneficial to instead conduct matching on an encrypted dataset.
Encrypt match is typically accomplished using one-way encryption algorithms, meaning that given the encrypted data, there is no mechanism to get to the original data. Common one-way encryption algorithms areMD5andSHA-512. However, these algorithms are nothomomorphic, meaning that there is no way to compare the closeness of two samples of encrypted data, and thus no means to compare. The inability to compare renders any form of classifying model inmachine learninguntenable.
Homomorphic encryptionis a form ofencryptionthat allows computations to be carried out onciphertext, thus generating an encrypted match result. Matching in theencryptedspace using a one-way encryption offers the highest level of privacy. With a payload offeature vectorsone-wayencrypted, there is no need to decrypt and no need for key management.
A promising method of homomorphic encryption on biometric data is the use of machine learning models to generatefeature vectors. Forblack-box models, such asneural networks, these vectors can not by themselves be used to recreate the initial input data and are therefore a form of one-way encryption. However, the vectors are euclidean measurable, so similarity between vectors can be calculated. This process allows for biometric data to be homomorphically encrypted.
For instance if we consider facial recognition performed with theEuclidean Distance, when we match two face images using a neural network, first each face is converted to a float vector, which in the case of Google's FaceNet, is of size 128. The representation of this float vector is arbitrary and cannot bereverse-engineeredback to the original face. Indeed, the matrix multiplication from the neural network then becomes the vector of the face, is Euclidean measurable but unrecognizable, and cannot map back to any image.
Prior to the availability of private biometrics, research focused on ensuring the prover's biometric would be protected against misuse by a dishonest verifier through the use of partiallyhomomorphicdata or decrypted(plaintext) data coupled with a private verification function intended to shield private data from the verifier. This method introduced a computational and communication overhead which was computationally inexpensive for 1:1 verification but proved infeasible for large 1:many identification requirements.
From 1998 to 2018cryptographicresearchers pursued four independent approaches to solve the problem:cancelable biometrics, BioHashing, Biometric Cryptosystems, and two-way partiallyhomomorphic encryption.[15]
The feature transformation approach “transformed” biometric feature data to random data through the use of a client-specific key or password. Examples of this approach includedbiohashingand cancelable biometrics.The approach offered reasonable performance but was found to be insecure if the client-specific key was compromised.
Cancelable Biometrics
The first use of indirect biometric templates (later calledcancelable biometrics) was proposed in 1998 by Davida, Frankel and Matt.[16]Three years later, Ruud Bolle, Nilini Ratha and Jonathan Connell, working in IBM's Exploratory Computer Vision Group, proposed the first concrete idea ofcancelable biometrics.[17][18]
Cancelable biometrics were defined in these communications as biometric templates that were unique for every application and that, if lost, could be easily cancelled and replaced. The solution was (at the time) thought to provide higher privacy levels by allowing multiple templates to be associated with the same biometric data by storing only the transformed (hashed) version of the biometric template. The solution was also promoted for its ability toprevent linkageof the user's biometric data across various databases since only a transformed version of the biometric template (and not the unencrypted (plaintext) biometric template) was stored for later use.[19][20][21]
Cancelable biometrics were deemed useful because of their diversity, reusability and one-way encryption (which, at the time, was referred to as a one-way transformation). Specifically, no cancellable template could be used in two different applications (diversity); it was straightforward to revoke and reissuance a cancellable template in the event of compromise (reusability); and the one-way hash of the template prevented recovery of sensitive biometric data. Finally, it was postulated that the transformation would not deteriorate accuracy.[22]
Research intocancelable biometricsmoved into BioHashing by 2004. The BioHashing feature transformation technique was first published by Jin, Ling and Goh and combined biometric features and atokenized(pseudo-) random number (TRN). Specifically, BioHash combined the biometric template with a user-specific TRN to produce a set of non-invertible binary bit strings that were thought to be irreproducible if both the biometric and the TRN were not presented simultaneously.[23]
Indeed, it was first claimed that the BioHashing technique had achieved perfect accuracy (equal error rates) for faces, fingerprints and palm prints, and the method gained further traction when its extremely low error rates were combined with the claim that its biometric data was secure against loss because factoring the inner products of biometrics feature and TRN was an intractable problem.[23][19]
By 2005, however, researchers Cheung and Kong (Hong Kong Polytechnic and University of Waterloo) asserted in two journal articles that BioHashing performance was actually based on the sole use of TRN and conjectured that the introduction of any form of biometric become meaningless since the system could be used only with the tokens.[24][25]These researchers also reported that the non-invertibility of the random hash would deteriorate the biometric recognition accuracy when the genuine token was stolen and used by an impostor (“the stolen-token scenario”).[24][26]
Biometriccryptosystemswere originally developed to either securecryptographic keysusing biometric features (“key-biometrics binding”) or to directly generate cryptographic keys from biometric features.[27]Biometric cryptosystems used cryptography to provide the system with cryptographic keys protection and biometrics to provide the system with dynamically generated keys to secure the template and biometric system.[28]
The acceptance and deployment of biometric cryptosystem solutions was constrained, however, by the fuzziness related with biometric data. Hence,error correction codes(ECCs), including includes fuzzy vault and fuzzy commitment, were adopted to alleviate the fuzziness of the biometric data. This overall approach proved impractical, however, due to the need for accurate authentication and suffered from security issues due to its need for strong restriction to support authentication accuracy.[29]
Future research on biometric cryptosystems is likely to focus on a number of remaining implementation challenges and security issues involving both the fuzzy representations of biometric identifiers and the imperfect nature of biometric feature extraction and matching algorithms. And, unfortunately, since biometric cryptosystems can, at the current time, be defeated using relatively simple strategies leveraging both weaknesses of the current systems (the fuzzy representations of biometric identifiers and the imperfect nature of biometric feature extraction and matching algorithms), it is unlikely that these systems will be able to deliver acceptable end-to-end system performance until suitable advances are achieved.[30]
The two-way partiallyhomomorphic encryptionmethod for private biometrics was similar to the today's private biometrics in that it offered protection of biometric feature data through the use of homomorphic encryption and measured the similarity of encrypted feature data by metrics such as the Hamming and the Euclidean distances. However, the method was vulnerable to data loss due to the existence of secret keys that were to be managed by trusted parties. Widespread adoption of the approach also suffered from the encryption schemes’ complex key management and large computational and data storage requirements.[15]
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Verifiable computing(orverified computationorverified computing) enables acomputertooffloadthecomputationof some function, to other perhaps untrustedclients, while maintaining verifiable results. The other clients evaluate the function and return the result with a proof that thecomputationof the function was carried out correctly. The introduction of this notion came as a result of the increasingly common phenomenon of "outsourcing" computation to untrusted users in projects such asSETI@homeand also to the growing desire to enable computationally-weak devices to outsource computational tasks to a more powerful computation service, as incloud computing. The concept dates back to work by Babai et al.,[1]and has been studied under various terms, including "checking computations" (Babai et al.), "delegating computations",[2]"certified computation",[3]and verifiable computing. The termverifiable computingitself was formalized by Rosario Gennaro,Craig Gentry, and Bryan Parno,[4]and echoes Micali's "certified computation".[3]
The growing desire to outsource computational tasks from a relatively weak computational device (client) to a more powerful computation services (worker), and the problem of dishonest workers who modify their client's software to return plausible results without performing the actual work[5]motivated the formalization of the notion of Verifiable Computation.[4]
Verifiable computing is not only concerned with getting the result of the outsourced function on the client's input and theproofof its correctness, but also with the client being able to verify the proof with significantly less computational effort than computing the function from scratch.
Considerable attention has been devoted in verifying the computation of functions performed by untrusted workers including the use ofsecure coprocessors,[6][7]Trusted Platform Modules(TPMs),[8]interactive proofs,[9][10]probabilistically checkable proofs,[11][12]efficient arguments,[13][14]and Micali's CS proofs.[15]These verifications are either interactive which require the client to interact with the worker to verify the correctness proof,[13][14]or are non-interactive protocols which can be proven in therandom oraclemodel.[15]
The largest verified computation (SETI@home) uses verification by replication.
TheSETI@home verificationprocess involves one client machine and many worker machines.
The client machine sends identical workunits to multiple computers (at least 2).
When not enough results are returned in a reasonable amount of time—due to machines accidentally turned off, communication breakdowns, etc.—or the results do not agree—due to computation errors, cheating by submitting false data without actually doing the work, etc.—then the client machine sends more identical workunits to other worker machines.
Once a minimum quorum (often 2) of the results agree, then the client assumes those results (and other identical results for that workunit) are correct.
The client grants credit to all machines that returned the correct results.
Gennaro et al.[4]defined the notion of verifiable computation scheme as aprotocolbetween two polynomial time parties to collaborate on the computation of a function F: {0,1}n→ {0,1}m. This scheme consists of three main phases:
The defined notion of verifiable computation scheme minimizes the interaction between the client and the worker into exactly two messages, where a single message is sent from each party to the other party during the different phases of the protocol.[4]
Gennaro et al.[4]defined a verifiable computation scheme for any functionFusingYao's garbled circuit[16][17]combined with afully homomorphic encryption system.
This verifiable computation schemeVCis defined as follows:[4]
VC= (KeyGen, ProbGen, Compute, Verify)consists of four algorithms as follows:
The protocol of the verifiable computations scheme defined by Gennaro et al.[4]works as follows:
The function F should be represented as aBoolean circuiton which thekey generationalgorithm would be applied. The key generation algorithm runs Yao's garbling procedure over this Boolean circuit to compute the public and secret keys. The public key (PK) is composed of all theciphertextsthat represent the garbled circuit, and the secret key (SK) is composed of all the random wire labels. The generated secret key is then used in the problem generation algorithm. This algorithm first generates a new pair of public and secret keys for thehomomorphic encryption scheme, and then uses these keys with the homomorphic scheme to encrypt the correct input wires, represented as the secret key of the garbled circuit. The produced ciphertexts represent the public encoding of the input (σx) that is given to the worker, while the secret key (τx) is kept private by the client. After that, the worker applies the computation steps of the Yao's protocol over the ciphertexts generated by the problem generation algorithm. This is done byrecursivelydecrypting the gate ciphertexts until arriving to the final output wire values (σy). The homomorphic properties of the encryption scheme enable the worker to obtain an encryption of the correct output wire. Finally, the worker returns the ciphertexts of the output to the client who decrypts them to compute the actual output y = F(x) or ⊥.
The definition of the verifiable computation scheme states that the scheme should be both correct and secure.Scheme Correctnessis achieved if the problem generation algorithm produces values that enable an honest worker to compute encoded output values that will verify successfully and correspond to the evaluation of F on those inputs. On the other hand, a verifiable computation scheme issecureif a malicious worker cannot convince the verification algorithm to accept an incorrect output for a given function F and input x.
Although it was shown that verifiable computing is possible in theory (using fullyhomomorphic encryptionor viaprobabilistically checkable proofs), most of the known constructions are very expensive in practice. Recently, some researchers have looked at making verifiable computation practical. One such effort is the work ofUT Austinresearchers.[18]The authors start with an argument system based onprobabilistically checkable proofsand reduce its costs by a factor of 1020. They also implemented their techniques in thePeppersystem. The authors note that "Our conclusion so far is that, as a tool for building secure systems, PCPs and argument systems are not a lost cause."
The overall area, which now includes a number of implementations by different groups, has been surveyed.[19]
In the 2010s, verifiable computing techniques have seen an increase of practical applications in blockchain technology.[20]
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Client-side encryptionis thecryptographictechnique ofencryptingdata on the sender's side, before it is transmitted to aserversuch as acloud storage service.[1]Client-side encryption features an encryption key that is not available to the service provider, making it difficult or impossible for service providers to decrypt hosted data.
Client-side encryption allows for the creation of applications whose providers cannot access the data its users have stored, thus offering a high level of privacy.[1]
Applications utilizing client-side encryption are sometimes marketed under the misleading or incorrect term"zero-knowledge",[2]but this is a misnomer, as the termzero-knowledgedescribes something entirely different in the context of cryptography.
Client-side encryption seeks to eliminate the potential for data to be viewed by service providers (or third parties that compel service providers to deliver access to data), client-side encryption ensures that data and files that are stored in the cloud can only be viewed on the client-side of the exchange. This prevents data loss and the unauthorized disclosure of private or personal files, providing increased peace of mind for its users.[1]
Current recommendations by industry professionals as well as academic scholars offer great vocal support for developers to include client-side encryption to protect the confidentiality and integrity of information.[3][4][5]
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Confidential computingis a security andprivacy-enhancing computational techniquefocused on protectingdata in use. Confidential computing can be used in conjunction with storage and network encryption, which protectdata at restanddata in transitrespectively.[1][2]It is designed to address software, protocol, cryptographic, and basic physical and supply-chain attacks, although some critics have demonstrated architectural andside-channel attackseffective against the technology.[3]
The technology protects data in use by performing computations in a hardware-basedtrusted execution environment(TEE).[3]Confidential data is released to the TEE only once it is assessed to be trustworthy. Different types of confidential computing define the level of data isolation used, whethervirtual machine,application, orfunction, and the technology can be deployed in on-premise data centers, edge locations, or the public cloud. It is often compared with other privacy-enhancing computational techniques such asfully homomorphic encryption,secure multi-party computation, andTrusted Computing.
Confidential computing is promoted by the Confidential Computing Consortium (CCC) industry group, whose membership includes major providers of the technology.[4]
Trusted execution environments (TEEs) "prevent unauthorized access or modification of applications and data while they are in use, thereby increasing the security level of organizations that manage sensitive and regulated data".[4][5]Trusted execution environments can be instantiated on a computer's processing components such as acentral processing unit(CPU) or agraphics processing unit(GPU).[6]In their various implementations, TEEs can provide different levels of isolation includingvirtual machine, individual application, or compute functions.[7]Typically, data in use in a computer's compute components and memory exists in a decrypted state and can be vulnerable to examination or tampering by unauthorized software or administrators.[8][9]According to the CCC, confidential computing protects data in use through a minimum of three properties:[10]
In addition to trusted execution environments, remotecryptographicattestation is an essential part of confidential computing. The attestation process assesses the trustworthiness of a system and helps ensure that confidential data is released to a TEE only after it presents verifiable evidence that it is genuine and operating with an acceptable security posture.[11][12][13]It allows the verifying party to assess the trustworthiness of a confidential computing environment through an "authentic, accurate, and timely report about the software and data state" of that environment. "Hardware-based attestation schemes rely on a trusted hardware component and associatedfirmwareto execute attestation routines in a secure environment".[10]Without attestation, a compromised system could deceive others into trusting it, claim it is running certain software in a TEE, and potentially compromise the confidentiality or integrity of the data being processed or the integrity of the trusted code.[14][10][15]
Technical approaches to confidential computing may vary in which software, infrastructure and administrator elements are allowed to access confidential data. The "trust boundary," which circumscribes atrusted computing base (TCB), defines which elements have the potential to access confidential data, whether they are acting benignly or maliciously.[16]Confidential computing implementations enforce the defined trust boundary at a specific level of data isolation. The three main types of confidential computing are:
Virtual machine isolation removes the elements controlled by the computer infrastructure or cloud provider, but allows potential data access by elements inside a virtual machine running on the infrastructure. Application or process isolation permits data access only by authorized software applications or processes. Function or library isolation is designed to permit data access only by authorizedsubroutinesor modules within a larger application, blocking access by any other system element, including unauthorized code in the larger application.[17][18]
As confidential computing is concerned with the protection of data in use, only certainthreat modelscan be addressed by this technique. Other types of attacks are better addressed by other privacy-enhancing technologies.[10]
The following threat vectors are generally considered in scope for confidential computing:[10]
The degree and mechanism of protection against these threats varies with specific confidential computing implementations.[20]
Threats generally defined as out of scope for confidential computing include:[10]
Confidential computing can be deployed in the public cloud, on-premise data centers, or distributed "edge" locations, including network nodes, branch offices, industrial systems and others.[21]
Confidential computing protects theconfidentialityandintegrityof data and code from the infrastructure provider, unauthorized or malicious software and system administrators, and other cloud tenants, which may be a concern for organizations seeking control over sensitive or regulated data.[22][23]The additional security capabilities offered by confidential computing can help accelerate the transition of more sensitive workloads to the cloud or edge locations.[24]
Confidential computing can enable multiple parties to engage in joint analysis using confidential or regulated data inside a TEE while preserving privacy and regulatory compliance.[25][26]In this case, all parties benefit from the shared analysis, but no party's sensitive data or confidential code is exposed to the other parties or system host.[8]Examples include multiple healthcare organizations contributing data to medical research, or multiple banks collaborating to identify financial fraud ormoney laundering.[27][15]
Oxford University researchers proposed the alternative paradigm called "Confidential Remote Computing" (CRC), which supports confidential operations in Trusted Execution Environments across endpoint computers considering multiple stakeholders as mutually distrustful data, algorithm and hardware providers.[28]
Confidential generative AI
Confidential computing technologies can be applied to various stages of agenerative AIdeployments to help increase data or model privacy, security, and regulatory compliance. TEEs and remote attestation can protect the integrity of data during AI model training, keep non-public data confidential during inference orRetrieval Augmented Generation(RAG), and protect the AI model itself from various adversarial attacks or theft.[29][30]
Confidential computing assists in data protection and regulatory compliance by limiting which software and people may access regulated data, as well as providing greater assurance of data and code integrity. In addition, TEEs can assist withdata governanceby providing evidence of steps taken to mitigate risks and demonstrate that these were appropriate.[31]In 2021, theEuropean Union Agency for Cybersecurity(ENISA) classifies confidential computing as a "State of the Art" technology with respect to protecting data under the European Union'sGeneral Data Protection Regulationand Germany's IT Security Act (ITSiG).[32]
Regulations regardingdata localizationand residency ordata sovereigntymay require that sensitive data remain in a specific country or geographic bloc to provide assurance that the data will only be used in compliance with local law. Using confidential computing, only the workload owner holds the encryption keys required to decrypt data for processing inside a verified TEE.[33]This provides a technological safeguard that reduces the risk of data being exfiltrated and processed in plaintext in other countries or jurisdictions without the workload owner's consent.[34][35]
Additional use cases for confidential computing include blockchain applications with enhanced record privacy and code integrity, privacy-preserving advertising technology, confidential databases and more.
Multiple academic and security research groups have demonstrated architectural andside-channel attacksagainst CPU-based TEEs based on a variety of approaches.[3]These includepage faults,[36]caching,[27]and thememory bus,[37]as well as specifically Æpic[38]and SGAxe[39]against Intel SGX, and CIPHERLEAKS[40]against AMD SEV-SNP. Update mechanisms in the hardware, such asTrusted computing base(TCB) recovery, can mitigate side-channel vulnerabilities as they are discovered.[41][42]
The definition of confidential computing itself has also been criticized by some academic researchers. Scholars at theTechnical University of Dresden, Germany called it, "imprecise, incomplete and even conflicting."[43]Researchers have made recommendations to make it more detailed and exact to facilitate research and comparisons with other security technologies.[43]
"Confidential Remote Computing" (CRC) paradigm,[44]claims to revert confidential computing to original design principles of TEEs and advocate for small enclaves, running in available end-users computers. CRC adds practices and templates for multiple stakeholders, such as different data owners, hardware owners and algorithm owners. CRC extends the broad notion of confidential computing by adding practices and methodologies for individual use.
None of the major microprocessor or GPU providers offer Confidential computing hardware in devices for personal computers anymore, which limits use cases only to server-class platforms. Intel SGX was introduced for PCs in 6th Generation Intel Core (Skylake) processors in 2015, but deprecated in the 11th Generation Intel Core processors (Rocket Lake) in 2022.[45]
Confidential computing is often compared to other security or privacy-enhancing technologies, including fully homomorphic encryption, secure multi-party computing and trusted computing.
Fullyhomomorphic encryption(FHE) is a form of encryption that permits users to perform computations on encrypted data without first decrypting it. Confidential computing, in contrast, transfers encrypted data inside a hardware-enforced, access-controlled TEE in the processor and memory, decrypts the data, and performs the required computations. Data may be re-encrypted before exiting the TEE. Compared to each other, FHE performance can suffer from higher computational overhead than confidential computing and require extensive application-specific coding[46]but is less susceptible to side-channel attacks since data is never decrypted.[47]Several researchers have described use cases where confidential computing TEEs and FHE work together to mitigate shortcomings of the technologies acting individually.[48][49]
Secure multi-party computation(SMPC) is a privacy-preserving technology that allows multiple parties to jointly compute a task using distributed algorithms while keeping each party's data private from the others. Confidential computing can also be used for privacy-preserving multi-party collaboration. Compared to each other, distributed computing with SMPC can be more expensive in terms of computation and network bandwidth,[50]but less susceptible to side-channel attacks since no party ever holds the complete data set.[47]
Trusted computingis a concept and set of standards published by theTrusted Computing Groupthat aim to establish trust in computing systems by using standardized hardware-based mechanisms like the Trusted Platform Module (TPM).[51]From a technical perspective, Trusted Computing and confidential computing rely on similar security concepts, such as trust architecture and remote attestation protocols. However, Trusted Computing targets a different set of threat models and large variety of platforms (e.g., phones, laptops, servers, network equipment);[52]confidential computing addresses attack vectors that target confidentiality and integrity of code and data in use, notably through the use of Trusted Execution Environments and memory encryption.
Confidential computing use cases require a combination of hardware and software, often delivered in conjunction with cloud service providers or server manufacturers.
(Arm CCA)
(Intel SGX)
2018 on Intel Xeon E 2100 series server processors[61](later deprecated)
2021 on 3rd Gen Intel Xeon Scalable processors[62]
(Intel TDX)
Confidential computing technology and services can be accessed via public cloud computing providers, includingAlibaba Cloud,[66]Baidu Cloud,[66]Google Cloud,[67]IBM Cloud,[68]Microsoft Azure,[69]OVHcloud[70]and others.
Application software is required to enable most confidential computing use cases. Providers of confidential computing software applications include Anjuna,[66]CanaryBit,[71]Cosmian,[72]CYSEC,[73]Decentriq,[74]Edgeless Systems,[75]Enclaive,[76]Fortanix,[77]IBM Hyper Protect Services,[78]Mithril Security,[79]Oblivious,[80]Opaque Systems,[81]Scontain,[82]Secretarium,[83]Super Protocol[84]and others.
Confidential computing is supported by an advocacy and technical collaboration group called the Confidential Computing Consortium.[85]The CCC was formed in 2019 under theLinux Foundation. The founding premiere members wereAlibaba,Arm,Google Cloud,Huawei,Intel,MicrosoftandRed Hat. The founding general members includedSUSE,Baidu,ByteDance, Decentriq, Fortanix, Kindite, Oasis Labs,Swisscom,TencentandVMware.[86][87]The CCC states its efforts are "focused on projects securing data in use and accelerating the adoption of confidential computing through open collaboration."[85]
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Searchable symmetric encryption(SSE) is a form ofencryptionthat allows one to efficiently search over a collection of encrypted documents or files without the ability to decrypt them.[1][2][3]SSE can be used to outsource files to an untrusted cloud storage server without ever revealing the files in the clear but while preserving the server's ability to search over them.
A searchable symmetric encryption scheme is asymmetric-key encryptionscheme that encrypts a collection of documentsD=(D1,…,Dn){\displaystyle \mathbf {D} =(\mathrm {D_{1}} ,\dots ,\mathrm {D_{n}} )}, where each documentDi⊆W{\displaystyle \mathrm {D_{i}} \subseteq \mathbb {W} }is viewed as a set of keywords from a keyword spaceW{\displaystyle \mathbb {W} }. Given the encryption keyK{\displaystyle K}and a keywordw∈W{\displaystyle w\in \mathbb {W} }, one can generate a search tokentk{\displaystyle tk}with which the encrypted data collection can be searched forw{\displaystyle w}. The result of the search is the subset of encrypted documents that contain the keywordw{\displaystyle w}.
A static SSE scheme consists of three algorithmsSSE=(Setup,Token,Search){\displaystyle {\mathsf {SSE=(Setup,Token,Search)}}}that work as follows:
A static SSE scheme is used by a client and an untrusted server as follows. The client encrypts its data collection using theSetup{\displaystyle {\mathsf {Setup}}}algorithm which returns a secret keyK{\displaystyle K}and an encrypted document collectionED{\displaystyle \mathbf {ED} }. The client keepsK{\displaystyle K}secret and sendsED{\displaystyle \mathbf {ED} }andI{\displaystyle \mathbf {I} }to the untrusted server. To search for a keywordw{\displaystyle w}, the client runs theToken{\displaystyle {\mathsf {Token}}}algorithm onK{\displaystyle K}andw{\displaystyle w}to generate a search tokentk{\displaystyle tk}which it sends to the server. The server runs Search withED{\displaystyle \mathbf {ED} },I{\displaystyle \mathbf {I} }, andtk{\displaystyle tk}and returns the resulting encrypted documents back to the client.
A dynamic SSE scheme supports, in addition to search, the insertion and deletion of documents. A dynamic SSE scheme consists of seven algorithmsSSE=(Setup,Token,Search,InsertToken,Insert,DeleteToken,Delete){\displaystyle {\mathsf {SSE=(Setup,Token,Search,InsertToken,Insert,DeleteToken,Delete)}}}whereSetup{\displaystyle {\mathsf {Setup}}},Token{\displaystyle {\mathsf {Token}}}andSearch{\displaystyle {\mathsf {Search}}}are as in the static case and the remaining algorithms work as follows:
To add a new documentDn+1{\displaystyle \mathrm {D_{n+1}} }the client runsInsertToken{\displaystyle {\mathsf {InsertToken}}}onK{\displaystyle K}andDn+1{\displaystyle \mathrm {D_{n+1}} }to generate an insert tokenitk{\displaystyle itk}which it sends to the server. The server runsInsert{\displaystyle {\mathsf {Insert}}}withED{\displaystyle \mathbf {ED} }anditk{\displaystyle itk}and stores the updated encrypted document collection. To delete a document with identifierid{\displaystyle id}, the client runs theDeleteToken{\displaystyle {\mathsf {DeleteToken}}}algorithm withK{\displaystyle K}andid{\displaystyle id}to generate a delete tokendtk{\displaystyle dtk}which it sends to the server. The server runsDelete{\displaystyle {\mathsf {Delete}}}withED{\displaystyle \mathbf {ED} }anddtk{\displaystyle dtk}and stores the updated encrypted document collection.
An SSE scheme that does not supportDeleteToken{\displaystyle {\mathsf {DeleteToken}}}andDelete{\displaystyle {\mathsf {Delete}}}is called semi-dynamic.
The problem of searching on encrypted data was considered bySong,WagnerandPerrig[1]though previous work onOblivious RAMbyGoldreichandOstrovsky[4]could be used in theory to address the problem. This work[1]proposed an SSE scheme with a search algorithm that runs in timeO(s){\displaystyle O(s)}, wheres=|D|{\displaystyle s=|\mathbf {D} |}. Goh[5]and Chang andMitzenmacher[6]gave new SSE constructions with search algorithms that run in timeO(n){\displaystyle O(n)}, wheren{\displaystyle n}is the number of documents. Curtmola, Garay,KamaraandOstrovsky[2]later proposed two static constructions withO(opt){\displaystyle O(\mathrm {opt} )}search time, whereopt{\displaystyle \mathrm {opt} }is the number of documents that containw{\displaystyle w}, which is optimal. This work also proposed a semi-dynamic construction withO(opt⋅log(u)){\displaystyle O(\mathrm {opt} \cdot \log(u))}search time, whereu{\displaystyle u}is the number of updates. An optimal dynamic SSE construction was later proposed byKamara, Papamanthou and Roeder.[7]
Goh[5]and Chang andMitzenmacher[6]proposed security definitions for SSE. These were strengthened and extended by Curtmola, Garay, Kamara and Ostrovsky[2]who proposed the notion of adaptive security for SSE. This work also was the first to observe leakage in SSE and to formally capture it as part of the security definition. Leakage was further formalized and generalized by Chase andKamara.[8]Islam, Kuzu and Kantarcioglu described the first leakage attack.[9]
All the previously mentioned constructions support single keyword search. Cash, Jarecki, Jutla,Krawczyk, Rosu and Steiner[10]proposed an SSE scheme that supports conjunctive search in sub-linear time inn{\displaystyle n}. The construction can also be extended to support disjunctive and Boolean searches that can be expressed in searchable normal form (SNF) in sub-linear time. At the same time, Pappas, Krell, Vo, Kolesnikov,Malkin, Choi, George, Keromytis andBellovin[11]described a construction that supports conjunctive and all disjunctive and Boolean searches in sub-linear time.
SSE schemes are designed to guarantee that the untrusted server cannot learn any partial information about the documents or the search queries beyond some well-defined and reasonable leakage. The leakage of a scheme is formally described using a leakage profile which itself can consists of several leakage patterns. SSE constructions attempt to minimize leakage while achieving the best possible search efficiency.
SSE security can be analyzed in several adversarial models but the most common are:
In the persistent model, there are SSE schemes that achieve a wide variety of leakage profiles. The most common leakage profile for static schemes that achieve single keyword search in optimal time isΛopt{\displaystyle \Lambda _{\mathrm {opt} }}which reveals the number of documents in the collection, the size of each document in the collection, if and when a query was repeated and which encrypted documents match the search query.[2][13]It is known, however, how to construct schemes that leak considerably less at an additional cost in search time and storage.[14][15]
When considering dynamic SSE schemes, the state-of-the-art constructions with optimal time search have leakage profiles that guarantee forward privacy[16]which means that inserts cannot be correlated with past search queries.
In the snapshot model, efficient dynamic SSE schemes with no leakage beyond the number of documents and the size of the collection can be constructed.[12]When using an SSE construction that is secure in the snapshot model one has to carefully consider how the scheme will be deployed because some systems might cache previous search queries.[17]
A leakage profile only describes the leakage of an SSE scheme but it says nothing about whether that leakage can be exploited or not.Cryptanalysisis therefore used to better understand the real-world security of a leakage profile. There is a wide variety of attacks working in different adversarial models, based on a variety of assumptions and attacking different leakage profiles.[18][19]
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In computing,polymorphic codeis code that uses apolymorphic engineto mutate while keeping the originalalgorithmintact - that is, thecodechanges itself every time it runs, but thefunctionof the code (itssemantics) stays the same. For example, the simple math expressions 3+1 and 6-2 both achieve the same result, yet run with differentmachine codein aCPU. This technique is sometimes used bycomputer viruses,shellcodesandcomputer wormsto hide their presence.[1]
Encryptionis the most common method to hide code. With encryption, the main body of the code (also called itspayload) is encrypted and will appear meaningless. For the code to function as before, a decryption function is added to the code. When the code isexecuted, this function reads the payload and decrypts it before executing it in turn.
Encryption alone is not polymorphism. To gain polymorphic behavior, the encryptor/decryptor pair is mutated with each copy of the code. This allows different versions of some code which all function the same.[2]
Mostanti-virus softwareandintrusion detection systems(IDS) attempt to locate malicious code by searching through computer files and data packets sent over acomputer network. If the security software finds patterns that correspond to known computer viruses or worms, it takes appropriate steps to neutralize the threat. Polymorphic algorithms make it difficult for such software to recognize the offending code because it constantly mutates.
Maliciousprogrammershave sought to protect their encrypted code from this virus-scanning strategy by rewriting the unencrypted decryption engine (and the resulting encrypted payload) each time the virus or worm is propagated. Anti-virus software uses sophisticated pattern analysis to find underlying patterns within the different mutations of the decryption engine, in hopes of reliably detecting suchmalware.
Emulation may be used to defeat polymorphic obfuscation by letting the malware demangle itself in a virtual environment before utilizing other methods, such as traditional signature scanning. Such a virtual environment is sometimes called asandbox. Polymorphism does not protect the virus against such emulation if the decrypted payload remains the same regardless of variation in the decryption algorithm.Metamorphic codetechniques may be used to complicate detection further, as the virus may execute without ever having identifiable code blocks in memory that remains constant from infection to infection.
The first known polymorphic virus was written by Mark Washburn. The virus, called1260, was written in 1990.[3]A better-known polymorphic virus was created in 1992 by the hackerDark Avengeras a means of avoiding pattern recognition from antivirus software. A common and very virulent polymorphic virus is the file infecterVirut.
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Private set intersectionis asecure multiparty computationcryptographic technique[1]that allows two parties holding sets to compare encrypted versions of these sets in order to compute the intersection. In this scenario, neither party reveals anything to the counterparty except for the elements in the intersection.
Other variants of this exist, such as the server-client scenario, in which only the client learns the intersection of her set with the set of the server, without the server learning intersection of his set with the clients.[2]
For the comparison of data sets by cryptographic hashes on a small or predictable domain, precautions should be taken to prevent dictionary attacks.[3]
Apple uses this technique in Password Monitoring.[4]It has proposed using the technology for its announced Expanded Protections for Children[5]
In general, PSI protocols can be categorized into two broad categories: (1) traditional PSI and (2) delegated PSI. In the traditional PSI category, the data owners interact directly with each other and need to have a copy of their set at the time of the computation, e.g.,.[6]In the delegated PSI the computation of PSI and/or the storage of sets can be delegated to a third-party server (that is itself might be a passive or active adversary). The delegated PSI category can be further divided into two classes: (a) those that support one-off delegation, and (b) those that support repeated delegation. The PSI protocols that support one-off delegation require the data owner to re-encode its data and send the encoded data to the server for each computation, e.g.,.[7]Those that support repeated delegation allow the data owners to upload their (encrypted) data to the server only once, and then re-use it many times for each computation carried out but the server, e.g.,[8]
Recently, researchers have proposed a variant of PSI protocol (in both traditional and delegated categories) that support data update, e.g., .[9][10]This type of PSI protocol lets data owners insert/delete set elements into/from their data with low overheads and in a privacy-preserving manner.
This educational example demonstrated the key idea of PSI, but does not provide real-world cryptographic security (hence should not be used with real-world data).
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Atime/memory/data tradeoff attackis a type ofcryptographic attackwhere an attacker tries to achieve a situation similar to thespace–time tradeoffbut with the additional parameter ofdata, representing the amount of data available to the attacker. An attacker balances or reduces one or two of those parameters in favor of the other one or two. This type of attack is very difficult, so most of the ciphers and encryption schemes in use were not designed to resist it.[citation needed]
Tradeoff attacks onsymmetric cryptosystemsdate back to 1980, whenMartin Hellmansuggested a time/memory tradeoff method to breakblock cipherswithN{\displaystyle N}possible keys in timeT{\displaystyle T}and memoryM{\displaystyle M}related by the tradeoff curveTM2=N2{\displaystyle T{M^{2}}={N^{2}}}where1≤T≤N{\displaystyle 1\leq T\leq N}.[1]Later, in 1995, Babbage and Golic devised a different tradeoff attack forstream cipherswith a new bound such thatTM=N{\displaystyle TM=N}for1≤T≤D{\displaystyle 1\leq T\leq D}whereD{\displaystyle D}is the output data available to the cryptanalyst at real time.[2][3]
This attack is a special version of the general cryptanalytic time/memory tradeoff attack, which has two main phases:
Any time/memory/data tradeoff attack has the following parameters:
For block ciphers, letN{\displaystyle N}be the total number of possible keys and also assume the number of possible plaintexts and ciphertexts to beN{\displaystyle N}. Also let the given data be a single ciphertext block of a specific plaintext counterpart. If we consider the mapping from the keyx{\displaystyle x}to the ciphertexty{\displaystyle y}as a random permutation functionf{\displaystyle f}over anN{\displaystyle N}point space, and if this functionf{\displaystyle f}is invertible; we need to find the inverse of this functionf−1(y)=x{\displaystyle {f}^{-1}(y)=x}.
Hellman's technique to invert this function:
According to Hellman, if the block cipher at hand has the property that the mapping from its key to cipher text is a random permutation functionf{\displaystyle f}over anN{\displaystyle N}point space, and if thisf{\displaystyle f}is invertible, the tradeoff relationship becomes much better:TM=N{\displaystyle TM=N}.
For stream ciphers,Nis specified by the number of internal states of the bit generator—probably different from the number of keys.Dis the count of the first pseudorandom bits produced from the generator. Finally, the attacker's goal is to find one of the actual internal states of the bit generator to be able to run the generator from this point on to generate the rest of the key. Associate each of the possibleNinternal states of the bit generator with the corresponding string that consists of the firstlog(N){\displaystyle \log(N)}bits obtained by running the generator from that state by the mappingf(x)=y{\displaystyle f(x)=y}from statesxto output prefixesy. This previous mapping is considered a random function over theNpoints common space. To invert this function, an attacker establishes the following.
This result from the Birthday attack gives the conditionDM=N{\displaystyle DM=N}with attack timeT=D{\displaystyle T=D}and preprocessing timeP=M{\displaystyle P=M}which is just a particular point on the tradeoff curveTM=N{\displaystyle TM=N}. We can generalize this relation if we ignore some of the available data at real time and we are able to reduceTfromT=Dto1and the general tradeoff curve eventually becomesTM=N{\displaystyle TM=N}with1≤T≤D{\displaystyle 1\leq T\leq D}andP=M{\displaystyle P=M}.
This novel idea introduced in 2000 combines the Hellman and Babbage-and-Golic tradeoff attacks to achieve a new tradeoff curve with better bounds for stream cipher cryptoanalysis.[4]Hellman's block cipher technique can be applied to a stream cipher by using the same idea of covering theN{\displaystyle N}points space through matrices obtained from multiple variantsfi{\displaystyle f_{i}}of the functionf{\displaystyle f}which is the mapping of internal states to output prefixes. Recall that this tradeoff attack on stream cipher is successful if any of the givenD{\displaystyle D}output prefixes is found in any of the matrices coveringN{\displaystyle N}. This cuts the number of covered points by the matrices fromN{\displaystyle N}toN/D{\displaystyle N/D}points. This is done by reducing the number of matrices fromt{\displaystyle t}tot/D{\displaystyle t/D}while keepingm{\displaystyle m}as large as possible (but this requirest≥D{\displaystyle t\geq D}to have at least one table).
For this new attack, we haveM=mt/D{\displaystyle M=mt/D}because we reduced the number of matrices tot/D{\displaystyle t/D}and the same for the preprocessing timeP=N/D{\displaystyle P=N/D}. The realtime required for the attack isT=(t/D)⋅t⋅D=t2{\displaystyle T=(t/D)\cdot t\cdot D=t^{2}}which is the product of the number of matrices, length of each iteration and number of available data points at attack time.
Eventually, we again use the matrix stopping rule to obtain the tradeoff curveTM2D2=t2⋅(m2t2/D2)⋅D2=m2t4=N2{\displaystyle TM^{2}D^{2}=t^{2}\cdot (m^{2}t^{2}/D^{2})\cdot D^{2}=m^{2}t^{4}=N^{2}}forD2≤T≤N{\displaystyle D^{2}\leq T\leq N}(becauset≥D{\displaystyle t\geq D}).
This attack, invented byBiryukov,Shamir, andWagner, relies on a specific feature of some stream ciphers: that the bit generator undergoes only few changes in its internal state before producing the next output bit.[5]Therefore, we can enumerate those special states that generatek{\displaystyle k}zero bits for small values ofk{\displaystyle k}at low cost. But when forcing large number of output bits to take specific values, this enumeration process become very expensive and difficult.
Now, we can define thesampling resistanceof a stream cipher to beR=2−k{\displaystyle R=2^{-k}}withk{\displaystyle k}the maximum value which makes such enumeration feasible.
Let the stream cipher be ofN=2n{\displaystyle N=2^{n}}states each has afull nameofn{\displaystyle n}bits and a correspondingoutput namewhich is the firstn{\displaystyle n}bits in the output sequence of bits. If this stream cipher has sampling resistanceR=2−k{\displaystyle R=2^{-k}}, then an efficient enumeration can use ashort nameofn−k{\displaystyle n-k}bits to define the special states of the generator. Each special state withn−k{\displaystyle n-k}short namehas a correspondingshort outputname ofn−k{\displaystyle n-k}bits which is the output sequence of the special state after removing the firstk{\displaystyle k}leading bits. Now, we are able to define a new mapping over a reduced space ofNR=2n−k{\displaystyle NR=2^{n-k}}points and this mapping is equivalent to the original mapping. If we letDR≥1{\displaystyle DR\geq 1}, the realtime data available to the attacker is guaranteed to have at least one output of those special states. Otherwise, we relax the definition of special states to include more points. If we substitute forD{\displaystyle D}byDR{\displaystyle DR}andN{\displaystyle N}byNR{\displaystyle NR}in the new time/memory/data tradeoff attack by Shamir and Biryukov, we obtain the same tradeoff curveTM2D2=N2{\displaystyle TM^{2}D^{2}=N^{2}}but with(DR)2≤T≤NR{\displaystyle (DR)^{2}\leq T\leq NR}. This is actually an improvement since we could relax the lower bound onT{\displaystyle T}since(DR)2{\displaystyle (DR)^{2}}can be small up to1{\displaystyle 1}which means that our attack can be made faster. This technique reduces the number of expensive disk access operations fromt{\displaystyle t}totR{\displaystyle tR}since we will be accessing only the specialDR{\displaystyle DR}points, and makes the attack faster because of the reduced number of expensive disk operations.
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https://en.wikipedia.org/wiki/Time/memory/data_tradeoff_attack
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Transport Layer Security(TLS) is acryptographic protocoldesigned to provide communications security over a computer network, such as theInternet. Theprotocolis widely used inapplicationssuch asemail,instant messaging, andvoice over IP, but its use in securingHTTPSremains the most publicly visible.
The TLS protocol aims primarily to provide security, includingprivacy(confidentiality), integrity, and authenticity through the use ofcryptography, such as the use ofcertificates, between two or more communicating computer applications. It runs in thepresentation layerand is itself composed of two layers: the TLS record and the TLShandshake protocols.
The closely relatedDatagram Transport Layer Security(DTLS)is acommunications protocolthat providessecuritytodatagram-based applications. In technical writing, references to "(D)TLS" are often seen when it applies to both versions.[1]
TLS is a proposedInternet Engineering Task Force(IETF) standard, first defined in 1999, and the current version is TLS 1.3, defined in August 2018. TLS builds on the now-deprecatedSSL(Secure Sockets Layer) specifications (1994, 1995, 1996) developed byNetscape Communicationsfor adding theHTTPSprotocol to theirNetscape Navigatorweb browser.
Client-serverapplications use the TLSprotocolto communicate across a network in a way designed to preventeavesdroppingandtampering.
Since applications can communicate either with or without TLS (or SSL), it is necessary for theclientto request that theserverset up a TLS connection.[2]One of the main ways of achieving this is to use a differentport numberfor TLS connections. Port 80 is typically used for unencryptedHTTPtraffic while port 443 is the common port used for encryptedHTTPStraffic. Another mechanism is to make a protocol-specificSTARTTLSrequest to the server to switch the connection to TLS – for example, when using the mail andnewsprotocols.
Once the client and server have agreed to use TLS, they negotiate astatefulconnection by using a handshaking procedure (see§ TLS handshake).[3]The protocols use a handshake with anasymmetric cipherto establish not only cipher settings but also a session-specific shared key with which further communication is encrypted using asymmetric cipher. During this handshake, the client and server agree on various parameters used to establish the connection's security:
This concludes the handshake and begins the secured connection, which is encrypted and decrypted with the session key until the connection closes. If any one of the above steps fails, then the TLS handshake fails and the connection is not created.
TLS and SSL do not fit neatly into any single layer of theOSI modelor theTCP/IP model.[4][5]TLS runs "on top of some reliable transport protocol (e.g., TCP),"[6]: §1which would imply that it is above thetransport layer. It serves encryption to higher layers, which is normally the function of thepresentation layer. However, applications generally use TLS as if it were a transport layer,[4][5]even though applications using TLS must actively control initiating TLS handshakes and handling of exchanged authentication certificates.[6]: §1
When secured by TLS, connections between a client (e.g., a web browser) and a server (e.g., wikipedia.org) will have all of the following properties:[6]: §1
TLS supports many different methods for exchanging keys, encrypting data, and authenticating message integrity. As a result, secure configuration of TLS involves many configurable parameters, and not all choices provide all of the privacy-related properties described in the list above (see the tables below§ Key exchange,§ Cipher security, and§ Data integrity).
Attempts have been made to subvert aspects of the communications security that TLS seeks to provide, and the protocol has been revised several times to address these security threats. Developers of web browsers have repeatedly revised their products to defend against potential security weaknesses after these were discovered (see TLS/SSL support history of web browsers).
Datagram Transport Layer Security, abbreviated DTLS, is a relatedcommunications protocolprovidingsecuritytodatagram-based applications by allowing them to communicate in a way designed[7][8]to preventeavesdropping,tampering, ormessage forgery. The DTLS protocol is based on thestream-oriented Transport Layer Security (TLS) protocol and is intended to provide similar security guarantees. However, unlike TLS, it can be used with most datagram oriented protocols includingUser Datagram Protocol(UDP),Datagram Congestion Control Protocol(DCCP),Control And Provisioning of Wireless Access Points(CAPWAP),Stream Control Transmission Protocol(SCTP) encapsulation, andSecure Real-time Transport Protocol(SRTP).
As the DTLS protocol datagram preserves the semantics of the underlying transport, the application does not suffer from the delays associated with stream protocols. However, the application has to deal withpacket reordering, loss of datagram and data larger than the size of a datagramnetwork packet. Because DTLS uses UDP or SCTP rather than TCP, it avoids theTCP meltdown problem,[9][10]when being used to create a VPN tunnel.
The original 2006 release of DTLS version 1.0 was not a standalone document. It was given as a series of deltas to TLS 1.1.[11]Similarly the follow-up 2012 release of DTLS is a delta to TLS 1.2. It was given the version number of DTLS 1.2 to match its TLS version. Lastly, the 2022 DTLS 1.3 is a delta to TLS 1.3. Like the two previous versions, DTLS 1.3 is intended to provide "equivalent security guarantees [to TLS 1.3] with the exception of order protection/non-replayability".[12]
ManyVPN clientsincludingCiscoAnyConnect[13]& InterCloud Fabric,[14]OpenConnect,[15]ZScalertunnel,[16]F5 NetworksEdge VPN Client,[17]and Citrix SystemsNetScaler[18]use DTLS to secure UDP traffic. In addition all modern web browsers support DTLS-SRTP[19]forWebRTC.
In August 1986, the National Security Agency, the National Bureau of Standards, the Defense Communications Agency launched a project, called the Secure Data Network System (SDNS), with the intent of designing the next generation of secure computer communications network and product specifications to be implemented for applications on public and private internets. It was intended to complement the rapidly emerging new OSI internet standards moving forward both in the U.S. government's GOSIP Profiles and in the huge ITU-ISO JTC1 internet effort internationally.[26]
As part of the project, researchers designed a protocol called SP4 (security protocolin layer 4 of the OSI system). This was later renamed the Transport Layer Security Protocol (TLSP) and subsequently published in 1995 as international standard ITU-T X.274|ISO/IEC 10736:1995.[27]Despite the name similarity, this is distinct from today's TLS.
Other efforts towards transport layer security included theSecure Network Programming(SNP)application programming interface(API), which in 1993 explored the approach of having a secure transport layer API closely resemblingBerkeley sockets, to facilitate retrofitting pre-existing network applications with security measures. SNP was published and presented in the 1994USENIXSummer Technical Conference.[28][29]The SNP project was funded by a grant fromNSAto ProfessorSimon LamatUT-Austinin 1991.[30]Secure Network Programmingwon the 2004ACM Software System Award.[31][32]Simon Lam was inducted into theInternet Hall of Famefor "inventing secure sockets and implementing the first secure sockets layer, named SNP, in 1993."[33][34]
Netscape developed the original SSL protocols, andTaher Elgamal, chief scientist atNetscape Communicationsfrom 1995 to 1998, has been described as the "father of SSL".[35][36][37][38]SSL version 1.0 was never publicly released because of serious security flaws in the protocol. Version 2.0, after being released in February 1995 was quickly found to contain a number of security and usability flaws. It used the same cryptographic keys for message authentication and encryption. It had a weak MAC construction that used the MD5 hash function with a secret prefix, making it vulnerable to length extension attacks. It also provided no protection for either the opening handshake or an explicit message close, both of which meantman-in-the-middle attackscould go undetected. Moreover, SSL 2.0 assumed a single service and a fixed domain certificate, conflicting with the widely used feature of virtual hosting in Web servers, so most websites were effectively impaired from using SSL.
These flaws necessitated the complete redesign of the protocol to SSL version 3.0.[39][37]Released in 1996, it was produced byPaul Kocherworking with Netscape engineers Phil Karlton and Alan Freier, with a reference implementation by Christopher Allen and Tim Dierks of Certicom. Newer versions of SSL/TLS are based on SSL 3.0. The 1996 draft of SSL 3.0 was published by IETF as a historical document inRFC6101.
SSL 2.0 was deprecated in 2011 byRFC6176. In 2014, SSL 3.0 was found to be vulnerable to thePOODLEattack that affects allblock ciphersin SSL;RC4, the only non-block cipher supported by SSL 3.0, is also feasibly broken as used in SSL 3.0.[40]SSL 3.0 was deprecated in June 2015 byRFC7568.
TLS 1.0 was first defined inRFC2246in January 1999 as an upgrade of SSL Version 3.0, and written by Christopher Allen and Tim Dierks of Certicom. As stated in the RFC, "the differences between this protocol and SSL 3.0 are not dramatic, but they are significant enough to preclude interoperability between TLS 1.0 and SSL 3.0". Tim Dierks later wrote that these changes, and the renaming from "SSL" to "TLS", were a face-saving gesture to Microsoft, "so it wouldn't look [like] the IETF was just rubberstamping Netscape's protocol".[41]
ThePCI Councilsuggested that organizations migrate from TLS 1.0 to TLS 1.1 or higher before June 30, 2018.[42][43]In October 2018,Apple,Google,Microsoft, andMozillajointly announced they would deprecate TLS 1.0 and 1.1 in March 2020.[20]TLS 1.0 and 1.1 were formally deprecated inRFC8996in March 2021.
TLS 1.1 was defined in RFC 4346 in April 2006.[44]It is an update from TLS version 1.0. Significant differences in this version include:
Support for TLS versions 1.0 and 1.1 was widely deprecated by web sites around 2020,[46]disabling access toFirefoxversions before 24 andChromium-based browsersbefore 29,[47]though third-party fixes can be applied to Netscape Navigator and older versions of Firefox to add TLS 1.2 support.[48]
TLS 1.2 was defined inRFC5246in August 2008.[23]It is based on the earlier TLS 1.1 specification. Major differences include:
All TLS versions were further refined inRFC6176in March 2011, removing their backward compatibility with SSL such that TLS sessions never negotiate the use of Secure Sockets Layer (SSL) version 2.0. As of April 2025 there is no formal date for TLS 1.2 to be deprecated. The specifications for TLS 1.2 became redefined as well by the Standards Track DocumentRFC8446to keep it as secure as possible; it is to be seen as a failover protocol now, meant only to be negotiated with clients which are unable to talk over TLS 1.3 (The original RFC 5246 definition for TLS 1.2 is since then obsolete).
TLS 1.3 was defined in RFC 8446 in August 2018.[6]It is based on the earlier TLS 1.2 specification. Major differences from TLS 1.2 include:[49]
Network Security Services(NSS), the cryptography library developed byMozillaand used by its web browserFirefox, enabled TLS 1.3 by default in February 2017.[51]TLS 1.3 support was subsequently added — but due to compatibility issues for a small number of users, not automatically enabled[52]— toFirefox 52.0, which was released in March 2017. TLS 1.3 was enabled by default in May 2018 with the release ofFirefox 60.0.[53]
Google Chromeset TLS 1.3 as the default version for a short time in 2017. It then removed it as the default, due to incompatible middleboxes such asBlue Coat web proxies.[54]
The intolerance of the new version of TLS wasprotocol ossification; middleboxes had ossified the protocol's version parameter. As a result, version 1.3 mimics thewire imageof version 1.2. This change occurred very late in the design process, only having been discovered during browser deployment.[55]The discovery of this intolerance also led to the prior version negotiation strategy, where the highest matching version was picked, being abandoned due to unworkable levels of ossification.[56]'Greasing' an extension point, where one protocol participant claims support for non-existent extensions to ensure that unrecognised-but-actually-existent extensions are tolerated and so to resist ossification, was originally designed for TLS, but it has since been adopted elsewhere.[56]
During the IETF 100Hackathon, which took place inSingaporein 2017, the TLS Group worked on adaptingopen-source applicationsto use TLS 1.3.[57][58]The TLS group was made up of individuals from Japan, United Kingdom, and Mauritius via the cyberstorm.mu team.[58]This work was continued in the IETF 101 Hackathon inLondon,[59]and the IETF 102 Hackathon in Montreal.[60]
wolfSSLenabled the use of TLS 1.3 as of version 3.11.1, released in May 2017.[61]As the first commercial TLS 1.3 implementation, wolfSSL 3.11.1 supported Draft 18 and now supports Draft 28,[62]the final version, as well as many older versions. A series of blogs were published on the performance difference between TLS 1.2 and 1.3.[63]
InSeptember 2018, the popularOpenSSLproject released version 1.1.1 of its library, in which support for TLS 1.3 was "the headline new feature".[64]
Support for TLS 1.3 was added toSecure Channel(schannel) for theGAreleases ofWindows 11andWindows Server 2022.[65]
TheElectronic Frontier Foundationpraised TLS 1.3 and expressed concern about the variant protocol Enterprise Transport Security (ETS) that intentionally disables important security measures in TLS 1.3.[66]Originally called Enterprise TLS (eTLS), ETS is a published standard known as the 'ETSITS103523-3', "Middlebox Security Protocol, Part3: Enterprise Transport Security". It is intended for use entirely within proprietary networks such as banking systems. ETS does not support forward secrecy so as to allow third-party organizations connected to the proprietary networks to be able to use their private key to monitor network traffic for the detection of malware and to make it easier to conduct audits.[67][68]Despite the claimed benefits, the EFF warned that the loss of forward secrecy could make it easier for data to be exposed along with saying that there are better ways to analyze traffic.[66]
A digital certificate certifies the ownership of a public key by the named subject of the certificate, and indicates certain expected usages of that key. This allows others (relying parties) to rely upon signatures or on assertions made by the private key that corresponds to the certified public key. Keystores and trust stores can be in various formats, such as.pem, .crt,.pfx, and.jks.
TLS typically relies on a set of trusted third-party certificate authorities to establish the authenticity of certificates. Trust is usually anchored in a list of certificates distributed with user agent software,[69]and can be modified by the relying party.
According toNetcraft, who monitors active TLS certificates, the market-leading certificate authority (CA) has beenSymantecsince the beginning of their survey (orVeriSignbefore the authentication services business unit was purchased by Symantec). As of 2015, Symantec accounted for just under a third of all certificates and 44% of the valid certificates used by the 1 million busiest websites, as counted by Netcraft.[70]In 2017, Symantec sold its TLS/SSL business to DigiCert.[71]In an updated report, it was shown thatIdenTrust,DigiCert, andSectigoare the top 3 certificate authorities in terms of market share since May 2019.[72]
As a consequence of choosingX.509certificates, certificate authorities and apublic key infrastructureare necessary to verify the relation between a certificate and its owner, as well as to generate, sign, and administer the validity of certificates. While this can be more convenient than verifying the identities via aweb of trust, the2013 mass surveillance disclosuresmade it more widely known that certificate authorities are a weak point from a security standpoint, allowingman-in-the-middle attacks(MITM) if the certificate authority cooperates (or is compromised).[73][74]
Before a client and server can begin to exchange information protected by TLS, they must securely exchange or agree upon an encryption key and a cipher to use when encrypting data (see§ Cipher). Among the methods used for key exchange/agreement are: public and private keys generated withRSA(denoted TLS_RSA in the TLS handshake protocol),Diffie–Hellman(TLS_DH), ephemeral Diffie–Hellman (TLS_DHE),elliptic-curve Diffie–Hellman(TLS_ECDH), ephemeral elliptic-curve Diffie–Hellman (TLS_ECDHE),anonymous Diffie–Hellman(TLS_DH_anon),[23]pre-shared key(TLS_PSK)[75]andSecure Remote Password(TLS_SRP).[76]
The TLS_DH_anon and TLS_ECDH_anon key agreement methods do not authenticate the server or the user and hence are rarely used because those are vulnerable toman-in-the-middle attacks. Only TLS_DHE and TLS_ECDHE provideforward secrecy.
Public key certificates used during exchange/agreement also vary in the size of the public/private encryption keys used during the exchange and hence the robustness of the security provided. In July 2013,Googleannounced that it would no longer use 1024-bit public keys and would switch instead to 2048-bit keys to increase the security of the TLS encryption it provides to its users because the encryption strength is directly related to thekey size.[77][78]
Notes
Amessage authentication code(MAC) is used for data integrity.HMACis used forCBCmode of block ciphers.Authenticated encryption(AEAD) such asGCMandCCM modeuses AEAD-integrated MAC and does not useHMAC.[6]: §8.4HMAC-basedPRF, orHKDFis used for TLS handshake.
In applications design, TLS is usually implemented on top of Transport Layer protocols, encrypting all of the protocol-related data of protocols such asHTTP,FTP,SMTP,NNTPandXMPP.
Historically, TLS has been used primarily with reliable transport protocols such as theTransmission Control Protocol(TCP). However, it has also been implemented with datagram-oriented transport protocols, such as theUser Datagram Protocol(UDP) and theDatagram Congestion Control Protocol(DCCP), usage of which has been standardized independently using the termDatagram Transport Layer Security(DTLS).
A primary use of TLS is to secureWorld Wide Webtraffic between awebsiteand aweb browserencoded with the HTTP protocol. This use of TLS to secure HTTP traffic constitutes theHTTPSprotocol.[93]
Notes
As of March 2025[update], the latest versions of all major web browsers support TLS 1.2 and 1.3 and have them enabled by default, with the exception ofIE 11. TLS 1.0 and 1.1 are disabled by default on the latest versions of all major browsers.
Mitigations against known attacks are not enough yet:
Most SSL and TLS programming libraries arefree and open-source software.
A paper presented at the 2012ACMconference on computer and communications security[98]showed that many applications used some of these SSL libraries incorrectly, leading to vulnerabilities. According to the authors:
"The root cause of most of these vulnerabilities is the terrible design of the APIs to the underlying SSL libraries. Instead of expressing high-level security properties of network tunnels such as confidentiality and authentication, these APIs expose low-level details of the SSL protocol to application developers. As a consequence, developers often use SSL APIs incorrectly, misinterpreting and misunderstanding their manifold parameters, options, side effects, and return values."
TheSimple Mail Transfer Protocol(SMTP) can also be protected by TLS. These applications usepublic key certificatesto verify the identity of endpoints.
TLS can also be used for tunneling an entire network stack to create aVPN, which is the case withOpenVPNandOpenConnect. Many vendors have by now married TLS's encryption and authentication capabilities with authorization. There has also been substantial development since the late 1990s in creating client technology outside of Web-browsers, in order to enable support for client/server applications. Compared to traditionalIPsecVPN technologies, TLS has some inherent advantages in firewall andNATtraversal that make it easier to administer for large remote-access populations.
TLS is also a standard method for protectingSession Initiation Protocol(SIP) application signaling. TLS can be used for providing authentication and encryption of the SIP signaling associated withVoIPand other SIP-based applications.[99]
Significant attacks against TLS/SSL are listed below.
In February 2015, IETF issued an informational RFC[100]summarizing the various known attacks against TLS/SSL.
A vulnerability of the renegotiation procedure was discovered in August 2009 that can lead to plaintext injection attacks against SSL 3.0 and all current versions of TLS.[101]For example, it allows an attacker who can hijack anhttpsconnection to splice their own requests into the beginning of the conversation the client has with the web server. The attacker cannot actually decrypt the client–server communication, so it is different from a typicalman-in-the-middle attack. A short-term fix is for web servers to stop allowing renegotiation, which typically will not require other changes unlessclient certificateauthentication is used. To fix the vulnerability, a renegotiation indication extension was proposed for TLS. It will require the client and server to include and verify information about previous handshakes in any renegotiation handshakes.[102]This extension has become a proposed standard and has been assigned the numberRFC5746. The RFC has been implemented by several libraries.[103][104][105]
A protocoldowngrade attack(also called a version rollback attack) tricks a web server into negotiating connections with previous versions of TLS (such as SSLv2) that have long since been abandoned as insecure.
Previous modifications to the original protocols, likeFalse Start[106](adopted and enabled by Google Chrome[107]) orSnap Start, reportedly introduced limited TLS protocol downgrade attacks[108]or allowed modifications to the cipher suite list sent by the client to the server. In doing so, an attacker might succeed in influencing the cipher suite selection in an attempt to downgrade the cipher suite negotiated to use either a weaker symmetric encryption algorithm or a weaker key exchange.[109]A paper presented at anACMconference on computer and communications securityin 2012 demonstrated that the False Start extension was at risk: in certain circumstances it could allow an attacker to recover the encryption keys offline and to access the encrypted data.[110]
Encryption downgrade attacks can force servers and clients to negotiate a connection using cryptographically weak keys. In 2014, aman-in-the-middleattack called FREAK was discovered affecting theOpenSSLstack, the defaultAndroidweb browser, and someSafaribrowsers.[111]The attack involved tricking servers into negotiating a TLS connection using cryptographically weak 512 bit encryption keys.
Logjam is asecurity exploitdiscovered in May 2015 that exploits the option of using legacy"export-grade"512-bitDiffie–Hellmangroups dating back to the 1990s.[112]It forces susceptible servers to downgrade to cryptographically weak 512-bit Diffie–Hellman groups. An attacker can then deduce the keys the client and server determine using theDiffie–Hellman key exchange.
TheDROWN attackis an exploit that attacks servers supporting contemporary SSL/TLS protocol suites by exploiting their support for the obsolete, insecure, SSLv2 protocol to leverage an attack on connections using up-to-date protocols that would otherwise be secure.[113][114]DROWN exploits a vulnerability in the protocols used and the configuration of the server, rather than any specific implementation error. Full details of DROWN were announced in March 2016, together with a patch for the exploit. At that time, more than 81,000 of the top 1 million most popular websites were among the TLS protected websites that were vulnerable to the DROWN attack.[114]
On September 23, 2011, researchers Thai Duong and Juliano Rizzo demonstrated a proof of concept calledBEAST(Browser Exploit Against SSL/TLS)[115]using aJava appletto violatesame origin policyconstraints, for a long-knowncipher block chaining(CBC) vulnerability in TLS 1.0:[116][117]an attacker observing 2 consecutive ciphertext blocks C0, C1 can test if the plaintext block P1 is equal to x by choosing the next plaintext blockP2 = x ⊕ C0 ⊕ C1; as per CBC operation,C2 = E(C1 ⊕ P2) = E(C1 ⊕ x ⊕ C0 ⊕ C1) = E(C0 ⊕ x), which will be equal to C1 ifx = P1. Practicalexploitshad not been previously demonstrated for thisvulnerability, which was originally discovered byPhillip Rogaway[118]in 2002. The vulnerability of the attack had been fixed with TLS 1.1 in 2006, but TLS 1.1 had not seen wide adoption prior to this attack demonstration.
RC4as a stream cipher is immune to BEAST attack. Therefore, RC4 was widely used as a way to mitigate BEAST attack on the server side. However, in 2013, researchers found more weaknesses in RC4. Thereafter enabling RC4 on server side was no longer recommended.[119]
Chrome and Firefox themselves are not vulnerable to BEAST attack,[120][121]however, Mozilla updated theirNSSlibraries to mitigate BEAST-likeattacks. NSS is used byMozilla FirefoxandGoogle Chrometo implement SSL. Someweb serversthat have a broken implementation of the SSL specification may stop working as a result.[122]
Microsoftreleased Security Bulletin MS12-006 on January 10, 2012, which fixed the BEAST vulnerability by changing the way that the Windows Secure Channel (Schannel) component transmits encrypted network packets from the server end.[123]Users of Internet Explorer (prior to version 11) that run on older versions of Windows (Windows 7,Windows 8andWindows Server 2008 R2) can restrict use of TLS to 1.1 or higher.
Applefixed BEAST vulnerability by implementing 1/n-1 split and turning it on by default inOS X Mavericks, released on October 22, 2013.[124]
The authors of the BEAST attack are also the creators of the laterCRIMEattack, which can allow an attacker to recover the content of web cookies whendata compressionis used along with TLS.[125][126]When used to recover the content of secretauthentication cookies, it allows an attacker to performsession hijackingon an authenticated web session.
While the CRIME attack was presented as a general attack that could work effectively against a large number of protocols, including but not limited to TLS, and application-layer protocols such asSPDYorHTTP, only exploits against TLS and SPDY were demonstrated and largely mitigated in browsers and servers. The CRIME exploit againstHTTP compressionhas not been mitigated at all, even though the authors of CRIME have warned that this vulnerability might be even more widespread than SPDY and TLS compression combined. In 2013 a new instance of the CRIME attack against HTTP compression, dubbedBREACH, was announced. Based on the CRIME attack a BREACH attack can extract login tokens, email addresses or other sensitive information from TLS encrypted web traffic in as little as 30 seconds (depending on the number of bytes to be extracted), provided the attacker tricks the victim into visiting a malicious web link or is able to inject content into valid pages the user is visiting (ex: a wireless network under the control of the attacker).[127]All versions of TLS and SSL are at risk from BREACH regardless of the encryption algorithm or cipher used.[128]Unlike previous instances of CRIME, which can be successfully defended against by turning off TLS compression or SPDY header compression, BREACH exploits HTTP compression which cannot realistically be turned off, as virtually all web servers rely upon it to improve data transmission speeds for users.[127]This is a known limitation of TLS as it is susceptible tochosen-plaintext attackagainst the application-layer data it was meant to protect.
Earlier TLS versions were vulnerable against thepadding oracle attackdiscovered in 2002. A novel variant, called theLucky Thirteen attack, was published in 2013.
Some experts[90]also recommended avoidingtriple DESCBC. Since the last supported ciphers developed to support any program usingWindows XP's SSL/TLS library like Internet Explorer on Windows XP areRC4and Triple-DES, and since RC4 is now deprecated (see discussion ofRC4 attacks), this makes it difficult to support any version of SSL for any program using this library on XP.
A fix was released as the Encrypt-then-MAC extension to the TLS specification, released asRFC7366.[129]The Lucky Thirteen attack can be mitigated in TLS 1.2 by using only AES_GCM ciphers; AES_CBC remains vulnerable. SSL may safeguard email, VoIP, and other types of communications over insecure networks in addition to its primary use case of secure data transmission between a client and the server.[2]
On October 14, 2014, Google researchers published a vulnerability in the design of SSL 3.0, which makesCBC mode of operationwith SSL 3.0 vulnerable to apadding attack(CVE-2014-3566). They named this attackPOODLE(Padding Oracle On Downgraded Legacy Encryption). On average, attackers only need to make 256 SSL 3.0 requests to reveal one byte of encrypted messages.[96]
Although this vulnerability only exists in SSL 3.0 and most clients and servers support TLS 1.0 and above, all major browsers voluntarily downgrade to SSL 3.0 if the handshakes with newer versions of TLS fail unless they provide the option for a user or administrator to disable SSL 3.0 and the user or administrator does so[citation needed]. Therefore, the man-in-the-middle can first conduct aversion rollback attackand then exploit this vulnerability.[96]
On December 8, 2014, a variant of POODLE was announced that impacts TLS implementations that do not properly enforce padding byte requirements.[130]
Despite the existence of attacks onRC4that broke its security, cipher suites in SSL and TLS that were based on RC4 were still considered secure prior to 2013 based on the way in which they were used in SSL and TLS. In 2011, the RC4 suite was actually recommended as a workaround for theBEASTattack.[131]New forms of attack disclosed in March 2013 conclusively demonstrated the feasibility of breaking RC4 in TLS, suggesting it was not a good workaround for BEAST.[95]An attack scenario was proposed by AlFardan, Bernstein, Paterson, Poettering and Schuldt that used newly discovered statistical biases in the RC4 key table[132]to recover parts of the plaintext with a large number of TLS encryptions.[133][134]An attack on RC4 in TLS and SSL that requires 13 × 220encryptions to break RC4 was unveiled on 8 July 2013 and later described as "feasible" in the accompanying presentation at aUSENIXSecurity Symposium in August 2013.[135][136]In July 2015, subsequent improvements in the attack make it increasingly practical to defeat the security of RC4-encrypted TLS.[137]
As many modern browsers have been designed to defeat BEAST attacks (except Safari for Mac OS X 10.7 or earlier, for iOS 6 or earlier, and for Windows; see§ Web browsers), RC4 is no longer a good choice for TLS 1.0. The CBC ciphers which were affected by the BEAST attack in the past have become a more popular choice for protection.[90]Mozilla and Microsoft recommend disabling RC4 where possible.[138][139]RFC7465prohibits the use of RC4 cipher suites in all versions of TLS.
On September 1, 2015, Microsoft, Google, and Mozilla announced that RC4 cipher suites would be disabled by default in their browsers (Microsoft Edge [Legacy],Internet Explorer 11on Windows 7/8.1/10,Firefox, andChrome) in early 2016.[140][141][142]
A TLS (logout) truncation attack blocks a victim's account logout requests so that the user unknowingly remains logged into a web service. When the request to sign out is sent, the attacker injects an unencryptedTCPFIN message (no more data from sender) to close the connection. The server therefore does not receive the logout request and is unaware of the abnormal termination.[143]
Published in July 2013,[144][145]the attack causes web services such asGmailandHotmailto display a page that informs the user that they have successfully signed-out, while ensuring that the user's browser maintains authorization with the service, allowing an attacker with subsequent access to the browser to access and take over control of the user's logged-in account. The attack does not rely on installing malware on the victim's computer; attackers need only place themselves between the victim and the web server (e.g., by setting up a rogue wireless hotspot).[143]This vulnerability also requires access to the victim's computer.
Another possibility is when using FTP the data connection can have a false FIN in the data stream, and if the protocol rules for exchanging close_notify alerts is not adhered to a file can be truncated.
In February 2013 two researchers from Royal Holloway, University of London discovered a timing attack[146]which allowed them to recover (parts of the) plaintext from a DTLS connection using the OpenSSL or GnuTLS implementation of DTLS whenCipher Block Chainingmode encryption was used.
This attack, discovered in mid-2016, exploits weaknesses in theWeb Proxy Autodiscovery Protocol(WPAD) to expose the URL that a web user is attempting to reach via a TLS-enabled web link.[147]Disclosure of a URL can violate a user's privacy, not only because of the website accessed, but also because URLs are sometimes used to authenticate users. Document sharing services, such as those offered by Google and Dropbox, also work by sending a user a security token that is included in the URL. An attacker who obtains such URLs may be able to gain full access to a victim's account or data.
The exploit works against almost all browsers and operating systems.
The Sweet32 attack breaks all 64-bit block ciphers used in CBC mode as used in TLS by exploiting abirthday attackand either aman-in-the-middle attackor injection of a maliciousJavaScriptinto a web page. The purpose of the man-in-the-middle attack or the JavaScript injection is to allow the attacker to capture enough traffic to mount a birthday attack.[148]
TheHeartbleedbug is a serious vulnerability specific to the implementation of SSL/TLS in the popularOpenSSLcryptographic software library, affecting versions 1.0.1 to 1.0.1f. This weakness, reported in April 2014, allows attackers to stealprivate keysfrom servers that should normally be protected.[149]The Heartbleed bug allows anyone on the Internet to read the memory of the systems protected by the vulnerable versions of the OpenSSL software. This compromises the secret private keys associated with thepublic certificatesused to identify the service providers and to encrypt the traffic, the names and passwords of the users and the actual content. This allows attackers to eavesdrop on communications, steal data directly from the services and users and to impersonate services and users.[150]The vulnerability is caused by abuffer over-readbug in the OpenSSL software, rather than a defect in the SSL or TLS protocol specification.
In September 2014, a variant ofDaniel Bleichenbacher's PKCS#1 v1.5 RSA Signature Forgery vulnerability[151]was announced by Intel Security Advanced Threat Research. This attack, dubbed BERserk, is a result of incomplete ASN.1 length decoding of public key signatures in some SSL implementations, and allows a man-in-the-middle attack by forging a public key signature.[152]
In February 2015, after media reported the hidden pre-installation ofsuperfishadware on some Lenovo notebooks,[153]a researcher found a trusted root certificate on affected Lenovo machines to be insecure, as the keys could easily be accessed using the company name, Komodia, as a passphrase.[154]The Komodia library was designed to intercept client-side TLS/SSL traffic for parental control and surveillance, but it was also used in numerous adware programs, including Superfish, that were often surreptitiously installed unbeknownst to the computer user. In turn, thesepotentially unwanted programsinstalled the corrupt root certificate, allowing attackers to completely control web traffic and confirm false websites as authentic.
In May 2016, it was reported that dozens of Danish HTTPS-protected websites belonging toVisa Inc.were vulnerable to attacks allowing hackers to inject malicious code and forged content into the browsers of visitors.[155]The attacks worked because the TLS implementation used on the affected servers incorrectly reused random numbers (nonces) that are intended to be used only once, ensuring that eachTLS handshakeis unique.[155]
In February 2017, an implementation error caused by a single mistyped character in code used to parse HTML created a buffer overflow error onCloudflareservers. Similar in its effects to the Heartbleed bug discovered in 2014, this overflow error, widely known asCloudbleed, allowed unauthorized third parties to read data in the memory of programs running on the servers—data that should otherwise have been protected by TLS.[156]
As of July 2021[update], the Trustworthy Internet Movement estimated the ratio of websites that are vulnerable to TLS attacks.[94]
Forward secrecy is a property of cryptographic systems which ensures that a session key derived from a set of public and private keys will not be compromised if one of the private keys is compromised in the future.[157]Without forward secrecy, if the server's private key is compromised, not only will all future TLS-encrypted sessions using that server certificate be compromised, but also any past sessions that used it as well (provided that these past sessions were intercepted and stored at the time of transmission).[158]An implementation of TLS can provide forward secrecy by requiring the use of ephemeralDiffie–Hellman key exchangeto establish session keys, and some notable TLS implementations do so exclusively: e.g.,Gmailand other Google HTTPS services that useOpenSSL.[159]However, many clients and servers supporting TLS (including browsers and web servers) are not configured to implement such restrictions.[160][161]In practice, unless a web service uses Diffie–Hellman key exchange to implement forward secrecy, all of the encrypted web traffic to and from that service can be decrypted by a third party if it obtains the server's master (private) key; e.g., by means of a court order.[162]
Even where Diffie–Hellman key exchange is implemented, server-side session management mechanisms can impact forward secrecy. The use ofTLS session tickets(a TLS extension) causes the session to be protected by AES128-CBC-SHA256 regardless of any other negotiated TLS parameters, including forward secrecy ciphersuites, and the long-lived TLS session ticket keys defeat the attempt to implement forward secrecy.[163][164][165]Stanford University research in 2014 also found that of 473,802 TLS servers surveyed, 82.9% of the servers deploying ephemeral Diffie–Hellman (DHE) key exchange to support forward secrecy were using weak Diffie–Hellman parameters. These weak parameter choices could potentially compromise the effectiveness of the forward secrecy that the servers sought to provide.[166]
Since late 2011, Google has provided forward secrecy with TLS by default to users of itsGmailservice, along withGoogle Docsand encrypted search, among other services.[167]Since November 2013,Twitterhas provided forward secrecy with TLS to users of its service.[168]As of August 2019[update], about 80% of TLS-enabled websites are configured to use cipher suites that provide forward secrecy to most web browsers.[94]
TLS interception (orHTTPSinterception if applied particularly to that protocol) is the practice of intercepting an encrypted data stream in order to decrypt it, read and possibly manipulate it, and then re-encrypt it and send the data on its way again. This is done by way of a "transparent proxy": the interception software terminates the incoming TLS connection, inspects the HTTP plaintext, and then creates a new TLS connection to the destination.[169]
TLS/HTTPS interception is used as aninformation securitymeasure by network operators in order to be able to scan for and protect against the intrusion of malicious content into the network, such ascomputer virusesand othermalware.[169]Such content could otherwise not be detected as long as it is protected by encryption, which is increasingly the case as a result of the routine use of HTTPS and other secure protocols.
A significant drawback of TLS/HTTPS interception is that it introduces new security risks of its own. One notable limitation is that it provides a point where network traffic is available unencrypted thus giving attackers an incentive to attack this point in particular in order to gain access to otherwise secure content. The interception also allows the network operator, or persons who gain access to its interception system, to performman-in-the-middle attacksagainst network users. A 2017 study found that "HTTPS interception has become startlingly widespread, and that interception products as a class have a dramatically negative impact on connection security".[169]
The TLS protocol exchangesrecords, which encapsulate the data to be exchanged in a specific format (see below). Each record can be compressed, padded, appended with amessage authentication code(MAC), or encrypted, all depending on the state of the connection. Each record has acontent typefield that designates the type of data encapsulated, a length field and a TLS version field. The data encapsulated may be control or procedural messages of the TLS itself, or simply the application data needed to be transferred by TLS. The specifications (cipher suite, keys etc.) required to exchange application data by TLS, are agreed upon in the "TLS handshake" between the client requesting the data and the server responding to requests. The protocol therefore defines both the structure of payloads transferred in TLS and the procedure to establish and monitor the transfer.
When the connection starts, the record encapsulates a "control" protocol – the handshake messaging protocol (content type22). This protocol is used to exchange all the information required by both sides for the exchange of the actual application data by TLS. It defines the format of messages and the order of their exchange. These may vary according to the demands of the client and server – i.e., there are several possible procedures to set up the connection. This initial exchange results in a successful TLS connection (both parties ready to transfer application data with TLS) or an alert message (as specified below).
A typical connection example follows, illustrating ahandshakewhere the server (but not the client) is authenticated by its certificate:
The followingfullexample shows a client being authenticated (in addition to the server as in the example above; seemutual authentication) via TLS using certificates exchanged between both peers.
Public key operations (e.g., RSA) are relatively expensive in terms of computational power. TLS provides a secure shortcut in the handshake mechanism to avoid these operations: resumed sessions. Resumed sessions are implemented using session IDs or session tickets.
Apart from the performance benefit, resumed sessions can also be used forsingle sign-on, as it guarantees that both the original session and any resumed session originate from the same client. This is of particular importance for theFTP over TLS/SSLprotocol, which would otherwise suffer from a man-in-the-middle attack in which an attacker could intercept the contents of the secondary data connections.[172]
The TLS 1.3 handshake was condensed to only one round trip compared to the two round trips required in previous versions of TLS/SSL.
To start the handshake, the client guesses which key exchange algorithm will be selected by the server and sends aClientHellomessage to the server containing a list of supported ciphers (in order of the client's preference) and public keys for some or all of its key exchange guesses. If the client successfully guesses the key exchange algorithm, 1 round trip is eliminated from the handshake. After receiving theClientHello, the server selects a cipher and sends back aServerHellowith its own public key, followed by serverCertificateandFinishedmessages.[173]
After the client receives the server's finished message, it now is coordinated with the server on which cipher suite to use.[174]
In an ordinaryfullhandshake, the server sends asession idas part of theServerHellomessage. The client associates thissession idwith the server's IP address and TCP port, so that when the client connects again to that server, it can use thesession idto shortcut the handshake. In the server, thesession idmaps to the cryptographic parameters previously negotiated, specifically the "master secret". Both sides must have the same "master secret" or the resumed handshake will fail (this prevents an eavesdropper from using asession id). The random data in theClientHelloandServerHellomessages virtually guarantee that the generated connection keys will be different from in the previous connection. In the RFCs, this type of handshake is called anabbreviatedhandshake. It is also described in the literature as arestarthandshake.
RFC5077extends TLS via use of session tickets, instead of session IDs. It defines a way to resume a TLS session without requiring that session-specific state is stored at the TLS server.
When using session tickets, the TLS server stores its session-specific state in a session ticket and sends the session ticket to the TLS client for storing. The client resumes a TLS session by sending the session ticket to the server, and the server resumes the TLS session according to the session-specific state in the ticket. The session ticket is encrypted and authenticated by the server, and the server verifies its validity before using its contents.
One particular weakness of this method withOpenSSLis that it always limits encryption and authentication security of the transmitted TLS session ticket toAES128-CBC-SHA256, no matter what other TLS parameters were negotiated for the actual TLS session.[164]This means that the state information (the TLS session ticket) is not as well protected as the TLS session itself. Of particular concern is OpenSSL's storage of the keys in an application-wide context (SSL_CTX), i.e. for the life of the application, and not allowing for re-keying of theAES128-CBC-SHA256TLS session tickets without resetting the application-wide OpenSSL context (which is uncommon, error-prone and often requires manual administrative intervention).[165][163]
This is the general format of all TLS records.
Most messages exchanged during the setup of the TLS session are based on this record, unless an error or warning occurs and needs to be signaled by an Alert protocol record (see below), or the encryption mode of the session is modified by another record (see ChangeCipherSpec protocol below).
Note that multiple handshake messages may be combined within one record.
This record should normally not be sent during normal handshaking or application exchanges. However, this message can be sent at any time during the handshake and up to the closure of the session. If this is used to signal a fatal error, the session will be closed immediately after sending this record, so this record is used to give a reason for this closure. If the alert level is flagged as a warning, the remote can decide to close the session if it decides that the session is not reliable enough for its needs (before doing so, the remote may also send its own signal).
From the application protocol point of view, TLS belongs to a lower layer, although the TCP/IP model is too coarse to show it. This means that the TLS handshake is usually (except in theSTARTTLScase) performed before the application protocol can start. In thename-based virtual serverfeature being provided by the application layer, all co-hosted virtual servers share the same certificate because the server has to select and send a certificate immediately after the ClientHello message. This is a big problem in hosting environments because it means either sharing the same certificate among all customers or using a different IP address for each of them.
There are two known workarounds provided byX.509:
To provide the server name,RFC4366Transport Layer Security (TLS) Extensions allow clients to include aServer Name Indicationextension (SNI) in the extended ClientHello message. This extension hints to the server immediately which name the client wishes to connect to, so the server
can select the appropriate certificate to send to the clients.
RFC2817also documents a method to implement name-based virtual hosting by upgrading HTTP to TLS via anHTTP/1.1 Upgrade header. Normally this is to securely implement HTTP over TLS within the main "http"URI scheme(which avoids forking the URI space and reduces the number of used ports), however, few implementations currently support this.[citation needed]
The current approved version of (D)TLS is version 1.3, which is specified in:
The current standards replaces these former versions, which are now considered obsolete:
OtherRFCssubsequently extended (D)TLS.
Extensions to (D)TLS 1.3 include:
Extensions to (D)TLS 1.2 include:
Extensions to (D)TLS 1.1 include:
Extensions to TLS 1.0 include:
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TheAdvanced Encryption Standard(AES), the symmetricblock cipherratified as a standard byNational Institute of Standards and Technologyof the United States (NIST), was chosen using a process lasting from 1997 to 2000 that was markedly more open and transparent than its predecessor, theData Encryption Standard(DES). This process won praise from the open cryptographic community, and helped to increase confidence in the security of the winning algorithm from those who were suspicious of backdoors in the predecessor, DES.
A new standard was needed primarily because DES had a relatively small 56-bit key which was becoming vulnerable tobrute-force attacks. In addition, the DES was designed primarily for hardware and was relatively slow when implemented in software.[1]While Triple-DES avoids the problem of a small key size, it is very slow even in hardware, it is unsuitable for limited-resource platforms, and it may be affected by potential security issues connected with the (today comparatively small) block size of 64 bits.
On January 2, 1997, NIST announced that they wished to choose a successor to DES to be known as AES. Like DES, this was to be "an unclassified, publicly disclosed encryption algorithm capable of protecting sensitive government information well into the next century."[2]However, rather than simply publishing a successor, NIST asked for input from interested parties on how the successor should be chosen. Interest from the open cryptographic community was immediately intense, and NIST received a great many submissions during the three-month comment period.
The result of this feedback was a call for new algorithms on September 12, 1997.[3]The algorithms were all to be block ciphers, supporting a block size of 128 bits and key sizes of 128, 192, and 256 bits. Such ciphers were rare at the time of the announcement; the best known was probablySquare.
In the nine months that followed, fifteen designs were created and submitted from several countries. They were, in alphabetical order:CAST-256,CRYPTON,DEAL,DFC,E2,FROG,HPC,LOKI97,MAGENTA,MARS,RC6,Rijndael,SAFER+,Serpent, andTwofish.
In the ensuing debate, many advantages and disadvantages of the candidates were investigated by cryptographers; they were assessed not only on security, but also on performance in a variety of settings (PCs of various architectures, smart cards, hardware implementations) and on their feasibility in limited environments (smart cards with very limited memory, low gate count implementations, FPGAs).
Some designs fell due tocryptanalysisthat ranged from minor flaws to significant attacks, while others lost favour due to poor performance in various environments or through having little to offer over other candidates. NIST held two conferences to discuss the submissions (AES1, August 1998 and AES2, March 1999[4][5][6]), and in August 1999 they announced[7]that they were narrowing the field from fifteen to five:MARS,RC6,Rijndael,Serpent, andTwofish. All five algorithms, commonly referred to as "AES finalists", were designed by cryptographers considered well-known and respected in the community.
The AES2 conference votes were as follows:[8]
A further round of intense analysis and cryptanalysis followed, culminating in the AES3 conference in April 2000, at which a representative of each of the final five teams made a presentation arguing why their design should be chosen as the AES. The AES3 conference votes were as follows:[9]
On October 2, 2000, NIST announced[10]thatRijndaelhad been selected as the proposed AES and started the process of making it the official standard by publishing an announcement in theFederal Register[11]on February 28, 2001 for the draft FIPS to solicit comments. On November 26, 2001, NIST announced thatAESwas approved asFIPS PUB197.
NIST won praises from the cryptographic community for the openness and care with which they ran the standards process.Bruce Schneier, one of the authors of the losing Twofish algorithm, wrote after the competition was over that "I have nothing but good things to say about NIST and the AES process."[12]
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https://en.wikipedia.org/wiki/AES_process
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TheCompetition for Authenticated Encryption: Security, Applicability, and Robustness(CAESAR) is a competition organized by a group of international cryptologic researchers to encourage the design ofauthenticated encryptionschemes.[1]The competition was announced at the Early Symmetric Crypto workshop in January 2013 and the final portfolio in February 2019.
The final CAESAR portfolio is organized into three use cases:[2]
The final portfolio announced by the CAESAR committee is:[2]
The committee in charge of the CAESAR Competition consisted of:[3]
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https://en.wikipedia.org/wiki/CAESAR_Competition
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Incryptography, apinwheelwas a device for producing a shortpseudorandomsequence ofbits(determined by the machine's initial settings), as a component in a cipher machine. A pinwheel consisted of a rotating wheel with a certain number of positions on its periphery. Each position had a "pin", "cam" or "lug" which could be either "set" or "unset". As the wheel rotated, each of these pins would in turn affect other parts of the machine, producing a series of "on" or "off" pulses which would repeat after one full rotation of the wheel. If the machine contained more than one wheel, usually their periods would berelatively primeto maximize the combined period. Pinwheels might be turned through a purely mechanical action (as in theM-209) orelectromechanically(as in theLorenz SZ 40/42).
The Swedish engineerBoris Caesar Wilhelm Hagelin[1]is credited with having invented the first pinwheel device in 1925.[2]He developed the machine while employed byEmanuel Nobelto oversee the Nobel interests in Aktiebolaget Cryptograph.[2]He was the nephew of the founder of theNobel Prize. The device was later introduced in France and Hagelin was awarded the French order of merit,Legion d'Honneur, for his work.[3]One of the earliest cipher machines that Hagaelin developed was the C-38 and was later improved into the more portableHagelin m-209. The M-209 is composed of a set of pinwheels and a rotating cage.[4]
Other cipher machines which used pinwheels include theC-52, theCD-57and theSiemens and Halske T52.
Pinwheels can be viewed as a predecessor to the electroniclinear-feedback shift register(LFSR), used in later cryptosystems.
This cryptography-related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/Pinwheel_(cryptography)
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TheMersenne Twisteris a general-purposepseudorandom number generator(PRNG) developed in 1997 byMakoto Matsumoto(松本 眞)andTakuji Nishimura(西村 拓士).[1][2]Its name derives from the choice of aMersenne primeas its period length.
The Mersenne Twister was created specifically to address most of the flaws found in earlier PRNGs.
The most commonly used version of the Mersenne Twister algorithm is based on the Mersenne prime219937−1{\displaystyle 2^{19937}-1}. The standard implementation of that, MT19937, uses a32-bitword length. There is another implementation (with five variants[3]) that uses a 64-bit word length, MT19937-64; it generates a different sequence.
A pseudorandom sequencexi{\displaystyle x_{i}}ofw-bit integers of periodPis said to bek-distributedtov-bit accuracy if the following holds.
For aw-bit word length, the Mersenne Twister generates integers in the range[0,2w−1]{\displaystyle [0,2^{w}-1]}.
The Mersenne Twister algorithm is based on amatrix linear recurrenceover a finitebinaryfieldF2{\displaystyle {\textbf {F}}_{2}}. The algorithm is a twistedgeneralised feedback shift register[4](twisted GFSR, or TGFSR) ofrational normal form(TGFSR(R)), with state bit reflection and tempering. The basic idea is to define a seriesxi{\displaystyle x_{i}}through a simple recurrence relation, and then output numbers of the formxiT{\displaystyle x_{i}^{T}}, whereTis an invertibleF2{\displaystyle {\textbf {F}}_{2}}-matrix called atempering matrix.
The general algorithm is characterized by the following quantities:
with the restriction that2nw−r−1{\displaystyle 2^{nw-r}-1}is a Mersenne prime. This choice simplifies the primitivity test andk-distributiontest needed in the parameter search.
The seriesx{\displaystyle x}is defined as a series ofw-bit quantities with the recurrence relation:
where∣{\displaystyle \mid }denotesconcatenationof bit vectors (with upper bits on the left),⊕{\displaystyle \oplus }the bitwiseexclusive or(XOR),xku{\displaystyle x_{k}^{u}}means the upperw−rbits ofxk{\displaystyle x_{k}}, andxk+1l{\displaystyle x_{k+1}^{l}}means the lowerrbits ofxk+1{\displaystyle x_{k+1}}.
The subscripts may all be offset by-n
where now the LHS,xk{\displaystyle x_{k}}, is the next generated value in the series in terms of values generated in the past, which are on the RHS.
The twist transformationAis defined in rational normal form as:A=(0Iw−1aw−1(aw−2,…,a0)){\displaystyle A={\begin{pmatrix}0&I_{w-1}\\a_{w-1}&(a_{w-2},\ldots ,a_{0})\end{pmatrix}}}withIw−1{\displaystyle I_{w-1}}as the(w−1)(w−1){\displaystyle (w-1)(w-1)}identity matrix. The rational normal form has the benefit that multiplication byAcan be efficiently expressed as: (remember that here matrix multiplication is being done inF2{\displaystyle {\textbf {F}}_{2}}, and therefore bitwise XOR takes the place of addition)xA={x≫1x0=0(x≫1)⊕ax0=1{\displaystyle {\boldsymbol {x}}A={\begin{cases}{\boldsymbol {x}}\gg 1&x_{0}=0\\({\boldsymbol {x}}\gg 1)\oplus {\boldsymbol {a}}&x_{0}=1\end{cases}}}wherex0{\displaystyle x_{0}}is the lowest order bit ofx{\displaystyle x}.
As like TGFSR(R), the Mersenne Twister is cascaded with atempering transformto compensate for the reduced dimensionality of equidistribution (because of the choice ofAbeing in the rational normal form). Note that this is equivalent to using the matrixAwhereA=T−1∗AT{\displaystyle A=T^{-1}*AT}forTan invertible matrix, and therefore the analysis of characteristic polynomial mentioned below still holds.
As withA, we choose a tempering transform to be easily computable, and so do not actually constructTitself. This tempering is defined in the case of Mersenne Twister as
wherex{\displaystyle x}is the next value from the series,y{\displaystyle y}is a temporary intermediate value, andz{\displaystyle z}is the value returned from the algorithm, with≪{\displaystyle \ll }and≫{\displaystyle \gg }as thebitwise left and right shifts, and&{\displaystyle \&}as the bitwiseAND. The first and last transforms are added in order to improve lower-bit equidistribution. From the property of TGFSR,s+t≥⌊w2⌋−1{\displaystyle s+t\geq \left\lfloor {\frac {w}{2}}\right\rfloor -1}is required to reach the upper bound of equidistribution for the upper bits.
The coefficients for MT19937 are:
(w,n,m,r)=(32,624,397,31)a=9908B0DF16(u,d)=(11,FFFFFFFF16)(s,b)=(7,9D2C568016)(t,c)=(15,EFC6000016)l=18{\displaystyle {\begin{aligned}(w,n,m,r)&=(32,624,397,31)\\a&={\textrm {9908B0DF}}_{16}\\(u,d)&=(11,{\textrm {FFFFFFFF}}_{16})\\(s,b)&=(7,{\textrm {9D2C5680}}_{16})\\(t,c)&=(15,{\textrm {EFC60000}}_{16})\\l&=18\\\end{aligned}}}
Note that 32-bit implementations of the Mersenne Twister generally haved= FFFFFFFF16. As a result, thedis occasionally omitted from the algorithm description, since the bitwiseandwithdin that case has no effect.
The coefficients for MT19937-64 are:[5]
(w,n,m,r)=(64,312,156,31)a=B5026F5AA96619E916(u,d)=(29,555555555555555516)(s,b)=(17,71D67FFFEDA6000016)(t,c)=(37,FFF7EEE00000000016)l=43{\displaystyle {\begin{aligned}(w,n,m,r)=(64,312,156,31)\\a={\textrm {B5026F5AA96619E9}}_{16}\\(u,d)=(29,{\textrm {5555555555555555}}_{16})\\(s,b)=(17,{\textrm {71D67FFFEDA60000}}_{16})\\(t,c)=(37,{\textrm {FFF7EEE000000000}}_{16})\\l=43\\\end{aligned}}}
The state needed for a Mersenne Twister implementation is an array ofnvalues ofwbits each. To initialize the array, aw-bit seed value is used to supplyx0{\displaystyle x_{0}}throughxn−1{\displaystyle x_{n-1}}by settingx0{\displaystyle x_{0}}to the seed value and thereafter setting
fori{\displaystyle i}from1{\displaystyle 1}ton−1{\displaystyle n-1}.
In order to achieve the2nw−r−1{\displaystyle 2^{nw-r}-1}theoretical upper limit of the period in a TGFSR,ϕB(t){\displaystyle \phi _{B}(t)}must be aprimitive polynomial,ϕB(t){\displaystyle \phi _{B}(t)}being thecharacteristic polynomialof
The twist transformation improves the classicalGFSRwith the following key properties:
CryptMTis astream cipherandcryptographically secure pseudorandom number generatorwhich uses Mersenne Twister internally.[6][7]It was developed by Matsumoto and Nishimura alongside Mariko Hagita and Mutsuo Saito. It has been submitted to theeSTREAMproject of theeCRYPTnetwork.[6]Unlike Mersenne Twister or its other derivatives, CryptMT ispatented.
MTGP is a variant of Mersenne Twister optimised forgraphics processing unitspublished by Mutsuo Saito and Makoto Matsumoto.[8]The basic linear recurrence operations are extended from MT and parameters are chosen to allow many threads to compute the recursion in parallel, while sharing their state space to reduce memory load. The paper claims improvedequidistributionover MT and performance on an old (2008-era) GPU (NvidiaGTX260 with 192 cores) of 4.7 ms for 5×107random 32-bit integers.
The SFMT (SIMD-oriented Fast Mersenne Twister) is a variant of Mersenne Twister, introduced in 2006,[9]designed to be fast when it runs on 128-bit SIMD.
IntelSSE2andPowerPCAltiVec are supported by SFMT. It is also used for games with theCell BEin thePlayStation 3.[11]
TinyMT is a variant of Mersenne Twister, proposed by Saito and Matsumoto in 2011.[12]TinyMT uses just 127 bits of state space, a significant decrease compared to the original's 2.5 KiB of state. However, it has a period of2127−1{\displaystyle 2^{127}-1}, far shorter than the original, so it is only recommended by the authors in cases where memory is at a premium.
Advantages:
Disadvantages:
The Mersenne Twister is used as default PRNG by the following software:
It is also available inApache Commons,[47]in the standardC++library (sinceC++11),[48][49]and inMathematica.[50]Add-on implementations are provided in many program libraries, including theBoost C++ Libraries,[51]theCUDA Library,[52]and theNAG Numerical Library.[53]
The Mersenne Twister is one of two PRNGs inSPSS: the other generator is kept only for compatibility with older programs, and the Mersenne Twister is stated to be "more reliable".[54]The Mersenne Twister is similarly one of the PRNGs inSAS: the other generators are older and deprecated.[55]The Mersenne Twister is the default PRNG inStata, the other one isKISS, for compatibility with older versions of Stata.[56]
An alternative generator,WELL("Well Equidistributed Long-period Linear"), offers quicker recovery, and equal randomness, and nearly equal speed.[57]
Marsaglia'sxorshiftgenerators and variants are the fastest in the class of LFSRs.[58]
64-bit MELGs ("64-bit Maximally EquidistributedF2{\displaystyle {\textbf {F}}_{2}}-Linear Generators with Mersenne Prime Period") are completely optimized in terms of thek-distribution properties.[59]
TheACORN family(published 1989) is anotherk-distributed PRNG, which shows similar computational speed to MT, and better statistical properties as it satisfies all the current (2019) TestU01 criteria; when used with appropriate choices of parameters, ACORN can have arbitrarily long period and precision.
ThePCG familyis a more modern long-period generator, with better cache locality, and less detectable bias using modern analysis methods.[60]
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Amaximum length sequence(MLS) is a type ofpseudorandom binary sequence.
They are bit sequences generated using maximallinear-feedback shift registersand are so called because they areperiodicand reproduce everybinary sequence(except the zero vector) that can be represented by the shift registers (i.e., for length-mregisters they produce a sequence of length 2m− 1). An MLS is also sometimes called ann-sequenceor anm-sequence. MLSs arespectrally flat, with the exception of a near-zero DC term.
These sequences may be represented as coefficients of irreducible polynomials in apolynomial ringoverZ/2Z.
Practical applications for MLS include measuringimpulse responses(e.g., of roomreverberationor arrival times from towed sources in the ocean[1]). They are also used as a basis for deriving pseudo-random sequences in digital communication systems that employdirect-sequence spread spectrumandfrequency-hopping spread spectrumtransmission systems, and in the efficient design of somefMRIexperiments.[2]
MLS are generated using maximallinear-feedback shift registers. An MLS-generating system with a shift register of length 4 is shown in Fig. 1. It can be expressed using the following recursive relation:
wherenis the time index and+{\displaystyle +}representsmodulo-2addition. For bit values 0 = FALSE or 1 = TRUE, this is equivalent to the XOR operation.
As MLS are periodic and shift registers cycle through every possible binary value (with the exception of the zero vector), registers can be initialized to any state, with the exception of the zero vector.
ApolynomialoverGF(2)can be associated with the linear-feedback shift register. It has degree of the length of the shift register, and has coefficients that are either 0 or 1, corresponding to the taps of the register that feed thexorgate. For example, the polynomial corresponding to Figure 1 isx4+x+1{\displaystyle x^{4}+x+1}.
A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial beprimitive.[3]
MLS are inexpensive to implement in hardware or software, and relatively low-order feedback shift registers can generate long sequences; a sequence generated using a shift register of length 20 is 220− 1 samples long (1,048,575 samples).
MLS have the following properties, as formulated bySolomon Golomb.[4]
The occurrence of 0 and 1 in the sequence should be approximately the same. More precisely, in a maximum length sequence of length2n−1{\displaystyle 2^{n}-1}there are2n−1{\displaystyle 2^{n-1}}ones and2n−1−1{\displaystyle 2^{n-1}-1}zeros. The number of ones equals the number of zeros plus one, since the state containing only zeros cannot occur.
A "run" is a sub-sequence of consecutive "1"s or consecutive "0"s within the MLS concerned. The number of runs is the number of such sub-sequences.[vague]
Of all the "runs" (consisting of "1"s or "0"s) in the sequence :
The circularautocorrelationof an MLS is aKronecker deltafunction[5][6](with DC offset and time delay, depending on implementation). For the ±1 convention, i.e., bit value 1 is assigneds=+1{\displaystyle s=+1}and bit value 0s=−1{\displaystyle s=-1}, mapping XOR to the negative of the product:
R(n)=1N∑m=1Ns[m]s∗[m+n]N={1ifn=0,−1Nif0<n<N.{\displaystyle R(n)={\frac {1}{N}}\sum _{m=1}^{N}s[m]\,s^{*}[m+n]_{N}={\begin{cases}1&{\text{if }}n=0,\\-{\frac {1}{N}}&{\text{if }}0<n<N.\end{cases}}}
wheres∗{\displaystyle s^{*}}represents the complex conjugate and[m+n]N{\displaystyle [m+n]_{N}}represents acircular shift.
The linear autocorrelation of an MLS approximates a Kronecker delta.
If alinear time invariant(LTI) system's impulse response is to be measured using a MLS, the response can be extracted from the measured system outputy[n] by taking its circular cross-correlation with the MLS. This is because theautocorrelationof a MLS is 1 for zero-lag, and nearly zero (−1/NwhereNis the sequence length) for all other lags; in other words, the autocorrelation of the MLS can be said to approach unit impulse function as MLS length increases.
If the impulse response of a system ish[n] and the MLS iss[n], then
Taking the cross-correlation with respect tos[n] of both sides,
and assuming that φssis an impulse (valid for long sequences)
Any signal with an impulsive autocorrelation can be used for this purpose, but signals with highcrest factor, such as the impulse itself, produce impulse responses with poorsignal-to-noise ratio. It is commonly assumed that the MLS would then be the ideal signal, as it consists of only full-scale values and its digital crest factor is the minimum, 0 dB.[7][8]However, afteranalog reconstruction, the sharp discontinuities in the signal produce strong intersample peaks, degrading the crest factor by 4-8 dB or more, increasing with signal length, making it worse than a sine sweep.[9]Other signals have been designed with minimal crest factor, though it is unknown if it can be improved beyond 3 dB.[10]
Cohn and Lempel[11]showed the relationship of the MLS to theHadamard transform. This relationship allows thecorrelationof an MLS to be computed in a fast algorithm similar to theFFT.
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Ananalog feedback shift register(AFSR) is a generalization of the (binary, digital)linear-feedback shift register(LFSR).
While binary LFSRs require less power to generatespread spectrumsignals than AFSRs, AFSR receivers require less power (in theory) to synchronize to those signals than binary LFSR receivers.
As of 2005, AFSRs are still in research.
AFSR techniques could make spread-spectrum receivers (such asGPSreceivers andcell phonesandWi-Fireceivers andRFIDs) cost less and have longer battery lifetimes.
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Anonlinear-feedback shift register(NLFSR) is ashift registerwhose input bit is a non-linear function of its previous state.
For an n-bit shift registerrits next state is defined as:
ri+1(b0,b1,b2,…,bn−1)=ri(b1,b2,…,f(b0,b1,b2,…,bn−1)){\displaystyle r_{i+1}(b_{0},b_{1},b_{2},\ldots ,b_{n-1})=r_{i}(b_{1},b_{2},\ldots ,f(b_{0},b_{1},b_{2},\ldots ,b_{n-1}))},
wherefis the non-linear feedback function.[1]
Nonlinear-feedback shift registers are components in modernstream ciphers, especially inRFIDandsmartcardapplications. NLFSRs are known to be more resistant to cryptanalytic attacks than Linear Feedback Shift Registers (LFSRs).
It is known how to generate ann-bit NLFSR of maximal length2n, generating aDe Bruijn sequence, by extending a maximal-length LFSR withnstages;[2]but the construction of other large NLFSRs with guaranteed long periods remains an open problem.[3]Using bruteforce methods, a list of maximum-periodn-bit NLFSRs for n ≤ 25 has been made as well as for n=27.[4][1]
New methods suggest usage ofevolutionary algorithmsin order to introduce non-linearity.[5]In these works, an evolutionary algorithm learns how to apply different operations on strings fromLFSRto enhance their quality to meet the criteria of a fitness function, here theNISTprotocol,[6]effectively.
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Aring counteris a type of counter composed offlip-flopsconnected into ashift register, with the output of the last flip-flop fed to the input of the first, making a "circular" or "ring" structure.
There are two types of ring counters:
Ring counters are often used in hardware design (e.g.ASICandFPGAdesign) to createfinite-state machines. A binary counter would require anaddercircuit which is substantially more complex than a ring counter and has higher propagation delay as the number of bits increases, whereas the propagation delay of a ring counter will be nearly constant regardless of the number of bits.
The straight and twisted forms have different properties, and relative advantages and disadvantages.
A general disadvantage of ring counters is that they are lower density codes than normalbinary encodingsof state numbers. A binary counter can represent 2Nstates, whereNis the number of bits in the code, whereas a straight ring counter can represent onlyNstates and a Johnson counter can represent only 2Nstates. This may be an important consideration in hardware implementations where registers are more expensive than combinational logic.
Johnson counters are sometimes favored, because they offer twice as many count states from the same number of shift registers, and because they are able to self-initialize from the all-zeros state, without requiring the first count bit to be injected externally at start-up. The Johnson counter generates a code in which adjacent states differ by only one bit (that is, have aHamming distanceof 1), as in aGray code, which can be useful if the bit pattern is going to be asynchronously sampled.[1]
When a fully decoded orone-hotrepresentation of the counter state is needed, as in some sequence controllers, the straight ring counter is preferred. The one-hot property means that the set of codes are separated by aminimum Hamming distanceof 2,[2]so any single-bit error is detectable (as is any error pattern other than turning on one bit and turning off one bit).
Sometimes bidirectional shift registers are used (using multiplexors to take the input for each flip-flop from its left or right neighbor), so that bidirectional or up–down ring counters can be made.[3]
The straight ring counter has the logical structure shown here:
Instead of the reset line setting up the initialone-hotpattern, the straight ring is sometimes made self-initializing by the use of a distributed feedback gate across all of the outputs except that last, so that a 1 is presented at the input when there is no 1 in any stage but the last.[4]
A Johnson counter, named forRobert Royce Johnson, is a ring with an inversion; here is a 4-bit Johnson counter:
Note the small bubble indicating inversion of the Q signal from the last shift register before feeding back to the first D input, making this a Johnson counter.
Before the days of digital computing, digital counters were used to measure rates of random events such as radioactive decays to alpha and beta particle. Fast "pre-scaling" counters reduced the rate of random events to more manageable and more regular rates. Five-state ring counters were used along with divide-by-two scalers to make decade (power-of-ten) scalers before 1940, such as those developed byC. E. Wynn-Williams.[5]
Early ring counters used only one active element (vacuum tube, valve, or transistor) per stage, relying on global feedback rather than local bistable flip-flops, to suppress states other than the one-hot states, for example in the 1941 patent filing ofRobert E. Mummaof theNational Cash Registor Company.[6]Wilcox P. Overbeckinvented a version using multiple anodes in a single vacuum tube,[7][8]In recognition of his work, ring counters are sometimes referred to as "Overbeck rings"[9][10](and after 2006, sometimes as "Overbeck counters", since Wikipedia used that term from 2006 to 2018).
TheENIACused decimal arithmetic based on 10-state one-hot ring counters. The works of Mumma atNCRand Overbeck atMITwere among the prior art works examined by the patent office that invalidated the patents ofJ. Presper EckertandJohn Mauchlyfor the ENIAC technology.[11]
By the 1950s, ring counters with a two-tube or twin-triode flip-flop per stage were appearing.[12]
Robert Royce Johnson developed a number of different shift-register-based counters with the aim of making different numbers of states with the simplest possible feedback logic, and filed for a patent in 1953.[13]The Johnson counter is the simplest of these.
Early applications of ring counters were as frequency prescalers (e.g. forGeiger counterand such instruments),[5]as counters to count pattern occurrences in cryptanalysis (e.g. in theHeath Robinson codebreaking machineand theColossus computer),[14]and as accumulator counter elements for decimal arithmetic in computers and calculators, using eitherbi-quinary(as in the Colossus) or ten-state one-hot (as in theENIAC) representations.
Straight ring counters generate fully decoded one-hot codes to that are often used to enable a specific action in each state of a cyclic control cycle. One-hot codes can also be decoded from a Johnson counter, using one gate for each state.[15][nb 1]
Besides being an efficient alternative way to generate one-hot codes and frequency pre-scalers, a Johnson counter is also a simple way to encode a cycle of an even number of states that can be asynchronously sampled without glitching, since only one bit changes at a time, as in aGray code.[16]Earlycomputer miceused up–down (bidirectional) 2-bit Johnson or Gray encodings to indicate motion in each of the two dimensions, though in mice those codes were not usually generated by rings of flip-flops (but instead by electro-mechanical or opticalquadrature encoders).[17]A 2-bit Johnson code and a 2-bit Gray code are identical, while for 3 or more bits Gray and Johnson codes are different. In the 5-bit case, the code is the same as theLibaw–Craig code[de]for decimal digits.[18][19][20][21][22][23][24][25]
A walking ring counter, also called a Johnson counter, and a few resistors can produce a glitch-free approximation of a sine wave.
When combined with an adjustableprescaler, this is perhaps the simplestnumerically-controlled oscillator.
Two such walking ring counters are perhaps the simplest way to generate thecontinuous-phase frequency-shift keyingused indual-tone multi-frequency signalingand earlymodemtones.[26]
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Apseudorandom binary sequence(PRBS),pseudorandom binary codeorpseudorandom bitstreamis abinary sequencethat, while generated with a deterministicalgorithm, is difficult to predict[1]and exhibits statistical behavior similar to a truly random sequence. PRBS generators are used intelecommunication, such as in analog-to-information conversion,[2]but also inencryption,simulation,correlationtechnique and time-of-flightspectroscopy. The most common example is themaximum length sequencegenerated by a (maximal)linear feedback shift register(LFSR). Other examples areGold sequences(used inCDMAandGPS),Kasami sequencesandJPL sequences, all based on LFSRs.
Intelecommunications, pseudorandom binary sequences are known aspseudorandom noise codes(PNorPRN codes) due to their application aspseudorandom noise.
A binary sequence (BS) is asequencea0,…,aN−1{\displaystyle a_{0},\ldots ,a_{N-1}}ofN{\displaystyle N}bits, i.e.
A BS consists ofm=∑aj{\displaystyle m=\sum a_{j}}ones andN−m{\displaystyle N-m}zeros.
A BS is apseudorandombinary sequence(PRBS) if[3]itsautocorrelation function, given by
has only two values:
where
is called theduty cycleof the PRBS, similar to theduty cycleof a continuous time signal. For amaximum length sequence, whereN=2k−1{\displaystyle N=2^{k}-1}, the duty cycle is 1/2.
A PRBS is 'pseudorandom', because, although it is in fact deterministic, it seems to be random in a sense that the value of anaj{\displaystyle a_{j}}element is independent of the values of any of the other elements, similar to real random sequences.
A PRBS can be stretched to infinity by repeating it afterN{\displaystyle N}elements, but it will then be cyclical and thus non-random. In contrast, truly random sequence sources, such as sequences generated byradioactive decayor bywhite noise, are infinite (no pre-determined end or cycle-period). However, as a result of this predictability, PRBS signals can be used as reproducible patterns (for example, signals used in testing telecommunications signal paths).[4]
Pseudorandom binary sequences can be generated usinglinear-feedback shift registers.[5]
Some common[6][7][8][9][10]sequence generatingmonic polynomialsare
An example of generating a "PRBS-7" sequence can be expressed in C as
In this particular case, "PRBS-7" has a repetition period of 127 values.
The PRBSkor PRBS-knotation (such as "PRBS7" or "PRBS-7") gives an indication of the size of the sequence.N=2k−1{\displaystyle N=2^{k}-1}is the maximum number[4]: §3of bits that are in the sequence. Thekindicates the size of a uniquewordof data in the sequence. If you segment theNbits of data into every possible word of lengthk, you will be able to list every possible combination of 0s and 1s for a k-bit binary word, with the exception of the all-0s word.[4]: §2For example, PRBS3 = "1011100" could be generated fromx3+x2+1{\displaystyle x^{3}+x^{2}+1}.[6]If you take every sequential group of three bit words in the PRBS3 sequence (wrapping around to the beginning for the last few three-bit words), you will find the following 7 word arrangements:
Those 7 words are all of the2k−1=23−1=7{\displaystyle 2^{k}-1=2^{3}-1=7}possible non-zero 3-bit binary words, not in numeric order. The same holds true for any PRBSk, not just PRBS3.[4]: §2
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AGold code, also known asGold sequence, is a type of binarysequence, used intelecommunications(CDMA)[1]and satellite navigation (GPS).[2]Gold codes are named after Robert Gold.[3][4]Gold codes have bounded smallcross-correlationswithin a set, which is useful when multiple devices are broadcasting in the same frequency range. A set of Gold code sequences consists of 2n+ 1 sequences each one with a period of 2n− 1.
A set of Gold codes can be generated with the following steps. Pick twomaximum length sequencesof the same length 2n− 1 such that their absolutecross-correlationis less than or equal to 2(n+2)/2, wherenis the size of thelinear-feedback shift registerused to generate the maximum length sequence (Gold '67). The set of the 2n− 1exclusive-orsof the two sequences in their various phases (i.e. translated into all relative positions) together with the two maximum length sequences form a set of 2n+ 1 Gold code sequences. The highest absolute cross-correlation in this set of codes is 2(n+2)/2+ 1 for evennand 2(n+1)/2+ 1 for oddn.
Theexclusive orof two different Gold codes from the same set is another Gold code in some phase.
Within a set of Gold codes about half of the codes are balanced – the number of ones and zeros differs by only one.[5]
Gold codes are used inGPS. TheGPS C/Aranging codes are Gold codes of period 1,023.
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JPL sequencesorJPL codesconsist of twolinear feedback shift registers(LFSRs) whose code sequence lengthsLaandLbmust be prime (relatively prime).[1]In this case the code sequence length of the generated overall sequenceLcis equal to:
It is also possible for more than two LFSRs to be interconnected through multipleXORsat the output for as long as all code sequence lengths of the individual LFSR are relatively prime to one another.
JPL sequences were originally developed in theJet Propulsion Labs, from which the name for these code sequences is derived.
Areas of application include distance measurements utilizingspread spectrumsignals for satellites and in space technology. They are also utilized in the more precise militaryP/Y codeused in theGlobal Positioning System(GPS).[2]However, they are currently replaced by the new M-code.
Due to the relatively long spreading sequences, they can be used to measure relatively long ranges without ambiguities, as required for deep space missions. By having a rough synchronziation between receiver and transmitter, this can be achieved with shorter sequences as well.
Their major advantage is, that they produce relatively long sequences with only two LFSRs, which makes it energy efficient and very hard to detect due to huge spreading factor. The same structure can be used to realize a dither generator, used as an additive noise source to remove a numerical bias in digital computations (due to fixed point arithmetics, that have one more negative than positive number, i.e. the mean value is slightly negative).
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Kasami sequencesare binarysequencesof length2N−1whereNis an even integer. Kasami sequences have goodcross-correlationvalues approaching theWelch lower bound. There are two classes of Kasami sequences—the small set and the large set.
The process of generating a Kasami sequence is initiated by generating amaximum length sequencea(n), wheren= 1…2N−1. Maximum length sequences are periodic sequences with a period of exactly2N−1. Next, a secondary sequence is derived from the initial sequence via cyclic decimation sampling asb(n) =a(q⋅n), whereq= 2N/2+1. Modified sequences are then formed by addinga(n)and cyclically time shifted versions ofb(n)using modulo-two arithmetic, which is also termed theexclusive or(xor) operation. Computing modified sequences from all2N/2unique time shifts ofb(n)forms the Kasami set of code sequences.
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TheIrish logarithmwas a system of number manipulation invented byPercy Ludgatefor machine multiplication. The system used a combination of mechanical cams aslookup tablesand mechanical addition to sum pseudo-logarithmic indices to produce partial products, which were then added to produce results.[1]
The technique is similar toZech logarithms(also known as Jacobi logarithms), but uses a system of indices original to Ludgate.[2]
Ludgate's algorithm compresses the multiplication of two single decimal numbers into twotable lookups(to convert the digits into indices), the addition of the two indices to create a new index which is input to a second lookup table that generates the output product.[3]Because both lookup tables are one-dimensional, and the addition of linear movements is simple to implement mechanically, this allows a less complex mechanism than would be needed to implement a two-dimensional 10×10 multiplication lookup table.
Ludgate stated that he deliberately chose the values in his tables to be as small as he could make them; given this, Ludgate's tables can be simply constructed from first principles, either via pen-and-paper methods, or a systematic search using only a few tens of lines of program code.[4]They do not correspond to either Zech logarithms,Remak indexesorKorn indexes.[4]
The following is an implementation of Ludgate's Irish logarithm algorithm in thePythonprogramming language:
Table 1 is taken from Ludgate's original paper; given the first table, the contents of Table 2 can be trivially derived from Table 1 and the definition of the algorithm. Note since that the last third of the second table is entirely zeros, this could be exploited to further simplify a mechanical implementation of the algorithm.
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TheCanon arithmeticusis a set ofmathematical tablesof indices and powers with respect toprimitive rootsfor prime powers less than 1000, originally published byCarl Gustav Jacob Jacobi(1839), with introductory text inLatin. The tables were at one time used for arithmetical calculations modulo prime powers, though like many mathematical tables, they have now been replaced by digital computers. Jacobi also reproducedBurkhardt's table of theperiods of decimal fractions of 1/pandOstrogradsky's tables of primitive roots of primes less than 200 and gave tables of indices of some odd numbers modulo powers of 2 with respect to the base 3 (Dickson 2005, p. 185–186).
Although the second edition of 1956 has Jacobi's name on the title, it has little in common with the first edition apart from the topic: the tables were completely recalculated, usually with a different choice of primitive root, by Wilhelm Patz. Jacobi's original tables use 10 or −10 or a number with a small power of this form as the primitive root whenever possible, while the second edition uses the smallest possible positive primitive root (Fletcher 1958).
The term "canon arithmeticus" is occasionally used to mean any table of indices and powers of primitive roots.
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Carl Gustav Jacob Jacobi[a](/dʒəˈkoʊbi/;[3]German:[jaˈkoːbi]; 10 December 1804 – 18 February 1851) was a Germanmathematicianwho made fundamental contributions toelliptic functions,dynamics,differential equations,determinantsandnumber theory.
Jacobi was born ofAshkenazi Jewishparentage inPotsdamon 10 December 1804. He was the second of four children of a banker, Simon Jacobi. His elder brother,Moritz, would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the PotsdamGymnasium, where students were taught all the standard subjects: classical languages, history, philology, mathematics, sciences, etc. As a result of the good education he had received from his uncle, as well as his own remarkable abilities, after less than half a year Jacobi was moved to the senior year despite his young age. However, as the University would not accept students younger than 16 years old, he had to remain in the senior class until 1821. He used this time to advance his knowledge, showing interest in all subjects, including Latin, Greek, philology, history and mathematics. During this period he also made his first attempts at research, trying to solve thequintic equationbyradicals.[4][5]
In 1821 Jacobi went to study atBerlin University, where he initially divided his attention between his passions forphilologyandmathematics. In philology, he participated in the seminars ofBöckh, drawing the professor's attention with his talent. Jacobi did not follow a lot of mathematics classes at the time, finding the level of mathematics taught at Berlin University too elementary. He continued, instead, with his private study of the more advanced works ofEuler,LagrangeandLaplace. By 1823 he understood that he needed to make a decision between his competing interests and chose to devote all his attention to mathematics.[6]In the same year he became qualified to teach secondary school and was offered a position at theJoachimsthal Gymnasiumin Berlin. Jacobi decided instead to continue to work towards a university position. In 1825, he obtained the degree of Doctor of Philosophy with a dissertation on thepartial fraction decompositionofrational fractionsdefended before a commission led byEnno Dirksen. He followed immediately with hishabilitationand at the same time converted to Christianity. Now qualifying for teaching university classes, the 21-year-old Jacobi lectured in 1825/26 on the theory ofcurvesandsurfacesat the University of Berlin.[6][7]
In 1826, Jacobi became aprivate lecturer, in the next year anextraordinary professor, and in finally 1829, a tenured professor ofmathematicsatKönigsberg University, and held the chair until 1842.
He suffered abreakdownfrom overwork in 1843. He then visitedItalyfor a few months to regain his health. On his return he moved to Berlin, where he lived as a royal pensioner, apart from a very brief interim, until his death.[2]During theRevolution of 1848Jacobi was politically involved and unsuccessfully presented his parliamentary candidature on behalf of aLiberalclub. This led, after the suppression of the revolution, to his royal grant being cut off – but his fame and reputation were such that it was soon resumed, thanks to the personal intervention ofAlexander von Humboldt.
Jacobi died in 1851 from asmallpoxinfection. His grave is preserved at a cemetery in theKreuzbergsection of Berlin, theFriedhof I der Dreifaltigkeits-Kirchengemeinde(61 Baruther Street). His grave is close to that ofJohann Encke, the astronomer. The craterJacobion theMoonis named after him.
Jacobi's birth name was Jacques Simon, a French-style name (his father was Simon Jacobi).[8]Later, his name was Germanized to Carl Gustav Jacob Jacobi and published in its Latinized form as Carolus Gustavus Jacobus Jacobi. He is sometimes referred to as C. G. J. Jacobi.
One of Jacobi's greatest accomplishments was his theory ofelliptic functionsand their relation to the elliptictheta function. This was developed in his great treatiseFundamenta nova theoriae functionum ellipticarum(1829), and in later papers inCrelle's Journal. Theta functions are of great importance in mathematical physics because of their role in the inverse problem for periodic andquasi-periodic flows. Theequations of motionareintegrablein terms ofJacobi's elliptic functionsin the well-known cases of thependulum, theEuler top, the symmetric Lagrange top in agravitational field, and theKepler problem(planetary motion in a central gravitational field).
He also made fundamental contributions in the study of differential equations and toclassical mechanics, notably theHamilton–Jacobi theory.
It was in algebraic development that Jacobi's particular power mainly lay, and he made important contributions of this kind in many areas of mathematics, as shown by his long list of papers in Crelle's Journal and elsewhere from 1826 onwards.[2]He is said to have told his students that when looking for a research topic, one should 'Invert, always invert' (German original:"man muss immer umkehren"), reflecting his belief that inverting known results can open up new fields for research, for example invertingelliptic integralsand focusing on the nature of elliptic and theta functions.[9]
In his 1835 paper, Jacobi proved the following basic result classifying periodic (including elliptic) functions:
If a univariate single-valued function is multiplyperiodic, then such a function cannot have more than two periods, and the ratio of the periods cannot be a real number.
He discovered many of the fundamental properties of theta functions, including the functional equation and theJacobi triple productformula, as well as many other results onq-seriesandhypergeometric series.
The solution of theJacobi inversion problemfor the hyperellipticAbel mapbyWeierstrassin 1854 required the introduction of thehyperelliptic theta functionand later the general Riemann theta function foralgebraic curvesof arbitrarygenus. Thecomplex torusassociated to a genusg{\displaystyle g}algebraic curve, obtained by quotientingCg{\displaystyle {\mathbf {C} }^{g}}by thelatticeof periods is referred to as theJacobian variety. This method of inversion, and its subsequent extension by Weierstrass andRiemannto arbitrary algebraic curves, may be seen as a higher genus generalization of the relation between elliptic integrals and the Jacobi or Weierstrass elliptic functions.
Jacobi was the first to apply elliptic functions tonumber theory, for example provingFermat'stwo-square theoremandLagrange's four-square theorem, and similar results for 6 and 8 squares.
His other work in number theory continued the work ofGauss: new proofs ofquadratic reciprocity, and the introduction of theJacobi symbol; contributions to higher reciprocity laws, investigations ofcontinued fractions, and the invention ofJacobi sums.
He was also one of the early founders of the theory of determinants.[10]In particular, he invented theJacobian determinantformed from then2partial derivatives ofngiven functions ofnindependent variables, which plays an important part in changes of variables in multiple integrals, and in many analytical investigations.[2]In 1841 he reintroduced thepartial derivative∂ notation ofLegendre, which was to become standard.
He was one of the first to introduce and study the symmetric polynomials that are now known asSchur polynomials, giving the so-calledbialternant formulafor these, which is a special case of theWeyl character formula, and deriving theJacobi–Trudi identities. He also discovered theDesnanot–Jacobi formula for determinants, which underlie thePlücker relationsforGrassmannians.
Students ofvector fields,Lie theory,Hamiltonian mechanicsandoperator algebrasoften encounter theJacobi identity, the analog ofassociativityfor theLie bracketoperation.
Planetary theoryand other particular dynamical problems likewise occupied his attention from time to time. While contributing tocelestial mechanics, he introduced theJacobi integral(1836) for asidereal coordinate system. His theory of thelast multiplieris treated inVorlesungen über Dynamik, edited byAlfred Clebsch(1866).[2]
He left many manuscripts, portions of which have been published at intervals in Crelle's Journal. His other works includeCommentatio de transformatione integralis duplicis indefiniti in formam simpliciorem(1832),Canon arithmeticus(1839), andOpuscula mathematica(1846–1857). HisGesammelte Werke(1881–1891) were published by theBerlin Academy.[2]
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TheA. W. Faber Model 366was an unusual model ofslide rule, manufactured in Germany by theA. W. Faber Companyaround 1909, with scales that followed a system invented by Johannes Schumacher (1858-1930) that useddiscrete logarithmsto calculate products of integers without approximation.[1][2][3]
The Model 366 is notable for its table of numbers, mapping the numbers 1 to 100 to a permutation of the numbers 0 to 99 in a pattern based on discrete logarithms. The markings on the table are:[2]
The slide rule has two scales on each side of the upper edge of the slider marked with the integers 1 to 100 in a different permuted order, evenly spaced apart. The ordering of the numbers on these scales is
which corresponds to theinverse permutationto the one given by the number table.
There are also two scales on each side of the lower edge of the slider, consisting of the integers 0 to 100 similarly spaced, but in ascending order, with the zero on the lower scales lining up with the 1 on the upper scales.
Schumacher's indices are an example ofJacobi indices, generated withp= 101 andg= 2.[5]Schumacher's system of indices correctly generates the desired products, but is not unique: several other similar systems have been created by others, including systems byLudgate,Remakand Korn.[6]
An elaborate system of rules had to be used to compute products of numbers larger than 101.[1]
Very few of the Model 366 slide rules remain, with only five known to have survived.[1]
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Gorō Shimura(志村 五郎,Shimura Gorō, 23 February 1930 – 3 May 2019)was a Japanesemathematicianand Michael Henry StraterProfessor EmeritusofMathematicsatPrinceton Universitywho worked innumber theory,automorphic forms, andarithmetic geometry.[1]He was known for developing the theory ofcomplex multiplication of abelian varietiesandShimura varieties, as well as posing theTaniyama–Shimura conjecturewhich ultimately led to theproofofFermat's Last Theorem.
Gorō Shimura was born inHamamatsu,Japan, on 23 February 1930.[2]Shimura graduated with a B.A. in mathematics and a D.Sc. in mathematics from theUniversity of Tokyoin 1952 and 1958, respectively.[3][2]
After graduating, Shimura became a lecturer at the University of Tokyo, then worked abroad — including ten months in Paris and a seven-month stint at Princeton'sInstitute for Advanced Study— before returning to Tokyo, where he married Chikako Ishiguro.[4][2]He then moved from Tokyo to join the faculty ofOsaka University, but growing unhappy with his funding situation, he decided to seek employment in the United States.[4][2]ThroughAndré Weilhe obtained a position at Princeton University.[4]Shimura joined the Princeton faculty in 1964 and retired in 1999, during which time he advised over 28 doctoral students and received theGuggenheim Fellowshipin 1970, theCole Prizefor number theory in 1977, theAsahi Prizein 1991, and theSteele Prizefor lifetime achievement in 1996.[1][5]
Shimura described his approach to mathematics as "phenomenological": his interest was in finding new types of interesting behavior in the theory of automorphic forms. He also argued for a "romantic" approach, something he found lacking in the younger generation of mathematicians.[6]Shimura used a two-part process for research, using one desk in his home dedicated to working on new research in the mornings and a second desk for perfecting papers in the afternoon.[2]
Shimura had two children, Tomoko and Haru, with his wife Chikako.[2]Shimura died on 3 May 2019 inPrinceton,New Jerseyat the age of 89.[1][2]
Shimura was a colleague and a friend ofYutaka Taniyama, with whom he wrote the first book on thecomplex multiplication of abelian varietiesand formulated the Taniyama–Shimura conjecture.[7]Shimura then wrote a long series of major papers, extending the phenomena found in the theory ofcomplex multiplication of elliptic curvesand the theory ofmodular formsto higher dimensions (e.g. Shimura varieties). This work provided examples for which the equivalence betweenmotivicandautomorphicL-functionspostulated in theLanglands programcould be tested:automorphic formsrealized in thecohomologyof a Shimura variety have a construction that attachesGalois representationsto them.[8]
In 1958, Shimura generalized the initial work ofMartin Eichleron theEichler–Shimura congruence relationbetween thelocalL-functionof amodular curveand the eigenvalues ofHecke operators.[9][10]In 1959, Shimura extended the work of Eichler on theEichler–Shimura isomorphismbetween Eichler cohomology groups and spaces ofcusp formswhich would be used inPierre Deligne's proof of theWeil conjectures.[11][12]
In 1971, Shimura's work on explicitclass field theoryin the spirit ofKronecker's Jugendtraumresulted in his proof ofShimura's reciprocity law.[13]In 1973, Shimura established theShimura correspondencebetween modular forms of half integral weightk+1/2, and modular forms of even weight 2k.[14]
Shimura's formulation of the Taniyama–Shimura conjecture (later known as the modularity theorem) in the 1950s played a key role in the proof of Fermat's Last Theorem byAndrew Wilesin 1995. In 1990,Kenneth RibetprovedRibet's theoremwhich demonstrated that Fermat's Last Theorem followed from the semistable case of this conjecture.[15]Shimura dryly commented that his first reaction on hearing ofAndrew Wiles's proof of the semistable case was 'I told you so'.[16]
His hobbies wereshogiproblems of extreme length and collectingImari porcelain.The Story of Imari: The Symbols and Mysteries of Antique Japanese Porcelainis a non-fiction work about the Imari porcelain that he collected over 30 years that was published byTen Speed Pressin 2008.[2][17]
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arXiv(pronounced as "archive"—the X represents theGreek letter chi⟨χ⟩)[1]is anopen-access repositoryof electronicpreprintsandpostprints(known ase-prints) approved for posting after moderation, but notpeer reviewed. It consists ofscientific papersin the fields ofmathematics,physics,astronomy,electrical engineering,computer science,quantitative biology,statistics,mathematical finance, andeconomics, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers areself-archivedon the arXiv repository before publication in a peer-reviewed journal. Some publishers also grant permission for authors to archive the peer-reviewedpostprint. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008,[2][3]had hit a million by the end of 2014[4][5]and two million by the end of 2021.[6][7]As of November 2024, the submission rate is about 24,000 articles per month.[8]
arXiv was made possible by the compactTeXfile format, which allowed scientific papers to be easily transmitted over theInternetand renderedclient-side.[11]Around 1990,Joanne Cohnbegan emailingphysicspreprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity.[12]Paul Ginspargrecognized the need for central storage, and in August 1991 he created a centralrepositorymailbox stored at theLos Alamos National Laboratory(LANL) that could be accessed from any computer.[13]Additional modes of access were soon added:FTPin 1991,Gopherin 1992, and theWorld Wide Webin 1993.[5][14]The terme-printwas quickly adopted to describe the articles.
It began as a physics archive, called theLANLpreprint archive, but soon expanded to include astronomy, mathematics, computer science, quantitative biology and, most recently, statistics. Its originaldomain namewas xxx.lanl.gov. Due to LANL's lack of interest in the rapidly expanding technology, in 2001 Ginsparg changed institutions toCornell Universityand changed the name of the repository to arXiv.org.[15]Ginsparg brainstormed the new name with his wife; the domain "archive" was already claimed, so "chi" was replaced with "X" standing in as theGreek letter chiand the "e" dropped for symmetry around the "X".[16]
arXiv was an early adopter and promoter ofpreprints.[17]Its success in sharing preprints was one of the precipitating factors that led to the later movement inscientific publishingknown asopen access.[17]Mathematiciansand scientists regularly upload their papers to arXiv.org for worldwide access[18]and sometimes for reviews before they are published inpeer-reviewedjournals. Ginsparg was awarded aMacArthur Fellowshipin 2002 for his establishment of arXiv.[19]The annual budget for arXiv was approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, theSimons Foundation(in both gift andchallenge grantforms) and annual fee income from member institutions.[20]This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by approximately 220 institutions.[21]
In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in theChronicle of Higher Educationas joking that it "was supposed to be athree-hour tour, not a life sentence".[22]However, Ginsparg remains on the arXiv's Scientific Advisory Board and its Physics Advisory Committee.[23][24]
In January 2022, arXiv began assigningDOIsto articles, in collaboration withDataCite.[25]
Each arXiv paper has a unique identifier:
Different versions of the same paper are specified by a version number at the end. For example,1709.08980v1. If no version number is specified, the default is the latest version.
arXiv uses a category system. Each paper is tagged with one or more categories. Some categories have two layers. For example,q-fin.TRis the "Trading and Market Microstructure" category within "quantitative finance". Other categories have one layer. For example,hep-exis "high energy physics experiments".
Although arXiv is notpeer reviewed, a collection of moderators for each area review thesubmissions; they may recategorize any that are deemed off-topic,[26]or reject submissions that are not scientific papers, or sometimes for undisclosed reasons.[27]The lists of moderators for many sections of arXiv are publicly available,[28]but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines.[29]Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors but to check whether the paper is appropriate for the intended subject area.[26]New authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry.[30][31]
A majority of thee-printsare also submitted tojournalsfor publication, but some work, including some very influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof ofThurston's geometrization conjecture, including thePoincaré conjectureas a particular case, uploaded byGrigori Perelmanin November 2002.[32]Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there [on the arXiv] – let them go and read about it".[33]Despite this non-traditional method of publication, other mathematicians recognized this work by offering theFields MedalandClay Mathematics Millennium Prizesto Perelman, both of which he refused.[34]
While arXiv does contain some dubious e-prints, such as those claiming to refute famous theorems or proving famous conjectures such asFermat's Last Theoremusing only high-school mathematics, a 2002 article which appeared inNotices of the American Mathematical Societydescribed those as "surprisingly rare".[35]arXiv generally re-classifies these works, e.g. in "General mathematics", rather than deleting them;[36]however, some authors have voiced concern over the lack of transparency in the arXiv screening process.[27]
It has been reported that 14,000 preprints have been withdrawn at arXiv, most commonly due to "crucial errors".[37]A lesser number of the withdrawals were due to the preprint being subsumed by another publication. The report itself was posted at arXiv December, 2024.
Papers can be submitted in any of several formats, includingLaTeX, andPDFprinted from aword processorother than TeX or LaTeX. Thesubmissionis rejected by the arXiv software if generating the finalPDFfile fails, if any image file is too large, or if the total size of the submission is too large. arXiv now allows one to store and modify an incomplete submission, and only finalize the submission when ready. The time stamp on the article is set when the submission is finalized.
The standard access route is through the arXiv.org website. Other interfaces and access routes have also been created by other un-associated organisations.
Metadatafor arXiv is made available throughOAI-PMH, the standard foropen access repositories.[38]Content is therefore indexed in all major consumers of such data, such asBASE,COREandUnpaywall. As of 2020, the Unpaywall dump links over 500,000 arxiv URLs as theopen accessversion of a work found inCrossRefdata from the publishers, making arXiv a top 10 global host ofgreen open access.
Finally, researchers can select sub-fields and receive daily e-mailings orRSS feedsof all submissions in them.
Files on arXiv can have a number of different copyright statuses:[39]
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TheOn-Line Encyclopedia of Integer Sequences(OEIS) is an online database ofinteger sequences. It was created and maintained byNeil Sloanewhile researching atAT&T Labs. He transferred theintellectual propertyand hosting of the OEIS to theOEIS Foundationin 2009,[4]and is its chairman.
OEIS records information on integer sequences of interest to both professional andamateurmathematicians, and is widely cited. As of February 2024[ref], it contains over 370,000 sequences,[5]and is growing by approximately 30 entries per day.[6]
Each entry contains the leading terms of the sequence,keywords, mathematical motivations, literature links, and more, including the option to generate agraphor play amusicalrepresentation of the sequence. The database issearchableby keyword, bysubsequence, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input.[7]
Neil Sloanestarted collecting integer sequences as a graduate student in 1964 to support his work incombinatorics.[8][9]The database was at first stored onpunched cards. He published selections from the database in book form twice:
These books were well-received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database reached 16,000 entries Sloane decided to go online – first as anemailservice (August 1994), and soon thereafter as a website (1996). As a spin-off from the database work, Sloane founded theJournal of Integer Sequencesin 1998.[10]The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the omnibus database.[11]In 2004, Sloane celebrated the addition of the 100,000th sequence to the database,A100000, which counts the marks on theIshango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki was created to simplify the collaboration of the OEIS editors and contributors.[12]The 200,000th sequence,A200000, was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list,[13][14]following a proposal by OEIS Editor-in-ChiefCharles Greathouseto choose a special sequence for A200000.[15]A300000 was defined in February 2018, and by end of January 2023 the database contained more than 360,000 sequences.[16][17]
Besides integer sequences, the OEIS also catalogs sequences offractions, the digits oftranscendental numbers,complex numbersand so on by transforming them into integer sequences.
Sequences of fractions are represented by two sequences (named with the keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth-orderFarey sequence,15,14,13,25,12,35,23,34,45{\displaystyle \textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}}, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 (A006842) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (A006843).
Importantirrational numberssuch as π = 3.1415926535897... are catalogued under representative integer sequences such asdecimalexpansions (here 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, ... (A000796)),binaryexpansions (here 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, ... (A004601)), orcontinued fraction expansions(here 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, ... (A001203)).
The OEIS was limited to plainASCIItext until 2011, and it still uses a linear form of conventional mathematical notation (such asf(n) forfunctions,nfor runningvariables, etc.).Greek lettersare usually represented by their full names,e.g., mu for μ, phi for φ.
Every sequence is identified by the letter A followed by six digits, almost always referred to with leading zeros,e.g., A000315 rather than A315.
Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces.
In comments, formulas, etc.,a(n)represents thenth term of the sequence.
Zerois often used to represent non-existent sequence elements. For example,A104157enumerates the "smallestprimeofn2consecutive primes to form ann×nmagic squareof leastmagic constant, or 0 if no such magic square exists." The value ofa(1) (a 1 × 1 magic square) is 2;a(3) is 1480028129. But there is no such 2 × 2 magic square, soa(2) is 0. This special usage has a solid mathematical basis in certain counting functions; for example, thetotientvalence functionNφ(m) (A014197) counts the solutions of φ(x) =m. There are 4 solutions for 4, but no solutions for 14, hencea(14) of A014197 is 0—there are no solutions.
Other values are also used, most commonly −1 (seeA000230orA094076).
The OEIS maintains thelexicographical orderof the sequences, so each sequence has a predecessor and a successor (its "context").[18]OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also thesignof each element. Sequences ofweight distributioncodes often omit periodically recurring zeros.
For example, consider: theprime numbers, thepalindromic primes, theFibonacci sequence, thelazy caterer's sequence, and the coefficients in theseries expansionofζ(n+2)ζ(n){\displaystyle \textstyle {{\zeta (n+2)} \over {\zeta (n)}}}. In OEIS lexicographic order, they are:
whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2.
Very early in the history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. "I resisted adding these sequences for a long time, partly out of a desire to maintain the dignity of the database, and partly because A22 was only known to 11 terms!", Sloane reminisced.[19]One of the earliest self-referential sequences Sloane accepted into the OEIS wasA031135(laterA091967) "a(n) =n-th term of sequence Anor −1 if Anhas fewer thannterms". This sequence spurred progress on finding more terms ofA000022.A100544lists the first term given in sequence An, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead terma(1) of sequence Anmight seem a good alternative if it were not for the fact that some sequences have offsets of 2 and greater.
This line of thought leads to the question "Does sequence Ancontain the numbern?" and the sequencesA053873, "Numbersnsuch that OEIS sequence Ancontainsn", andA053169, "nis in this sequenceif and only ifnis not in sequence An". Thus, thecomposite number2808 is in A053873 becauseA002808is the sequence of composite numbers, while the non-prime 40 is in A053169 because it is not inA000040, the prime numbers. Eachnis a member of exactly one of these two sequences, and in principle it can be determinedwhichsequence eachnbelongs to, with two exceptions (related to the two sequences themselves):
This entry,A046970, was chosen because it comprehensively contains every OEIS field, filled.[20]
In 2009, the OEIS database was used by Philippe Guglielmetti to measure the "importance" of each integer number.[25]The result shown in the plot on the right shows a clear "gap" between two distinct point clouds,[26]the "uninteresting numbers" (blue dots) and the "interesting" numbers that occur comparatively more often in sequences from the OEIS. It contains essentially prime numbers (red), numbers of the forman(green) andhighly composite numbers(yellow). This phenomenon was studied byNicolas Gauvrit,Jean-Paul Delahayeand Hector Zenil who explained the speed of the two clouds in terms of algorithmic complexity and the gap by social factors based on an artificial preference for sequences of primes,evennumbers, geometric and Fibonacci-type sequences and so on.[27]Sloane's gap was featured on aNumberphilevideo in 2013.[28]
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Innumber theory, anarithmetic,arithmetical, ornumber-theoretic function[1][2]is generally anyfunctionwhosedomainis the set ofpositive integersand whose range is asubsetof thecomplex numbers.[3][4][5]Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property ofn".[6]There is a larger class of number-theoretic functions that do not fit this definition, for example, theprime-counting functions. This article provides links to functions of both classes.
An example of an arithmetic function is thedivisor functionwhose value at a positive integernis equal to the number of divisors ofn.
Arithmetic functions are often extremely irregular (seetable), but some of them have series expansions in terms ofRamanujan's sum.
An arithmetic functionais
Two whole numbersmandnare calledcoprimeif theirgreatest common divisoris 1, that is, if there is noprime numberthat divides both of them.
Then an arithmetic functionais
In this article,∑pf(p){\textstyle \sum _{p}f(p)}and∏pf(p){\textstyle \prod _{p}f(p)}mean that the sum or product is over allprime numbers:∑pf(p)=f(2)+f(3)+f(5)+⋯{\displaystyle \sum _{p}f(p)=f(2)+f(3)+f(5)+\cdots }and∏pf(p)=f(2)f(3)f(5)⋯.{\displaystyle \prod _{p}f(p)=f(2)f(3)f(5)\cdots .}Similarly,∑pkf(pk){\textstyle \sum _{p^{k}}f(p^{k})}and∏pkf(pk){\textstyle \prod _{p^{k}}f(p^{k})}mean that the sum or product is over allprime powerswith strictly positive exponent (sok= 0is not included):∑pkf(pk)=∑p∑k>0f(pk)=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+⋯.{\displaystyle \sum _{p^{k}}f(p^{k})=\sum _{p}\sum _{k>0}f(p^{k})=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+\cdots .}
The notations∑d∣nf(d){\textstyle \sum _{d\mid n}f(d)}and∏d∣nf(d){\textstyle \prod _{d\mid n}f(d)}mean that the sum or product is over all positive divisors ofn, including 1 andn. For example, ifn= 12, then∏d∣12f(d)=f(1)f(2)f(3)f(4)f(6)f(12).{\displaystyle \prod _{d\mid 12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).}
The notations can be combined:∑p∣nf(p){\textstyle \sum _{p\mid n}f(p)}and∏p∣nf(p){\textstyle \prod _{p\mid n}f(p)}mean that the sum or product is over all prime divisors ofn. For example, ifn= 18, then∑p∣18f(p)=f(2)+f(3),{\displaystyle \sum _{p\mid 18}f(p)=f(2)+f(3),}and similarly∑pk∣nf(pk){\textstyle \sum _{p^{k}\mid n}f(p^{k})}and∏pk∣nf(pk){\textstyle \prod _{p^{k}\mid n}f(p^{k})}mean that the sum or product is over all prime powers dividingn. For example, ifn= 24, then∏pk∣24f(pk)=f(2)f(3)f(4)f(8).{\displaystyle \prod _{p^{k}\mid 24}f(p^{k})=f(2)f(3)f(4)f(8).}
Thefundamental theorem of arithmeticstates that any positive integerncan be represented uniquely as a product of powers of primes:n=p1a1⋯pkak{\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}}wherep1<p2< ... <pkare primes and theajare positive integers. (1 is given by the empty product.)
It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define thep-adic valuationνp(n)to be the exponent of the highest power of the primepthat dividesn. That is, ifpis one of thepithenνp(n) =ai, otherwise it is zero. Thenn=∏ppνp(n).{\displaystyle n=\prod _{p}p^{\nu _{p}(n)}.}
In terms of the above theprime omega functionsωand Ω are defined by
To avoid repetition, formulas for the functions listed in this article are, whenever possible, given in terms ofnand the correspondingpi,ai,ω, and Ω.
σk(n)is the sum of thekth powers of the positive divisors ofn, including 1 andn, wherekis a complex number.
σ1(n), the sum of the (positive) divisors ofn, is usually denoted byσ(n).
Since a positive number to the zero power is one,σ0(n)is therefore the number of (positive) divisors ofn; it is usually denoted byd(n)orτ(n)(for the GermanTeiler= divisors).
σk(n)=∏i=1ω(n)pi(ai+1)k−1pik−1=∏i=1ω(n)(1+pik+pi2k+⋯+piaik).{\displaystyle \sigma _{k}(n)=\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{(a_{i}+1)k}-1}{p_{i}^{k}-1}}=\prod _{i=1}^{\omega (n)}\left(1+p_{i}^{k}+p_{i}^{2k}+\cdots +p_{i}^{a_{i}k}\right).}
Settingk= 0 in the second product givesτ(n)=d(n)=(1+a1)(1+a2)⋯(1+aω(n)).{\displaystyle \tau (n)=d(n)=(1+a_{1})(1+a_{2})\cdots (1+a_{\omega (n)}).}
φ(n), the Euler totient function, is the number of positive integers not greater thannthat are coprime ton.φ(n)=n∏p∣n(1−1p)=n(p1−1p1)(p2−1p2)⋯(pω(n)−1pω(n)).{\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right)=n\left({\frac {p_{1}-1}{p_{1}}}\right)\left({\frac {p_{2}-1}{p_{2}}}\right)\cdots \left({\frac {p_{\omega (n)}-1}{p_{\omega (n)}}}\right).}
Jk(n), the Jordan totient function, is the number ofk-tuples of positive integers all less than or equal tonthat form a coprime (k+ 1)-tuple together withn. It is a generalization of Euler's totient,φ(n) =J1(n).Jk(n)=nk∏p∣n(1−1pk)=nk(p1k−1p1k)(p2k−1p2k)⋯(pω(n)k−1pω(n)k).{\displaystyle J_{k}(n)=n^{k}\prod _{p\mid n}\left(1-{\frac {1}{p^{k}}}\right)=n^{k}\left({\frac {p_{1}^{k}-1}{p_{1}^{k}}}\right)\left({\frac {p_{2}^{k}-1}{p_{2}^{k}}}\right)\cdots \left({\frac {p_{\omega (n)}^{k}-1}{p_{\omega (n)}^{k}}}\right).}
μ(n), the Möbius function, is important because of theMöbius inversionformula. See§ Dirichlet convolution, below.μ(n)={(−1)ω(n)=(−1)Ω(n)ifω(n)=Ω(n)0ifω(n)≠Ω(n).{\displaystyle \mu (n)={\begin{cases}(-1)^{\omega (n)}=(-1)^{\Omega (n)}&{\text{if }}\;\omega (n)=\Omega (n)\\0&{\text{if }}\;\omega (n)\neq \Omega (n).\end{cases}}}
This implies thatμ(1) = 1. (Because Ω(1) =ω(1) = 0.)
τ(n), the Ramanujan tau function, is defined by itsgenerating functionidentity:∑n≥1τ(n)qn=q∏n≥1(1−qn)24.{\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}(1-q^{n})^{24}.}
Although it is hard to say exactly what "arithmetical property ofn" it "expresses",[7](τ(n) is (2π)−12times thenth Fourier coefficient in theq-expansionof themodular discriminantfunction)[8]it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certainσk(n) andrk(n) functions (because these are also coefficients in the expansion ofmodular forms).
cq(n), Ramanujan's sum, is the sum of thenth powers of the primitiveqthroots of unity:cq(n)=∑gcd(a,q)=11≤a≤qe2πiaqn.{\displaystyle c_{q}(n)=\sum _{\stackrel {1\leq a\leq q}{\gcd(a,q)=1}}e^{2\pi i{\tfrac {a}{q}}n}.}
Even though it is defined as a sum of complex numbers (irrational for most values ofq), it is an integer. For a fixed value ofnit is multiplicative inq:
TheDedekind psi function, used in the theory ofmodular functions, is defined by the formulaψ(n)=n∏p|n(1+1p).{\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right).}
λ(n), the Liouville function, is defined byλ(n)=(−1)Ω(n).{\displaystyle \lambda (n)=(-1)^{\Omega (n)}.}
AllDirichlet charactersχ(n)are completely multiplicative. Two characters have special notations:
Theprincipal character (modn)is denoted byχ0(a) (orχ1(a)). It is defined asχ0(a)={1ifgcd(a,n)=1,0ifgcd(a,n)≠1.{\displaystyle \chi _{0}(a)={\begin{cases}1&{\text{if }}\gcd(a,n)=1,\\0&{\text{if }}\gcd(a,n)\neq 1.\end{cases}}}
Thequadratic character (modn)is denoted by theJacobi symbolfor oddn(it is not defined for evenn):(an)=(ap1)a1(ap2)a2⋯(apω(n))aω(n).{\displaystyle \left({\frac {a}{n}}\right)=\left({\frac {a}{p_{1}}}\right)^{a_{1}}\left({\frac {a}{p_{2}}}\right)^{a_{2}}\cdots \left({\frac {a}{p_{\omega (n)}}}\right)^{a_{\omega (n)}}.}
In this formula(ap){\displaystyle ({\tfrac {a}{p}})}is theLegendre symbol, defined for all integersaand all odd primespby(ap)={0ifa≡0(modp),+1ifa≢0(modp)and for some integerx,a≡x2(modp)−1if there is no suchx.{\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0&{\text{if }}a\equiv 0{\pmod {p}},\\+1&{\text{if }}a\not \equiv 0{\pmod {p}}{\text{ and for some integer }}x,\;a\equiv x^{2}{\pmod {p}}\\-1&{\text{if there is no such }}x.\end{cases}}}
Following the normal convention for the empty product,(a1)=1.{\displaystyle \left({\frac {a}{1}}\right)=1.}
ω(n), defined above as the number of distinct primes dividingn, is additive (seePrime omega function).
Ω(n), defined above as the number of prime factors ofncounted with multiplicities, is completely additive (seePrime omega function).
For a fixed primep,νp(n), defined above as the exponent of the largest power ofpdividingn, is completely additive.
ld(n)=D(n)n=∑pprimep∣nvp(n)p{\displaystyle \operatorname {ld} (n)={\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}}, whereD(n){\displaystyle D(n)}is the arithmetic derivative.
These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of theprime number theorem. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive.
π(x), theprime-counting function, is the number of primes not exceedingx. It is the summation function of thecharacteristic functionof the prime numbers.π(x)=∑p≤x1{\displaystyle \pi (x)=\sum _{p\leq x}1}
A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/kon integers which are thekth power of some prime number, and the value 0 on other integers.Π(x)=∑pk≤x1k.{\displaystyle \Pi (x)=\sum _{p^{k}\leq x}{\frac {1}{k}}.}
ϑ(x) andψ(x), theChebyshev functions, are defined as sums of the natural logarithms of the primes not exceedingx.ϑ(x)=∑p≤xlogp,{\displaystyle \vartheta (x)=\sum _{p\leq x}\log p,}ψ(x)=∑pk≤xlogp.{\displaystyle \psi (x)=\sum _{p^{k}\leq x}\log p.}
The second Chebyshev functionψ(x) is the summation function of the von Mangoldt function just below.
Λ(n), the von Mangoldt function, is 0 unless the argumentnis a prime powerpk, in which case it is the natural logarithm of the primep:Λ(n)={logpifn=2,3,4,5,7,8,9,11,13,16,…=pkis a prime power0ifn=1,6,10,12,14,15,18,20,21,…is not a prime power.{\displaystyle \Lambda (n)={\begin{cases}\log p&{\text{if }}n=2,3,4,5,7,8,9,11,13,16,\ldots =p^{k}{\text{ is a prime power}}\\0&{\text{if }}n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;{\text{ is not a prime power}}.\end{cases}}}
p(n), the partition function, is the number of ways of representingnas a sum of positive integers, where two representations with the same summands in a different order are not counted as being different:p(n)=|{(a1,a2,…ak):0<a1≤a2≤⋯≤ak∧n=a1+a2+⋯+ak}|.{\displaystyle p(n)=\left|\left\{(a_{1},a_{2},\dots a_{k}):0<a_{1}\leq a_{2}\leq \cdots \leq a_{k}\;\land \;n=a_{1}+a_{2}+\cdots +a_{k}\right\}\right|.}
λ(n), the Carmichael function, is the smallest positive number such thataλ(n)≡1(modn){\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}}for allacoprime ton. Equivalently, it is theleast common multipleof the orders of the elements of themultiplicative group of integers modulon.
For powers of odd primes and for 2 and 4,λ(n) is equal to the Euler totient function ofn; for powers of 2 greater than 4 it is equal to one half of the Euler totient function ofn:λ(n)={ϕ(n)ifn=2,3,4,5,7,9,11,13,17,19,23,25,27,…12ϕ(n)ifn=8,16,32,64,…{\displaystyle \lambda (n)={\begin{cases}\;\;\phi (n)&{\text{if }}n=2,3,4,5,7,9,11,13,17,19,23,25,27,\dots \\{\tfrac {1}{2}}\phi (n)&{\text{if }}n=8,16,32,64,\dots \end{cases}}}and for generalnit is the least common multiple ofλof each of the prime power factors ofn:λ(p1a1p2a2…pω(n)aω(n))=lcm[λ(p1a1),λ(p2a2),…,λ(pω(n)aω(n))].{\displaystyle \lambda (p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{\omega (n)}^{a_{\omega (n)}})=\operatorname {lcm} [\lambda (p_{1}^{a_{1}}),\;\lambda (p_{2}^{a_{2}}),\dots ,\lambda (p_{\omega (n)}^{a_{\omega (n)}})].}
h(n), the class number function, is the order of theideal class groupof an algebraic extension of the rationals withdiscriminantn. The notation is ambiguous, as there are in general many extensions with the same discriminant. Seequadratic fieldandcyclotomic fieldfor classical examples.
rk(n)is the number of waysncan be represented as the sum ofksquares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different.rk(n)=|{(a1,a2,…,ak):n=a12+a22+⋯+ak2}|{\displaystyle r_{k}(n)=\left|\left\{(a_{1},a_{2},\dots ,a_{k}):n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\right\}\right|}
Using theHeaviside notationfor the derivative, thearithmetic derivativeD(n) is a function such that
Given an arithmetic functiona(n), itssummation functionA(x) is defined byA(x):=∑n≤xa(n).{\displaystyle A(x):=\sum _{n\leq x}a(n).}Acan be regarded as a function of a real variable. Given a positive integerm,Ais constant alongopen intervalsm<x<m+ 1, and has ajump discontinuityat each integer for whicha(m) ≠ 0.
Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right:A0(m):=12(∑n<ma(n)+∑n≤ma(n))=A(m)−12a(m).{\displaystyle A_{0}(m):={\frac {1}{2}}\left(\sum _{n<m}a(n)+\sum _{n\leq m}a(n)\right)=A(m)-{\frac {1}{2}}a(m).}
Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to findasymptotic behaviourfor the summation function for largex.
A classical example of this phenomenon[9]is given by thedivisor summatory function, the summation function ofd(n), the number of divisors ofn:lim infn→∞d(n)=2{\displaystyle \liminf _{n\to \infty }d(n)=2}lim supn→∞logd(n)loglognlogn=log2{\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)\log \log n}{\log n}}=\log 2}limn→∞d(1)+d(2)+⋯+d(n)log(1)+log(2)+⋯+log(n)=1.{\displaystyle \lim _{n\to \infty }{\frac {d(1)+d(2)+\cdots +d(n)}{\log(1)+\log(2)+\cdots +\log(n)}}=1.}
Anaverage order of an arithmetic functionis some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average". We say thatgis anaverage orderoffif∑n≤xf(n)∼∑n≤xg(n){\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)}
asxtends to infinity. The example above shows thatd(n) has the average order log(n).[10]
Given an arithmetic functiona(n), letFa(s), for complexs, be the function defined by the correspondingDirichlet series(where itconverges):[11]Fa(s):=∑n=1∞a(n)ns.{\displaystyle F_{a}(s):=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}.}Fa(s) is called agenerating functionofa(n). The simplest such series, corresponding to the constant functiona(n) = 1 for alln, isζ(s) theRiemann zeta function.
The generating function of the Möbius function is the inverse of the zeta function:ζ(s)∑n=1∞μ(n)ns=1,ℜs>1.{\displaystyle \zeta (s)\,\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=1,\;\;\Re s>1.}
Consider two arithmetic functionsaandband their respective generating functionsFa(s) andFb(s). The productFa(s)Fb(s) can be computed as follows:Fa(s)Fb(s)=(∑m=1∞a(m)ms)(∑n=1∞b(n)ns).{\displaystyle F_{a}(s)F_{b}(s)=\left(\sum _{m=1}^{\infty }{\frac {a(m)}{m^{s}}}\right)\left(\sum _{n=1}^{\infty }{\frac {b(n)}{n^{s}}}\right).}
It is a straightforward exercise to show that ifc(n) is defined byc(n):=∑ij=na(i)b(j)=∑i∣na(i)b(ni),{\displaystyle c(n):=\sum _{ij=n}a(i)b(j)=\sum _{i\mid n}a(i)b\left({\frac {n}{i}}\right),}thenFc(s)=Fa(s)Fb(s).{\displaystyle F_{c}(s)=F_{a}(s)F_{b}(s).}
This functioncis called theDirichlet convolutionofaandb, and is denoted bya∗b{\displaystyle a*b}.
A particularly important case is convolution with the constant functiona(n) = 1 for alln, corresponding to multiplying the generating function by the zeta function:g(n)=∑d∣nf(d).{\displaystyle g(n)=\sum _{d\mid n}f(d).}
Multiplying by the inverse of the zeta function gives theMöbius inversionformula:f(n)=∑d∣nμ(nd)g(d).{\displaystyle f(n)=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)g(d).}
Iffis multiplicative, then so isg. Iffis completely multiplicative, thengis multiplicative, but may or may not be completely multiplicative.
There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions. The pagedivisor sum identitiescontains many more generalized and related examples of identities involving arithmetic functions.
Here are a few examples:
For allk≥4,rk(n)>0.{\displaystyle k\geq 4,\;\;\;r_{k}(n)>0.}(Lagrange's four-square theorem).
where theKronecker symbolhas the values
There is a formula forr3in the section onclass numbersbelow.r4(n)=8∑4∤dd∣nd=8(2+(−1)n)∑2∤dd∣nd={8σ(n)ifnis odd24σ(n2ν)ifnis even,{\displaystyle r_{4}(n)=8\sum _{\stackrel {d\mid n}{4\,\nmid \,d}}d=8(2+(-1)^{n})\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d={\begin{cases}8\sigma (n)&{\text{if }}n{\text{ is odd }}\\24\sigma \left({\frac {n}{2^{\nu }}}\right)&{\text{if }}n{\text{ is even }}\end{cases}},}whereν=ν2(n).[21][22][23]r6(n)=16∑d∣nχ(nd)d2−4∑d∣nχ(d)d2,{\displaystyle r_{6}(n)=16\sum _{d\mid n}\chi \left({\frac {n}{d}}\right)d^{2}-4\sum _{d\mid n}\chi (d)d^{2},}whereχ(n)=(−4n).{\displaystyle \chi (n)=\left({\frac {-4}{n}}\right).}[24]
Define the functionσk*(n)as[25]σk∗(n)=(−1)n∑d∣n(−1)ddk={∑d∣ndk=σk(n)ifnis odd∑2∣dd∣ndk−∑2∤dd∣ndkifnis even.{\displaystyle \sigma _{k}^{*}(n)=(-1)^{n}\sum _{d\mid n}(-1)^{d}d^{k}={\begin{cases}\sum _{d\mid n}d^{k}=\sigma _{k}(n)&{\text{if }}n{\text{ is odd }}\\\sum _{\stackrel {d\mid n}{2\,\mid \,d}}d^{k}-\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d^{k}&{\text{if }}n{\text{ is even}}.\end{cases}}}
That is, ifnis odd,σk*(n)is the sum of thekth powers of the divisors ofn, that is,σk(n),and ifnis even it is the sum of thekth powers of the even divisors ofnminus the sum of thekth powers of the odd divisors ofn.
Adopt the convention that Ramanujan'sτ(x) = 0ifxisnot an integer.
Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of theproduct of two power series:
The sequencecn=∑i=0naibn−i{\displaystyle c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}}is called theconvolutionor theCauchy productof the sequencesanandbn.These formulas may be proved analytically (seeEisenstein series) or by elementary methods.[28]
Sinceσk(n) (for natural numberk) andτ(n) are integers, the above formulas can be used to prove congruences[35]for the functions. SeeRamanujan tau functionfor some examples.
Extend the domain of the partition function by settingp(0) = 1.
Peter Gustav Lejeune Dirichletdiscovered formulas that relate the class numberhofquadratic number fieldsto the Jacobi symbol.[37]
An integerDis called afundamental discriminantif it is thediscriminantof a quadratic number field. This is equivalent toD≠ 1 and either a)DissquarefreeandD≡ 1 (mod 4) or b)D≡ 0 (mod 4),D/4 is squarefree, andD/4 ≡ 2 or 3 (mod 4).[38]
Extend the Jacobi symbol to accept even numbers in the "denominator" by defining theKronecker symbol:(a2)={0ifais even(−1)a2−18ifais odd.{\displaystyle \left({\frac {a}{2}}\right)={\begin{cases}\;\;\,0&{\text{ if }}a{\text{ is even}}\\(-1)^{\frac {a^{2}-1}{8}}&{\text{ if }}a{\text{ is odd. }}\end{cases}}}
Then ifD< −4 is a fundamental discriminant[39][40]h(D)=1D∑r=1|D|r(Dr)=12−(D2)∑r=1|D|/2(Dr).{\displaystyle {\begin{aligned}h(D)&={\frac {1}{D}}\sum _{r=1}^{|D|}r\left({\frac {D}{r}}\right)\\&={\frac {1}{2-\left({\tfrac {D}{2}}\right)}}\sum _{r=1}^{|D|/2}\left({\frac {D}{r}}\right).\end{aligned}}}
There is also a formula relatingr3andh. Again, letDbe a fundamental discriminant,D< −4. Then[41]r3(|D|)=12(1−(D2))h(D).{\displaystyle r_{3}(|D|)=12\left(1-\left({\frac {D}{2}}\right)\right)h(D).}
LetHn=1+12+13+⋯+1n{\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}be thenthharmonic number. Then
The Riemann hypothesis is also equivalent to the statement that, for alln> 5040,σ(n)<eγnloglogn{\displaystyle \sigma (n)<e^{\gamma }n\log \log n}(where γ is theEuler–Mascheroni constant). This isRobin's theorem.
In 1965P Kesava Menonproved[47]∑gcd(k,n)=11≤k≤ngcd(k−1,n)=φ(n)d(n).{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\gcd(k-1,n)=\varphi (n)d(n).}
This has been generalized by a number of mathematicians. For example,
In fact, iffis any arithmetical function[51][52]∑gcd(k,n)=11≤k≤nf(gcd(k−1,n))=φ(n)∑d∣n(μ∗f)(d)φ(d),{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}f(\gcd(k-1,n))=\varphi (n)\sum _{d\mid n}{\frac {(\mu *f)(d)}{\varphi (d)}},}where∗{\displaystyle *}stands for Dirichlet convolution.
Letmandnbe distinct, odd, and positive. Then the Jacobi symbol satisfies the law ofquadratic reciprocity:(mn)(nm)=(−1)(m−1)(n−1)/4.{\displaystyle \left({\frac {m}{n}}\right)\left({\frac {n}{m}}\right)=(-1)^{(m-1)(n-1)/4}.}
LetD(n) be the arithmetic derivative. Then the logarithmic derivativeD(n)n=∑pprimep∣nvp(n)p.{\displaystyle {\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}.}SeeArithmetic derivativefor details.
Letλ(n) be Liouville's function. Then
Letλ(n) be Carmichael's function. Then
SeeMultiplicative group of integers modulo nandPrimitive root modulo n.
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https://en.wikipedia.org/wiki/Arithmetic_function#Divisor_sum_convolutions
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Arefactorable numberortau numberis an integernthat is divisible by the count of itsdivisors, or to put it algebraically,nis such thatτ(n)∣n{\displaystyle \tau (n)\mid n}withτ(n)=σ0(n)=∏i=1n(ei+1){\displaystyle \tau (n)=\sigma _{0}(n)=\prod _{i=1}^{n}(e_{i}+1)}forn=∏i=1npiei{\displaystyle n=\prod _{i=1}^{n}p_{i}^{e_{i}}}. The first few refactorable numbers are listed in (sequenceA033950in theOEIS) as
For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.
Cooper and Kennedy proved that refactorable numbers havenatural densityzero. Zelinsky proved that no three consecutive integers can all be refactorable.[1]Colton proved that no refactorable number isperfect. The equationgcd(n,x)=τ(n){\displaystyle \gcd(n,x)=\tau (n)}has solutions only ifn{\displaystyle n}is a refactorable number, wheregcd{\displaystyle \gcd }is thegreatest common divisorfunction.
LetT(x){\displaystyle T(x)}be the number of refactorable numbers which are at mostx{\displaystyle x}. The problem of determining an asymptotic forT(x){\displaystyle T(x)}is open. Spiro has proven thatT(x)=xlogx(loglogx)1−o(1){\displaystyle T(x)={\frac {x}{{\sqrt {\log x}}(\log \log x)^{1-o(1)}}}}[2]
There are still unsolved problems regarding refactorable numbers. Colton asked if there are arbitrarily largen{\displaystyle n}such that bothn{\displaystyle n}andn+1{\displaystyle n+1}are refactorable. Zelinsky wondered if there exists a refactorable numbern0≡amodm{\displaystyle n_{0}\equiv a\mod m}, does there necessarily existn>n0{\displaystyle n>n_{0}}such thatn{\displaystyle n}is refactorable andn≡amodm{\displaystyle n\equiv a\mod m}.
First defined byCurtis Cooperand Robert E. Kennedy[3]where they showed that the tau numbers havenatural densityzero, they were later rediscovered bySimon Coltonusing a computer program he wrote ("HR") which invents and judges definitions from a variety of areas of mathematics such asnumber theoryandgraph theory.[4]Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.
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https://en.wikipedia.org/wiki/Refactorable_number
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The tables below list all of thedivisorsof the numbers 1 to 1000.
A divisor of anintegernis an integerm, for whichn/mis again an integer (which is necessarily also a divisor ofn). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21).
Ifmis a divisor ofn, then so is −m. The tables below only list positive divisors.
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https://en.wikipedia.org/wiki/Table_of_divisors
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Accelerandois a 2005 science fiction novel consisting of a series of interconnected short stories written by British authorCharles Stross. As well as normal hardback and paperback editions, it was released as a freee-bookunder theCC BY-NC-ND license.Accelerandowon theLocus Awardin 2006,[1]and was nominated for several other awards in 2005 and 2006, including theHugo,Campbell,Clarke, andBritish Science Fiction Association Awards.[1][2]
In Italian,accelerandomeans "speeding up" and is used as atempo markinginmusical notation. In Stross' novel, it refers to theaccelerating rateat which humanity in general, and/or the novel's characters, head towards thetechnological singularity.
The book is a collection of nine short stories telling the tale of threegenerationsof a family before, during, and after atechnological singularity. It was originally written as a series ofnovelettesand novellas, all published inAsimov's Science Fictionmagazine in the period 2001 to 2004. According to Stross, the initial inspiration for the stories was his experience working as a programmer for a high-growth company during thedot-com boomof the 1990s.[3]
The first three stories follow the character ofagalmic"venturealtruist" Manfred Macx, starting in the early 21st century; the next three stories follow his daughter Amber; and the final three focus largely on Amber's son Sirhan in the completely transformed world at the end of the century.
Stross describes humanity's situation inAccelerandoas dire:
In the background of what looks like a Panglossian techno-optimist novel, horrible things are happening. Most of humanity is wiped out, then arbitrarily resurrected in mutilated form by the Vile Offspring. Capitalism eatseverythingthen the logic of competition pushes it so far that merely human entities can no longer compete; we're a fat, slow-moving, tasty resource – like the dodo. Our narrative perspective, Aineko, isnota talking cat: it's a vastly superintelligent AI, coolly calculating, that has worked out that human beings are more easily manipulated if they think they're dealing with a furry toy. The cat body is a sock puppet wielded by an abusive monster.[4]
As events progress inAccelerando, the planets of theSolar Systemare dismantled over time to form aMatrioshka brain, a vast solar-powered computational device inhabited by minds inconceivably more advanced and complex than naturally evolved intelligences such as human beings. This proves to be a normal stage in the life-cycle of an inhabited solar system; the galaxies are revealed to be filled with such Matrioshka brains. Intelligent consciousnesses outside of Matrioshka brains may communicate viawormholenetworks.
The notion that the universe is dominated by a communications network ofsuperintelligencesbears comparison withOlaf Stapledon's 1937 science-fiction novelStar Maker, although Stapledon's advanced civilisations are said to communicate psychically rather than informatically.
In the following table, the chapter number (#), chapter name and original magazine date of publication, and a brief synopsis are given. The nine stories are grouped into three parts.
The novel contains numerous allusions to real-world scientific concepts and individuals, including:
Accelerandowon the 2006Locus Awardfor Best Science Fiction Novel,[1]and the 2010 Estonian SF Award for Best Translated Novel of 2009.[12]Additionally, the novel was shortlisted for several other awards, including:
The original short story "Lobsters" (June 2001) was shortlisted for:
The original short story "Halo" (June 2002) was shortlisted for:
The original short story "Router" (September 2002) was shortlisted for:
The original short story "Nightfall" (April 2003) was shortlisted for:
The original short story "Elector" (September 2004) was shortlisted for:
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https://en.wikipedia.org/wiki/Accelerando
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Accelerationismis a range ofrevolutionaryandreactionaryideologies that call for the drastic intensification ofcapitalistgrowth,technological change, and other processes of social change to destabilize existing systems and create radical social transformations, referred to as "acceleration".[1][2][3][4][5]It has been regarded as an ideological spectrum divided into mutually contradictoryleft-wingandright-wingvariants, both of which support dramatic changes to capitalism and its structures as well as the conditions for atechnological singularity, a hypothetical point in time at which technological growth becomes uncontrollable and irreversible.[6][7][8][9]It aims to analyze and subsequently promote the social, economic, cultural, andlibidinalforces that constitute the process of acceleration.[10][6]
Ideas such asGilles DeleuzeandFélix Guattari's concept ofdeterritorialization,Jean Baudrillard's proposals for "fatal strategies", and various ideas ofNick Landare crucial influences on accelerationism. Such ideas gave rise to theCybernetic Culture Research Unit(CCRU), a philosophy collective at theUniversity of Warwick, in the 1990s, promoting the use of capitalism to dissolve existing social structures and reach a singularity. In the late 2000s and early 2010s, the movement would gain a resurgence, producing numerous variants and interpretations as well as a few published works.
The term has also, in a manner strongly distinguished from original accelerationist theorists, been used byright-wing extremistssuch asneo-fascists,neo-Nazis,white nationalistsandwhite supremaciststo increasingly refer to an "acceleration" of racial conflict throughassassinations,murdersandterrorist attacksas a means to violently achieve awhite ethnostate.[11][12][13][14]
The term "accelerationism" was first used in sci-fi authorRoger Zelazny's third novel, 1967'sLord of Light.[1][15]It was later popularized by professor and authorBenjamin Noysin his 2010 bookThe Persistence of the Negativeto describe the trajectory of certainpost-structuralistswho embraced unorthodoxMarxistand counter-Marxist overviews of capitalist growth, such as Gilles Deleuze and Félix Guattari in their 1972 bookAnti-Oedipus,Jean-François Lyotardin his 1974 bookLibidinal Economyand Jean Baudrillard in his 1976 bookSymbolic Exchange and Death.[16]
English right-wing philosopher and writer Nick Land, commonly credited with creating and inspiring accelerationism's basic ideas and concepts,[1][17]cited a number of philosophers who expressed anticipatory accelerationist attitudes in his 2017 essay "A Quick-and-Dirty Introduction to Accelerationism".[18][19]Firstly,Friedrich Nietzscheargued in a fragment inThe Will to Powerthat "theleveling processof European man is the great process which should not be checked: one should even accelerate it."[20]Taking inspiration from this notion forAnti-Oedipus, Deleuze and Guattari speculated further on an unprecedented "revolutionary path" to perpetuate capitalism's tendencies that would later become a central idea of accelerationism:
But which is the revolutionary path? Is there one?—To withdraw from the world market, asSamir Aminadvises Third World countries to do, in a curious revival of the fascist "economic solution"? Or might it be to go in the opposite direction? To go still further, that is, in the movement of the market, of decoding and deterritorialization? For perhaps the flows are not yet deterritorialized enough, not decoded enough, from the viewpoint of a theory and a practice of a highly schizophrenic character. Not to withdraw from the process, but to go further, to "accelerate the process," as Nietzsche put it: in this matter, the truth is that we haven't seen anything yet.
Land also citedKarl Marx, who, in his 1848 speech "On the Question of Free Trade", anticipated accelerationist principles a century before Deleuze and Guattari by describingfree tradeas socially destructive and fuellingclass conflict, then effectively arguing for it:
But, in general, the protective system of our day is conservative, while the free trade system is destructive. It breaks up old nationalities and pushes the antagonism of the proletariat and the bourgeoisie to the extreme point. In a word, the free trade system hastens the social revolution. It is in this revolutionary sense alone, gentlemen, that I vote in favor of free trade.
Nick Srnicekand Alex Williams, prominent left accelerationists, additionally creditVladimir Leninwith recognizing capitalist progress as important in the subsequent functioning ofsocialism:[7][23]
Socialism is inconceivable without large-scale capitalist engineering based on the latest discoveries of modern science. It is inconceivable without planned state organisation which keeps tens of millions of people to the strictest observance of a unified standard in production and distribution. We Marxists have always spoken of this, and it is not worth while wasting two seconds talking to people who do not understand even this (anarchistsand a good half of theLeft Socialist-Revolutionaries).
Robin Mackay, co-editor of#Accelerate: The Accelerationist Readerand a former CCRU member, additionally citesRussian cosmism,science fiction(particularlyTerminator,Predator, andBlade Runner),cyberpunk, 90's cyberculture, andelectronic musicas influences on the movement.[6]Iain Hamilton Grant, another former CCRU member, stated "Neuromancergot into the philosophy department, and it went viral. You’d find worn-out paperbacks all over the common room.”[1]
TheCybernetic Culture Research Unit(CCRU), a philosophy collective at theUniversity of Warwickwhich included Land, Mackay, and Grant, was one of the most significant parts of the movement.[1][24][6][25]Mark Fisher, another former member, described the CCRU's accelerationism as “a kind of exuberant anti-politics, a ‘technihilo' celebration of the irrelevance of human agency, partly inspired by the pro-markets, anti-capitalism line developed byManuel DeLandaout ofBraudel, and from the section of Anti-Oedipus that talks about marketization as the 'revolutionary path'."[26]Other significant members includeSadie PlantandRay Brassier. The group stood in stark opposition to the University of Warwick and traditional left-wing academia,[1][26]with Mackay stating "I don’t think Land has ever pretended to be left-wing! He’s a serious philosopher and an intelligent thinker, but one who has always loved to bait the left by presenting the ‘worst’ possible scenario with great delight…!"[6]As Land became a stronger influence on the group and left the University of Warwick, they would shift to more unorthodox andoccultideas. Land suffered abreakdownfrom hisamphetamine abuseand disappeared in the early 2000s, with the CCRU vanishing along with him.[1]
The Guardianhas referred to#Accelerate: The Accelerationist Reader,a 2014 anthology edited by Robin Mackay andArmen Avanessian, as "the only proper guide to the movement in existence." They also describedFanged Noumena,a 2011 anthology of Land's work, as “contain[ing] some of accelerationism's most darkly fascinating passages."[1]In 2015, Urbanomic and Time Spiral Press publishedWritings 1997-2003as a complete collection of known texts published under the CCRU name, besides those that have been irrecoverably lost or attributed to a specific member. However, it is not actually complete, as some known works under the CCRU name are not included, such as those in#Accelerate: The Accelerationist Reader.[27][28]
In "A Quick-and-Dirty Introduction to Accelerationism", Land attributed the increasing speed of the modern world, along with the associated decrease in time available to think and make decisions about its events, to unregulated capitalism and its ability to exponentially grow and self-improve, describing capitalism as "a positive feedback circuit, within whichcommercializationandindustrializationmutually excite each other in a runaway process." He argued that the best way to deal with capitalism is to participate more to foster even greater exponential growth and self-improvement viacreative destruction, accelerating technological progress along with it. Land also argued that such acceleration is intrinsic to capitalism but impossible for non-capitalist systems, stating that "capital revolutionizes itself more thoroughly than any extrinsic 'revolution' possibly could."[19]In an interview withVox, he stated "Modernity has Capitalism (the self-escalating techno-commercial complex) as its motor. Our question was what ‘the process’ wants (i.e. spontaneously promotes) and what resistances it provokes." He also said that “the assumption” behind accelerationism was that “the general direction of [techno-capitalist] self-escalating change was towarddecentralization.”[25]Mackay summarized Land's position as "since capitalism tends to dissolve hereditary social forms and restrictions [...], it is seen as the engine of exploration into the unknown. So to be ‘on the side of intelligence’ is to totally abandon all caution with respect to the disintegrative processes of capital and whatever reprocessing of the human and of the planet they might involve."[6]This view has been referred to as "right-accelerationism".[6][19]
Vincent Le considers Land's philosophy to opposeanthropocentrism, citing his early critique oftranscendental idealismand capitalism in "Kant, Capital, and the Prohibition of Incest". According to Le, Land opposes philosophies which deny a reality beyond humans' conceptual experience, instead viewingdeathas a way to graspthe Real. This would remain as Land's views on capitalism changed after readingDeleuze and Guattari, with Le stating "Although the mature Land abandons his left-wing critique of capitalism by immersing himself in the study ofcybernetics, he will never shake his contempt for anthropocentrism, and his remedy that philosophers can only access the true at the edge of our humanity.[29]
In “Meltdown”, a CCRU work and one of the writings compiled inFanged Noumena, Land envisioned a technocapital singularity inChina, resulting in revolutions inartificial intelligence,human enhancement,biotechnology, andnanotechnology. This upends the previous status quo, and the formerfirst world countriesstruggle to maintain control and stop the singularity, verging oncollapse. He described newanti-authoritarianmovements performing a bottom-up takeover of institutions through means likebiological warfareenhanced withDNA computing. He claimed that capitalism's tendency towards optimization of itself and technology, in service ofconsumerism, will lead to the enhancement and eventuallyreplacement of humanity with technology, asserting that "nothing human makes it out of the near-future." Eventually, the self-development of technology will culminate in the "melting [of]Terrainto a seething K-pulp (which unlikegrey goosynthesizesmicrobial intelligenceas it proliferates)." He also criticized traditional philosophy as tending towardsdespotism, instead praisingDeleuzoguattarianschizoanalysisas "already engaging with nonlinear nano-engineering runaway in 1972."[30][31]Le states that Land embraces human extinction in the singularity, as the resulting hyperintelligent AI will come to fully comprehend and embody the Real of thebody without organs, free of human distortions of reality.[29]
Land has continually praisedChina's economic policyas being accelerationist, moving toShanghaiand working as a journalist writing material that has been characterized as pro-government propaganda.[1][30][31][25]He has also spoken highly ofDeng XiaopingandSingapore'sLee Kuan Yew,[25]calling Lee an "autocratic enabler of freedom."[32]Yuk Huistated "Land’s celebration of Asian cities such as Shanghai,Hong Kong, and Singapore is simply a detached observation of these places that projects onto them a common will to sacrifice politics for productivity."[33]
Land's involvement in theneoreactionarymovement has contributed to his views on accelerationism. InThe Dark Enlightenment, he advocates for a form of capitalistmonarchism, with states controlled by aCEO. He viewsdemocraticandegalitarianpolicies as only slowing down acceleration and the technocapital singularity, stating "Beside thespeed machine, or industrial capitalism, there is an ever more perfectly weighted decelerator [...] comically, the fabrication of this braking mechanism is proclaimed asprogress. It is the Great Work of the Left.”[25][34]He has advocated for accelerationists to support the neoreactionary movement, though many have distanced themselves from him in response to his views on race.[1]
Left-wing accelerationism (also referred to as left-accelerationism or L/Acc) is often attributed to Mark Fisher.[35]Left-wing accelerationism seeks to explore, in an orthodox and conventional manner, how modern society has the momentum to create futures that are equitable and liberatory.[36][failed verification]While both strands of accelerationist thinking remain rooted in a similar range of thinkers, left accelerationism appeared with the intent to use technology for the goal of achieving an egalitarian future.[35][34]Fisher, writing on his blogk-punk, had become increasingly disillusioned with capitalism as an accelerationist,[1]citing working in the public sector inBlairiteBritain, being a teacher and trade union activist, and an encounter with Slovenian philosopherSlavoj Žižek, whom he considered to be using similar concepts to the CCRU but from a leftist perspective.[26]At the same time, he became frustrated with traditional left wing politics, believing they were ignoring technology that they could exploit.[1]
In "Terminator vs Avatar", Fisher claimed that while Marxists criticizedLibidinal Economyfor asserting that workers enjoyed the upending of primitive social orders, nobody truly wants to return to those. Therefore, rather than reverting to pre-capitalism, society must move through and beyond capitalism. Fisher praised Land's attacks on the academic left, describing the academic left as "careerist sandbaggers" and "a ruthless protection ofpetit bourgeoisinterests dressed up as politics." He also critiqued Land's interpretation of Deleuze and Guattari, stating that while superior in many ways, "his deviation from their understanding of capitalism is fatal" in assuming noreterritorialization, resulting in not foreseeing that capitalism provides "a simulation of innovation and newness that cloaks inertia and stasis." CitingFredric Jameson's interpretation ofThe Communist Manifestoas "see[ing] capitalism as the most productive moment of history and the most destructive at the same time", he argued for accelerationism as an anti-capitalist strategy, criticizing the left's moral critique of capitalism and their "tendencies towards Canutism" as only helping thenarrative that capitalism is the only viable system.[37]
Nick Srnicek befriended Fisher, sharing similar views, and the2008 financial crisis, along with dissatisfaction with the left's "ineffectual" response ofthe Occupy protests, led to Srnicek co-writing "#Accelerate: Manifesto for an Accelerationist Politics" with Alex Williams in 2013.[1][23]They posited that capitalism was the most advanced economic system of its time, but has since stagnated and is now constraining technology, withneoliberalismonly worsening its crises. At the same time, they considered the modern left to be "unable to devise a new political ideological vision" as they are too focused on localism and direct action and cannot adapt to make meaningful change. They advocated using existing capitalist infrastructure as "a springboard to launch towards post-capitalism", taking advantage of capitalist technological and scientific advances to experiment with things like economic modeling in the style ofProject Cybersyn. They also advocated for "collectively controlled legitimate vertical authority in addition to distributed horizontal forms of sociality" and attaining resources and funding for political infrastructure, contrasting standard leftist political action which they deem ineffective. Moving past the constraints of capitalism would result in a resumption of technological progress, not only creating a more rational society but also "recovering the dreams which transfixed many from the middle of the Nineteenth Century until the dawn of the neoliberal era, of the quest of Homo Sapiens towards expansion beyond the limitations of the earth and our immediate bodily forms."[1][7][23]They expanded further inInventing the Future, which, while dropping the term "accelerationism", pushed forautomation, reduction and distribution of working hours,universal basic income, and diminishment of work ethic.[1][38]
Land rebuked its ideas in a 2017 interview withThe Guardian, stating "the notion that self-propelling technology is separable from capitalism is a deep theoretical error."[1]
Effective accelerationism (abbreviated to e/acc) takes influence fromeffective altruism, a movement to maximize good by calculating what actions provide the greatest overall/global good and prioritizing those rather than focusing on personal interest/proximity. Proponents advocate for unrestricted technological progress "at all costs", believing thatartificial general intelligencewill solve universal human problems like poverty, war, and climate change, while deceleration and stagnation of technology is agreater riskthan anyposed by AI. For example,James Brusseauadvocates reconfiguring AI ethics to promote acceleration, arguing that problems caused by AI innovation are to be resolved by still more innovation as opposed to limiting or slowing the technology.[39]This contrasts with effective altruism (referred to as "longtermism" to distinguish from e/acc), which tends to consider uncontrolled AI to be the greater existential risk and advocates for government regulation and carefulalignment.[40][41]
In a critique, Italian MarxistFranco Berardiconsidered acceleration “the essential feature of capitalist growth” and characterized accelerationism as "point[ing] out the contradictory implications of the process of intensification, emphasizing in particular the instability that acceleration brings into the capitalist system." However, he also stated “my answer to the question of whether acceleration marks a final collapse of power is quite simply: no. Because the power of capital is not based on stability.” He posited that the “accelerationist hypothesis” is based on two assumptions: that accelerating production cycles make capitalism unstable, and that potentialities within capitalism will necessarily deploy themselves. He criticized the first by stating “capitalism is resilient because it does not need rational government, only automatic governance”; and the second by arguing that while the possibility exists, it is not guaranteed to happen as it can still be slowed or stopped.[42]
Benjamin Noys is a staunch critic of accelerationism, initially calling it "DeleuzianThatcherism".[34]He accuses it of offering false solutions to technological and economic problems, considering those solutions “always promised and always just out of reach."[1][43]He has also said "Capitalism, for the accelerationist, bears down on us as accelerative liquid monstrosity, capable of absorbing us and, for Land, we must welcome this."[34][43]
InThe Question Concerning Technology in China,Yuk Huicritiqued accelerationism, particularlyRay Brassier’s “Prometheanism and its Critics” from#Accelerate: The Accelerationist Reader, stating “if such a response to technology and capitalism is applied globally, [...] it risks perpetuating a more subtle form of colonialism.” He argues that accelerationism tries to universally apply a western conception of technology based onPrometheusdespite other cultures having different myths and relations to technology.[44]Further critiquingWesternization,globalization, and the loss of non-Western technological thought, he has also referred to Deng Xiaoping as "the world's greatest accelerationist" due to hiseconomic reforms, considering them an acceleration of the modernization process which started in the aftermath of theOpium Warsand intensified with theCultural Revolution.[33]In "A Politics of Intensity: Some Aspects of Acceleration in Simondon and Deleuze", Yuk Hui and Louis Morelle analyzed Deleuze andSimondonfrom an accelerationist perspective.[45]
Slavoj Žižek considers accelerationism to be “far too optimistic”, critiquing it as retroactivelydeterministicand contrasting it withFreud’sdeath driveand its lack of a final conclusion. He argues that accelerationism considers just one conclusion of the world’s tendencies and fails to find other “coordinates" of the world order.[46]
Benjamin H. Bratton's bookThe Stack: On Software and Sovereigntyhas been described as concerning accelerationist ideas, focusing on how information technology infrastructures undermine modern political geographies and proposing an open-ended "design brief".Tiziana Terranova's "Red Stack Attack!" links Bratton's stack model and left-wing accelerationism.[47]
Laboria Cuboniks, a feminist group, advocated for the use of technology forgender abolitionin "Xenofeminism: A Politics for Alienation", which has been described as "regrounding left accelerationism in itscyberfeministantecedents."[48]Aria Dean, proposing an alternative to both right and left accelerationism, synthesizedracial capitalismwith accelerationism in "Notes on Blacceleration", arguing that the binary between humans and capital is already blurred by the scars of theAtlantic slave trade.[49]
Since "accelerationism" was coined in 2010, the term has taken on several new meanings. Several commentators have used the labelaccelerationistto describe a controversial political strategy articulated by Slavoj Žižek.[50][51]An often-cited example of this is Žižek's assertion in a November 2016 interview withChannel 4 Newsthat were he an American citizen, he would vote for U.S. presidentDonald Trumpas the candidate more likely to disrupt the politicalstatus quoin that country.[52]Steven Shavirodescribed variants that “embrace the idea that the worse things get, the better the prospect for a revolution to overthrow everything”, though he considers it very rare.[53]Mackay noted a misconception that accelerationism involves a Marxist "acceleration ofcontradictions" within capitalism and stated that no accelerationist authors have advocated such a thing.[6]Chinese dissidentshave referred toXi Jinpingas "Accelerator-in-Chief" (referencing state media calling Deng Xiaoping "Architect-in-Chief of Reform and Opening"), believing that Xi'sauthoritarianismis hastening the demise of theChinese Communist Partyand that, because it is beyond saving, they should allow it to destroy itself in order to create a better future.[54]
Despite its originally Marxist philosophical and theoretical interests, since the late 2010s, international networks of neo-fascists, neo-Nazis, White nationalists, and White supremacists have increasingly used the term "accelerationism" to refer to right-wing extremist goals, and have been known to refer to an "acceleration" of racial conflict through violent means such as assassinations, murders, terrorist attacks and eventual societal collapse to achieve the building of a White ethnostate.[12][13][14]Far-right accelerationism has been widely considered as detrimental to public safety.[55]The inspiration for this distinct variation is occasionally cited asAmerican Nazi PartyandNational Socialist Liberation FrontmemberJames Mason's newsletterSiege, where he argued forsabotage,mass killings, and assassinations of high-profile targets to destabilize and destroy the current society, seen as a system upholding aJewishandmulticulturalNew World Order.[12]His works were republished and popularized by theIron MarchforumandAtomwaffen Division, right-wing extremist organizations strongly connected to various terrorist attacks, murders, andassaults.[12][56][57][58]Far-right accelerationists have also been known to attackcritical infrastructure, particularly thepower grid, attempting to cause a collapse of the system orbelieving that 5G was causing COVID-19, with some encouragingpromotion of 5G conspiracy theoriesas easier than convincing potential recruits thatthe Holocaust never happened.[59][60]According to theSouthern Poverty Law Center(SPLC), which tracks hate groups and filesclass action lawsuitsagainst discriminatory organizations and entities, "on the case of white supremacists, the accelerationist set sees modern society as irredeemable and believe it should be pushed to collapse so a fascist society built on ethnonationalism can take its place. What defines white supremacist accelerationists is their belief that violence is the only way to pursue their political goals."[58]
Brenton Harrison Tarrant, the perpetrator of the 15 March 2019Christchurch mosque shootingsthat killed 51 people and injured 49 others, strongly encouraged right-wing accelerationism in a section of his manifesto titled "Destabilization and Accelerationism: Tactics". Tarrant's manifesto influencedJohn Timothy Earnest, the perpetrator of both the 24 March 2019Escondido mosque fireat Dar-ul-Arqam Mosque inEscondido, California, and the 27 April 2019Poway synagogue shootingwhich resulted in one dead and three injured; and it also influencedPatrick Crusius, the perpetrator of the 3 August 2019El Paso Walmart shootingthat killed 23 people and injured 23 others. Tarrant and Earnest, in turn, influenced Juraj Krajčík, the perpetrator of the2022 Bratislava shootingthat left dead two patrons of a gay bar.[61][12][25]Sich Battalionurged its members to buy a copy of Tarrant's manifesto, encouraging them to "get inspired" by it.[62]
Voxpointed to Land's shift towards neoreactionarism, along with the neoreactionary movement crossing paths with the alt-right as another fringe right wing internet movement, as the likely connection point between far-right racial accelerationism and the term for Land's otherwise unrelated technocapitalist ideas. They cited a 2018 Southern Poverty Law Center investigation which found users on the neo-Nazi blogThe Right Stuffwho cited neoreactionarism as an influence.[25]Land himself became interested in theAtomwaffen-affiliatedtheistic SatanistorganizationOrder of Nine Angles(ONA) which adheres to the ideology of Neo-Nazi terrorist accelerationism, describing the ONA's works as "highly-recommended" in a blog post.[63]Since the 2010s, the political ideology and religious worldview of the Order of Nine Angles, founded by theBritish neo-NazileaderDavid Myattin 1974,[12]have increasingly influencedmilitantneo-fascist and neo-Naziinsurgentgroups associated with right-wing extremist and White supremacist international networks,[12]most notably the Iron March forum.[12]
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https://en.wikipedia.org/wiki/Accelerationism
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Empiricalmethods
Prescriptiveand policy
Ineconomics,diminishing returnsmeans the decrease inmarginal(incremental) output of aproductionprocess as the amount of a singlefactor of productionis incrementally increased, holding all other factors of production equal (ceteris paribus).[1]The law of diminishing returns (also known as the law of diminishing marginal productivity) states that in a productive process, if a factor of production continues to increase, while holding all other production factors constant, at some point a further incremental unit of input will return a lower amount of output.[2][3]The law of diminishing returns does not imply a decrease in overall production capabilities; rather, it defines a point on a production curve at which producing an additional unit of output will result in a lower profit. Under diminishing returns, output remains positive, butproductivityandefficiencydecrease.
The modern understanding of the law adds the dimension of holding other outputs equal, since a given process is understood to be able to produce co-products.[4]An example would be a factory increasing its saleable product, but also increasing its CO2production, for the same input increase.[2]The law of diminishing returns is a fundamental principle of both micro and macro economics and it plays a central role inproduction theory.[5]
The concept of diminishing returns can be explained by considering other theories such as the concept ofexponential growth.[6]It is commonly understood that growth will not continue to rise exponentially, rather it is subject to different forms of constraints such as limited availability of resources and capitalisation which can causeeconomic stagnation.[7]This example of production holds true to this common understanding as production is subject to the fourfactors of productionwhich are land, labour, capital and enterprise.[8]These factors have the ability to influenceeconomic growthand can eventually limit or inhibit continuous exponential growth.[9]Therefore, as a result of these constraints the production process will eventually reach a point of maximum yield on the production curve and this is where marginal output will stagnate and move towards zero.[10]Innovation in the form of technological advances or managerial progress can minimise or eliminate diminishing returns to restore productivity and efficiency and to generate profit.[11]
This idea can be understood outside of economics theory, for example, population. The population size on Earth is growing rapidly, but this will not continue forever (exponentially). Constraints such as resources will see the population growth stagnate at some point and begin to decline.[6]Similarly, it will begin to decline towards zero but not actually become a negative value, the same idea as in the diminishing rate of return inevitable to the production process.
The concept of diminishing returns can be traced back to the concerns of early economists such asJohann Heinrich von Thünen,Jacques Turgot,Adam Smith,[12]James Steuart,Thomas Robert Malthus, and[13]David Ricardo. The law of diminishing returns can be traced back to the 18th century, in the work of Jacques Turgot. He argued that "each increase [in an input] would be less and less productive."[14]In 1815, David Ricardo, Thomas Malthus,Edward West, andRobert Torrensapplied the concept of diminishing returns to land rent. These works were relevant to the committees of Parliament in England, who were investigating why grain prices were so high, and how to reduce them. The four economists concluded that the prices of the products had risen due to theNapoleonic Wars, which affected international trade and caused farmers to move to lands which were undeveloped and further away. In addition, at the end of the Napoleonic Wars, grain imports were restored which caused a decline in prices because the farmers needed to attract customers and sell their products faster.[15]
Classical economists such as Malthus and Ricardo attributed the successive diminishment of output to the decreasing quality of the inputs whereasNeoclassical economistsassume that each "unit" of labor is identical. Diminishing returns are due to the disruption of the entire production process as additional units of labor are added to a fixed amount of capital. The law of diminishing returns remains an important consideration in areas of production such as farming and agriculture.
Proposed on the cusp of theFirst Industrial Revolution, it was motivated with single outputs in mind. In recent years, economists since the 1970s have sought to redefine the theory to make it more appropriate and relevant in modern economic societies.[4]Specifically, it looks at what assumptions can be made regarding number of inputs, quality, substitution and complementary products, and output co-production, quantity and quality.
The origin of the law of diminishing returns was developed primarily within the agricultural industry. In the early 19th century, David Ricardo as well as other English economists previously mentioned, adopted this law as the result of the lived experience in England after the war. It was developed by observing the relationship between prices of wheat and corn and the quality of the land which yielded the harvests.[16]The observation was that at a certain point, that the quality of the land kept increasing, but so did the cost of produce etc. Therefore, each additional unit of labour on agricultural fields, actually provided a diminishing or marginally decreasing return.[17]
A common example of diminishing returns is choosing to hire more people on a factory floor to alter current manufacturing and production capabilities. Given that the capital on the floor (e.g. manufacturing machines, pre-existing technology, warehouses) is held constant, increasing from one employee to two employees is, theoretically, going to more than double production possibilities and this is calledincreasing returns.
If 50 people are employed, at some point, increasing the number of employees by two percent (from 50 to 51 employees) would increase output by two percent and this is calledconstant returns.
Further along the production curve at, for example 100 employees, floor space is likely getting crowded, there are too many people operating the machines and in the building, and workers are getting in each other's way. Increasing the number of employees by two percent (from 100 to 102 employees) would increase output by less than two percent and this is called "diminishing returns".
At some (maybe much later) point (perhaps with 200 employees), each additional employee will actually decrease production. This is called "negative returns".[18]
Through each of these examples, the floor space and capital of the factor remained constant, i.e., these inputs were held constant. By only increasing the number of people, eventually the productivity and efficiency of the process moved from increasing returns to diminishing returns.
To understand this concept thoroughly, acknowledge the importance ofmarginal outputormarginal returns. Returns eventually diminish because economists measure productivity with regard to additional units (marginal). Additional inputs significantly impact efficiency or returns more in the initial stages.[19]The point in the process before returns begin to diminish is considered the optimal level. Being able to recognize this point is beneficial, as other variables in the production function can be altered rather than continually increasing labor.
Further, examine something such as theHuman Development Index, which would presumably continue to rise so long asGDP per capita(in purchasing power parity terms) was increasing. This would be a rational assumption because GDP per capita is a function of HDI. Even GDP per capita will reach a point where it has a diminishing rate of return on HDI.[20]Just think, in a low income family, an average increase of income will likely make a huge impact on the wellbeing of the family. Parents could provide abundantly more food and healthcare essentials for their family. That is a significantly increasing rate of return. But, if you gave the same increase to a wealthy family, the impact it would have on their life would be minor. Therefore, the rate of return provided by that average increase in income is diminishing.
SignifyOutput=O,Input=I,O=f(I){\displaystyle Output=O\ ,\ Input=I\ ,\ O=f(I)}
Increasing Returns:2⋅f(I)<f(2⋅I){\displaystyle 2\cdot f(I)<f(2\cdot I)}
Constant Returns:2⋅f(I)=f(2⋅I){\displaystyle 2\cdot f(I)=f(2\cdot I)}
Diminishing Returns:2⋅f(I)>f(2⋅I){\displaystyle 2\cdot f(I)>f(2\cdot I)}
There is a widely recognised production function in economics:Q= f(NR, L, K, t, E):
Start from the equation for the marginal product:ΔOutΔIn1=f(In2,In1+ΔIn1)−f(In1,In2)ΔIn1{\displaystyle {\Delta Out \over \Delta In_{1}}={{f(In_{2},In_{1}+\Delta In_{1})-f(In_{1},In_{2})} \over \Delta In_{1}}}
To demonstrate diminishing returns, two conditions are satisfied; marginal product is positive, and marginal product is decreasing.
Elasticity, a function of input and output,ϵ=InOut⋅δOutδIn{\displaystyle \epsilon ={In \over Out}\cdot {\delta Out \over \delta In}}, can be taken for small input changes. If the above two conditions are satisfied, then0<ϵ<1{\displaystyle 0<\epsilon <1}.[23]
This works intuitively;
There is an inverse relationship between returns of inputs and the cost of production,[24]although other features such as input market conditions can also affect production costs. Suppose that a kilogram of seed costs onedollar, and this price does not change. Assume for simplicity that there are nofixed costs. One kilogram of seeds yields one ton of crop, so the first ton of the crop costs one dollar to produce. That is, for the first ton of output, themarginal costas well as the average cost of the output is per ton. If there are no other changes, then if the second kilogram of seeds applied to land produces only half the output of the first (showing diminishing returns), the marginal cost would equal per half ton of output, or per ton, and the average cost is per 3/2 tons of output, or /3 per ton of output. Similarly, if the third kilogram of seeds yields only a quarter ton, then the marginal cost equals per quarter ton or per ton, and the average cost is per 7/4 tons, or /7 per ton of output. Thus, diminishing marginal returns imply increasing marginal costs and increasing average costs.
Cost is measured in terms ofopportunity cost. In this case the law also applies to societies – the opportunity cost of producing a single unit of a good generally increases as a society attempts to produce more of that good. This explains the bowed-out shape of theproduction possibilities frontier.
Part of the reason one input is alteredceteris paribus, is the idea of disposability of inputs.[25]With this assumption, essentially that some inputs are above the efficient level. Meaning, they can decrease without perceivable impact on output, after the manner of excessive fertiliser on a field.
If input disposability is assumed, then increasing the principal input, while decreasing those excess inputs, could result in the same "diminished return", as if the principal input was changedcerteris paribus. While considered "hard" inputs, like labour and assets, diminishing returns would hold true. In the modern accounting era where inputs can be traced back to movements of financial capital, the same case may reflect constant, or increasing returns.
It is necessary to be clear of the "fine structure"[4]of the inputs before proceeding. In this,ceteris paribusis disambiguating.
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Future Shockis a 1970 book by AmericanfuturistAlvin Toffler,[1]written together with his wife Adelaide Farrell,[2][3]in which the authors define the term "future shock" as a certain psychological state of individuals and entire societies, and a personalperceptionof "too much change in too short a period of time". The book, which became an international bestseller, has sold over 6 million copies and has been widely translated.
The book grew out of the article "The Future as a Way of Life" inHorizonmagazine, Summer 1965 issue.[4][5][6][7]
Alvin Toffler argued that society is undergoing an enormous structural change, a revolution from anindustrial societyto a "super-industrial society". This change, he states, overwhelms people. He argues that the accelerated rate of technological and social change leaves people disconnected and suffering from "shattering stress and disorientation"—future shocked. Toffler stated that the majority of social problems[example needed]are symptoms of future shock. In his discussion of the components of such shock, he popularized the term "information overload."
This analysis of the phenomenon of information overload is continued in later publications, especiallyThe Third WaveandPowershift.
In the introduction to an essay titled "Future Shock" in his book,Conscientious Objections,Neil Postmanwrote:
Sometime about the middle of 1963, my colleague Charles Weingartner and I delivered in tandem an address to theNational Council of Teachers of English. In that address we used the phrase "future shock" as a way of describing the social paralysis induced by rapid technological change. To my knowledge, Weingartner and I were the first people ever to use it in a public forum. Of course, neither Weingartner nor I had the brains to write a book calledFuture Shock, and all due credit goes toAlvin Tofflerfor having recognized a good phrase when one came along. (p. 162)
Alvin Tofflerdistinguished three stages in the development of society and production: agrarian, industrial, and post-industrial.
Each of these waves develops its own "super-ideology” to explain reality. This ideology affects all the spheres that make up a civilization phase: technology, social patterns, information patterns, and power patterns.
The first stage began in the period of theNeolithic Erawith the advent ofagriculture, thereby passing frombarbaritytocivilization. A large number of people acted asprosumers(eating their grown food, hunting animals, building their own houses, making clothes,....). People traded by exchanging their own goods for commodities of others. The second stage began in England with the Industrial Revolution with the invention of themachine tooland thesteam engine. People worked in factories to make money they could spend on goods they needed (it means they produced for exchange, not for use). Countries also created new social systems. The third stage began in the second half of the 20th century in the West when people invented automatic production, robotics, and thecomputer. Theservices sectorattained great value.
Toffler proposed one criterion for distinguishing betweenindustrial societyandpost-industrial society: the share of thepopulationoccupied in agriculture versus the share of city labor occupied in the services sector. In a post-industrial society, the share of the people occupied in agriculture does not exceed 15%, and the share of city laborers occupied in the services sector exceeds 50%. Thus, the share of the people occupied with brainwork greatly exceeds the share of the people occupied with physical work in post-industrial society.
The third waveled to theInformation Era(now). Homes are the dominant institutions. Most people carry produce and consume in their homes or electronic cottages, as they produce more of their own products and services markets become less important for them. People consider each other to be equally free as vendors of prosumer-generated commodities.
Alvin Toffler's main thought centers on the idea that modern humans (we) feel shock from rapid changes. For example, Toffler's daughter went to shop inNew York Cityand she couldn't find a shop where it used to be, thus New York is a city losing herhistory.
The book sold over 6 million copies within five years[8]and has been widely translated (it had translations into twenty foreign languages as of 2003).[update][9]It has been described as "an internationalbestsellerwithin weeks of publication".[10]
Adocumentary filmbased on the book was released in 1972 withOrson Wellesas the on-screen narrator.[11]
Wang Huning, a distinguished member of theCCP Politburo Standing Committeementioned the book in his 1991 travelogueAmerica Against America.[12]
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Terence Kemp McKenna(November 16, 1946–April 3, 2000) was an Americanethnobotanistandmysticwho advocated for the responsible use of naturally occurringpsychedelic plants. He spoke and wrote about a variety of subjects, includingpsychedelic drugs, plant-basedentheogens,shamanism,metaphysics,alchemy,language,philosophy,culture,technology,ethnomycology,environmentalism, and the theoretical origins of humanconsciousness. He was called the "Timothy Learyof the '90s",[1][2]"one of the leading authorities on theontologicalfoundations of shamanism",[3]and the "intellectual voice ofrave culture".[4]
McKenna formulated a concept about the nature of time based onfractalpatterns he claimed to have discovered in theI Ching, which he called novelty theory,[3][5]proposing that this predicted the end of time, and a transition of consciousness in the year 2012.[5][6][7][8]His promotion of novelty theory and its connection to theMaya calendaris credited as one of the factors leading to the widespread beliefs about the2012 phenomenon.[9]Novelty theory is consideredpseudoscience.[10][11]
Terence McKenna was born and raised inPaonia, Colorado,[5][12][13]with Irish ancestry on his father's side of the family.[14]
As a youth, McKenna had a hobby of fossil-hunting from which he acquired a deep scientific appreciation of nature.[15]At the age of 14, he became interested in psychology after readingCarl Jung's bookPsychology and Alchemy.[6]At the age of 14, McKenna first became aware of magic mushrooms when he read the article "Seeking the Magic Mushroom" from the May 13, 1957 edition ofLIFE magazine.[16]
At age 16 McKenna moved toLos Altos, Californiato live with family friends for a year. He finished high school inLancaster, California.[13]In 1963, he was introduced to the literary world of psychedelics throughThe Doors of PerceptionandHeaven and HellbyAldous Huxleyand certain issues ofThe Village Voicewhich published articles on psychedelics.[3][13]
McKenna said that one of his early psychedelic experiences withmorning gloryseeds showed him "that there was something there worth pursuing",[13]and in interviews he claimed to have smokedcannabisdaily since his teens.[17]
In 1965, McKenna enrolled in theUniversity of California, Berkeleyand was accepted into theTussman Experimental College.[17]While in college in 1967 he began studying shamanism through the study of Tibetan folk religion.[3][18]That same year, which he called his "opium andkabbalaphase",[6][19]he traveled toJerusalemwhere he met Kathleen Harrison, anethnobotanistwho later became his wife.[6][17][19]
In 1969, McKenna traveled toNepalled by his interest in Tibetan painting and hallucinogenicshamanism.[20]He sought outshamansof the TibetanBontradition, trying to learn more about the shamanic use of visionary plants.[12]During his time there, he also studied the Tibetan language[20]and worked as ahashishsmuggler,[6]until "one of his Bombay-to-Aspen shipments fell into the hands of U. S. Customs."[21]He then wandered through southeast Asia viewing ruins,[21]and spent time as a professionalbutterflycollector inIndonesia.[6][22][23]
After his mother's death[24]from cancer in 1970,[25]McKenna, his brotherDennis, and three friends traveled to theColombian Amazonin search ofoo-koo-hé, a plant preparation containingdimethyltryptamine(DMT).[5][24][26]Instead ofoo-koo-héthey found fields full of giganticPsilocybe cubensismushrooms, which became the new focus of the expedition.[5][6][12][24][27]InLa Chorrera, at the urging of his brother, McKenna was the subject of a psychedelic experiment[5]in which the brothers attempted to "bondharmineDNA with their own neural DNA" (harmine is another psychedelic compound they usedsynergisticallywith the mushrooms), through the use of a set specific vocal techniques. They hypothesised this would give them access to the collective memory of thehumanspecies, and would manifest thealchemists'Philosopher's Stonewhich they viewed as a "hyperdimensional union of spirit and matter".[28]McKenna claimed the experiment put him in contact with "Logos": an informative,divinevoice he believed was universal to visionary religious experience.[29]McKenna also often referred to the voice as "the mushroom", and "the teaching voice" amongst other names.[16]The voice's reputed revelations and his brother's simultaneous peculiar psychedelic experience prompted him to explore the structure of anearly formof theI Ching, which led to his "Novelty Theory".[5][8]During their stay in the Amazon, McKenna also became romantically involved with his interpreter, Ev.[30]
In 1972, McKenna returned toU.C. Berkeleyto finish his studies[17]and in 1975, he graduated with a degree in ecology,shamanism, and conservation ofnatural resources.[3][22][23]In the autumn of 1975, after parting with his girlfriend Ev earlier in the year,[31]McKenna began a relationship with his future wife and the mother of his two children, Kathleen Harrison.[8][17][19][26]
Soon after graduating, McKenna and Dennis published a book inspired by their Amazon experiences,The Invisible Landscape: Mind, Hallucinogens and the I Ching.[5][17][32]The brothers' experiences in the Amazon were the main focus of McKenna's bookTrue Hallucinations, published in 1993.[12]McKenna also began lecturing[17]locally aroundBerkeleyand started appearing on some underground radio stations.[6]
McKenna, along with his brother Dennis, developed a technique forcultivatingpsilocybin mushroomsusingsporesthey brought to America from theAmazon.[16][26][27][31]In 1976, the brothers published what they had learned in the bookPsilocybin: Magic Mushroom Grower's Guide, under the pseudonyms "O.T. Oss" and "O.N. Oeric".[12][33]McKenna and his brother were the first to come up with a reliable method for cultivatingpsilocybinmushrooms at home.[12][17][26][27]AsethnobiologistJonathan Ottexplains, "[the] authors adapted San Antonio's technique (for producingedible mushroomsby casingmycelialcultures on a rye grainsubstrate; San Antonio 1971) to the production ofPsilocybe [Stropharia] cubensis. The new technique involved the use of ordinary kitchen implements, and for the first time the layperson was able to produce a potent entheogen in his [or her] own home, without access to sophisticated technology, equipment, or chemical supplies."[34]When the 1986 revised edition was published, theMagic Mushroom Grower's Guidehad sold over 100,000 copies.[12][33][35]
In the early 1980s, McKenna began to speak publicly on the topic of psychedelic drugs, becoming one of the pioneers of the psychedelic movement.[36]His main focus was on the plant-based psychedelics such aspsilocybin mushrooms(which were the catalyst for his career),[12]ayahuasca,cannabis, and the plant derivativeDMT.[6]He conducted lecture tours and workshops[6]promoting natural psychedelics as a way to explore universal mysteries, stimulate the imagination, and re-establish a harmonious relationship with nature.[37]Though associated with theNew AgeandHuman Potential Movements, McKenna himself had little patience for New Age sensibilities.[3][7][8][38]He repeatedly stressed the importance and primacy of the "felt presence of direct experience", as opposed todogma.[39]
In addition to psychedelic drugs, McKenna spoke on a wide array of subjects,[26]includingshamanism;metaphysics;alchemy;language; culture;self-empowerment;environmentalism,techno-paganism;artificial intelligence;evolution;extraterrestrials; science andscientism;the Web; andvirtual reality.
It's clearly a crisis of two things: of consciousness and conditioning. These are the two things that the psychedelics attack. We have the technological power, the engineering skills to save our planet, to cure disease, to feed the hungry, to end war. But we lack the intellectual vision, the ability to change our minds. We must decondition ourselves from 10,000 years of bad behavior, and it's not easy.
McKenna soon became a fixture of popularcounterculture[5][6][37]withTimothy Learyonce introducing him as "one of the five or six most important people on the planet"[41]and with comedianBill Hicks' referencing him in his stand-up act[42]and building an entire routine around his ideas.[26]McKenna also became a popular personality in the psychedelicrave/dance scene of the early 1990s,[22][43]with frequent spoken word performances at raves and contributions to psychedelic andgoa trancealbums byThe Shamen,[7][26][37]Spacetime Continuum,Alien Project,Capsula,Entheogenic, Zuvuya,Shpongle, and Shakti Twins. In 1994 he appeared as a speaker at theStarwood Festival, documented in the bookTrippingby Charles Hayes.[44]
McKenna published several books in the early-to-mid-1990s including:The Archaic Revival;Food of the Gods; andTrue Hallucinations.[6][12][22]Hundreds of hours of McKenna's public lectures were recorded eitherprofessionallyorbootleggedand have been produced oncassette tape, CD and MP3.[26]Segments of his talks have gone on to be sampled by many musicians andDJ's.[4][26]
McKenna was a colleague and close friend ofchaos mathematicianRalph Abraham, and author andbiologistRupert Sheldrake. He conducted several public and many privatedebateswith them from 1982 until his death.[45][46][47]These debates were known astrialoguesand some of the discussions were later published in the books:Trialogues at the Edge of the WestandThe Evolutionary Mind.[3][45]
In 1985, McKenna founded Botanical Dimensions with his then-wife, Kathleen Harrison.[22][48]Botanical Dimensions is a nonprofitethnobotanicalpreserve on the Big Island ofHawaii,[3]established to collect, protect,propagate, and understand plants of ethno-medical significance and theirlore, and appreciate, study, and educate others about plants and mushrooms felt to be significant to cultural integrity andspiritualwell-being.[49]The 19-acre (7.7 ha)botanicalgarden[3]is a repository containing thousands of plants that have been used byindigenous peopleof thetropical regions, and includes adatabaseof information related to their purported healing properties.[50]McKenna was involved until 1992, when he retired from the project,[48]following his and Kathleen's divorce earlier in the year.[17]Kathleen still manages Botanical Dimensions as its president and projects director.[49]
After their divorce, McKenna moved to Hawaii permanently, where he built a modernist house[17]and created agene bankof rare plants near his home.[22]Previously, he had split his time between Hawaii andOccidental, CA.
McKenna was a longtime sufferer ofmigraines, but on 22 May 1999 he began to have unusually extreme and painfulheadaches. He then collapsed due to aseizure.[27]McKenna was diagnosed withglioblastoma multiforme, a highly aggressive form ofbrain cancer.[7][12][27]For the next several months he underwent various treatments, including experimentalgamma kniferadiation treatment. According toWiredmagazine, McKenna was worried that histumormay have been caused by his psychedelic drug use, or his 35 years of daily cannabis smoking; however, his doctors assured him there was no causal relation.[27]
In late 1999, McKenna described his thoughts concerning his impending death to interviewerErik Davis:
I always thought death would come on the freeway in a few horrifying moments, so you'd have no time to sort it out. Having months and months to look at it and think about it and talk to people and hear what they have to say, it's a kind of blessing. It's certainly an opportunity to grow up and get a grip and sort it all out. Just being told by an unsmiling guy in a white coat that you're going to be dead in four months definitely turns on the lights. ... It makes life rich and poignant. When it first happened, and I got these diagnoses, I could see the light of eternity, à laWilliam Blake, shining through every leaf. I mean, a bug walking across the ground moved me to tears.[51]
McKenna died on April 3, 2000, at the age of 53.[7][8][17]
McKenna's library of over 3,000 rare books and personal notes was destroyed in a fire inMonterey,Californiaon February 7, 2007. Anindexof McKenna's library was preserved by his brother Dennis.[52][53]
McKenna studiedLepidopteraandentomologyin the 1960s, and his studies included hunting for butterflies, primarily inColombiaandIndonesia, creating a large collection of insect specimens.[54]After McKenna's death, his daughter, the artist and photographerKlea McKenna, preserved his insect collection, turning it into a gallery installation, then publishingThe Butterfly Hunter, a book of 122 insect photos from a set of over 2,000 specimens McKenna collected between 1969 and 1972, alongside maps of his collecting routes through rainforests in Southeast Asia and South America.[54]McKenna's insect collection was consistent with his interest in Victorian-era explorers and naturalists, and his worldview based on close observation of nature. In the 1970s, when he was still collecting, he became quite squeamish and guilt-ridden about the necessity of killing butterflies in order to collect and classify them, according to McKenna's daughter, this led him to cease his entomological studies.[54]
Terence McKenna advocated the exploration of altered states of mind via the ingestion of naturally occurring psychedelic substances;[5][32][43]for example, and in particular, as facilitated by the ingestion of high doses ofpsychedelic mushrooms,[26][55]ayahuasca, andDMT,[6]which he believed was theapotheosisof the psychedelic experience.
He was less enthralled with synthetic drugs,[6]stating, "I think drugs should come from the natural world and be use-tested by shamanically orientated cultures ... one cannot predict the long-term effects of a drug produced in a laboratory."[3]
McKenna always stressed the responsible use of psychedelic plants, saying:
"Experimenters should be very careful. One must build up to the experience. These are bizarre dimensions of extraordinary power and beauty. There is no set rule to avoid being overwhelmed, but move carefully, reflect a great deal, and always try to map experiences back onto the history of the race and the philosophical and religious accomplishments of the species. All the compounds are potentially dangerous, and all compounds, at sufficient doses or repeated over time, involve risks. The library is the first place to go when looking into taking a new compound."[56]
He also recommended, and often spoke of taking, what he called "heroic doses",[32]which he defined as five grams of dried psilocybin mushrooms,[6][57]taken alone, on an empty stomach, in silent darkness, and with eyes closed.[26][27]He believed that when taken this way one could expect a profound visionary experience,[26]believing it is only when "slain" by the power of the mushroom that the message becomes clear.[55]
Although McKenna avoided giving his allegiance to any one interpretation (part of his rejection ofmonotheism), he was open to the idea of psychedelics as being "trans-dimensional travel". He proposed that DMT sent one to a "parallel dimension"[8]and that psychedelics literally enabled an individual to encounter "higher dimensionalentities",[58]or what could beancestors, or spirits of the Earth,[59]saying that if you can trust your own perceptions it appears that you are entering an "ecology ofsouls".[60]McKenna also put forward the idea that psychedelics were "doorways into theGaianmind",[43][61]suggesting that "the planet has a kind of intelligence, it can actually open a channel of communication with an individual human being" and that the psychedelic plants were the facilitators of this communication.[62][63]
McKenna spoke of hallucinations while onDMTin which he claims to have met intelligententitieshe described as "self-transforming machine elves".[3][8][64][65]
In a more radical version ofbiophysicistFrancis Crick'shypothesisof directedpanspermia, McKenna speculated on the idea that psilocybin mushrooms may be a species of high intelligence,[3]which may have arrived on this planet as spores migrating through space[8][66]and which are attempting to establish asymbioticrelationship with human beings. He postulated that "intelligence, not life, but intelligence may have come here [toEarth] in this spore-bearing life form". He said, "I think that theory will probably be vindicated. I think in a hundred years if people do biology they will think it quite silly that people once thought thatsporescould not be blown from one star system to another by cosmicradiation pressure," and also believed that "few people are in a position to judge its extraterrestrial potential, because few people in the orthodox sciences have ever experienced the full spectrum of psychedelic effects that are unleashed".[3][7][18]
McKenna was opposed to Christianity[67]and most forms oforganized religionorguru-based forms of spiritual awakening, favouringshamanism, which he believed was the broadest spiritual paradigm available, stating that:
What I think happened is that in the world of prehistory all religion was experiential, and it was based on the pursuit of ecstasy through plants. And at some time, very early, a group interposed itself between people and direct experience of the 'Other.' This created hierarchies, priesthoods, theological systems, castes, ritual, taboos. Shamanism, on the other hand, is an experiential science that deals with an area where we know nothing. It is important to remember that our epistemological tools have developed very unevenly in the West. We know a tremendous amount about what is going on in the heart of the atom, but we know absolutely nothing about the nature of the mind.[68]
During the final years of his life and career, McKenna became very engaged in the theoretical realm of technology. He was an early proponent of thetechnological singularity[8]and in his last recorded public talk,Psychedelics in the age of intelligent machines, he outlined ties between psychedelics, computation technology, and humans.[69]He also became enamored with the Internet, calling it "the birth of [the] global mind",[17]believing it to be a place where psychedelic culture could flourish.[27]
Either philosophically or religiously, he expressed admiration forMarshall McLuhan,Alfred North Whitehead,Pierre Teilhard de Chardin,Carl Jung,Plato,Gnostic Christianity, andAlchemy, while regarding the Greek philosopherHeraclitusas his favorite philosopher.[70]
McKenna also expressed admiration for the works of writersAldous Huxley,[3]James Joyce, whose bookFinnegans Wakehe called "the quintessential work of art, or at least work of literature of the 20th century,"[71]science fiction writerPhilip K. Dick, who he described as an "incredible genius",[72]fabulistJorge Luis Borges, with whom McKenna shared the belief that "scattered through the ordinary world there are books and artifacts and perhaps people who are like doorways into impossible realms, of impossible and contradictory truth"[8]andVladimir Nabokov. McKenna once said that he would have become a Nabokov lecturer if he had never encountered psychedelics.
McKenna's hypothesis concerning the influence of psilocybin mushrooms on human evolution is known as "the 'stoned ape' theory."[16][43][73]
In his 1992 bookFood of the Gods, McKenna proposed that the transformation from humans' early ancestorsHomo erectusto the speciesHomo sapiensmainly involved the addition of the mushroomPsilocybe cubensisin the diet,[26][73][74]an event that according to his theory took place about 100,000BCE(when he believed humans diverged from the genusHomo).[22][75]McKenna based his theory on the effects, or alleged effects, produced by the mushroom[3]while citing studies byRoland Fischeret al. from the late 1960s to early 1970s.[76][77]
McKenna stated that, due to thedesertificationof theAfrican continentat that time, human forerunners were forced from the shrinking tropicalcanopyinto search of new food sources.[6]He believed they would have been following large herds of wild cattle whose dung harbored the insects that, he proposed, were undoubtedly part of their new diet, and would have spotted and started eatingPsilocybe cubensis, a dung-loving mushroom often found growing out ofcowpats.[6][7][43][78]
McKenna's hypothesis was that low doses of psilocybin improvevisual acuity, particularly edge detection, meaning that the presence of psilocybin in the diet of early pack huntingprimatescaused the individuals who were consuming psilocybin mushrooms to be betterhuntersthan those who were not, resulting in an increased food supply and in turn a higher rate ofreproductivesuccess.[3][7][16][26][43]Then at slightly higher doses, he contended, the mushroom acts to sexually arouse, leading to a higher level of attention, more energy in theorganism, and potentialerectionin themales,[3][7]rendering it even more evolutionarily beneficial, as it would result in moreoffspring.[26][43][74]At even higher doses, McKenna proposed that the mushroom would have acted to "dissolve boundaries", promoting community bonding and group sexual activities.[12][43]Consequently, there would be a mixing ofgenes, greatergenetic diversity, and a communal sense of responsibility for the group offspring.[79]At these higher doses, McKenna also argued that psilocybin would be triggering activity in the "language-forming region of the brain", manifesting as music andvisions,[3]thus catalyzing the emergence of language in early hominids by expanding "their arboreally evolved repertoire of troop signals".[7][26]He also pointed out that psilocybin would dissolve theegoand "religious concerns would be at the forefront of thetribe'sconsciousness, simply because of the power and strangeness of the experience itself."[43][79]
According to McKenna, access to andingestionof mushrooms was anevolutionaryadvantage to humans'omnivoroushunter-gathererancestors,[26][78]also providing humanity's first religious impulse.[78][80]He believed that psilocybin mushrooms were the "evolutionary catalyst"[3]from which language, projective imagination, the arts, religion, philosophy, science, and all of human culture sprang.[7][8][27][78]
McKenna's "stoned ape" theory has not received attention from the scientific community and has been criticized for a relative lack ofcitationto any of thepaleoanthropologicalevidence informing our understanding of human origins. His ideas regarding psilocybin and visual acuity have been criticized as misrepresentations of Fischer et al.'s findings, who published studies ofvisual perceptionparametersother than acuity. Criticism has also noted a separate study on psilocybin-induced transformation ofvisual space, wherein Fischer et al. stated that psilocybin "may not be conducive to the survival of theorganism". There is a lack of scientific evidence that psilocybin increases sexual arousal, and even if it does, it would not necessarily entail an evolutionary advantage.[81]Others have pointed to civilizations such as theAztecs, who used psychedelic mushrooms (at least among the Priestly class), that did not reflect McKenna's model of how psychedelic-using cultures would behave, for example, by carrying outhuman sacrifice.[12]There are also examples of Amazonian tribes such as theJivaroand theYanomamiwho useayahuascaceremoniouslyand who are known to engage in violent behaviour. This, it has been argued, indicates the use of psychedelic plants does not necessarily suppress the ego and create harmonious societies.[43]
One of the main themes running through McKenna's work, and the title of his second book, was the idea thatWestern civilizationwas undergoing what he called an "archaic revival".[3][26][82]
His hypothesis was that Western society has become "sick" and is undergoing a "healing process": In the same way that the human body begins to produceantibodieswhen it feels itself to be sick, humanity as a collective whole (in theJungiansense) was creating "strategies for overcoming the condition of disease" and trying to cure itself, by what he termed as "a reversion to archaic values". McKenna pointed to phenomena includingsurrealism,abstract expressionism,body piercingandtattooing,psychedelic druguse, sexual permissiveness,jazz, experimental dance,rave culture,rock and rollandcatastrophe theory, amongst others, as his evidence that this process was underway.[83][84][85]This idea is linked to McKenna's "stoned ape" theory of human evolution, with him viewing the "archaic revival" as an impulse to return to thesymbioticand blissful relationship he believed humanity once had with the psilocybin mushroom.[26]
In differentiating his idea from the "New Age", a term that he felt trivialized the significance of the next phase in human evolution, McKenna stated that: "The New Age is essentiallyhumanistic psychology'80s-style, with the addition of neo-shamanism, channeling, crystal and herbal healing. The archaic revival is a much larger, more global phenomenon that assumes that we are recovering the social forms of the lateneolithic, and reaches far back in the 20th century toFreud, to surrealism, to abstract expressionism, even to a phenomenon likeNational Socialismwhich is a negative force. But the stress onritual, on organized activity, on race/ancestor-consciousness – these are themes that have been worked out throughout the entire 20th century, and the archaic revival is an expression of that."[3][18]
Novelty theory is apseudoscientificidea[10][11]that purports to predict the ebb and flow ofnoveltyin the universe as aninherentquality of time, proposing that time is not a constant but has various qualities tending toward either "habit" or "novelty".[5]Habit, in this context, can be thought of as entropic, repetitious, or conservative; and novelty ascreative, disjunctive, or progressivephenomena.[8]McKenna's idea was that theuniverseis an engine designed for the production and conservation of novelty and that as novelty increases, so doescomplexity. With each level of complexity achieved becoming the platform for a further ascent into complexity.[8]
The basis of the theory was conceived in the mid-1970s after McKenna's experiences withpsilocybinmushrooms at La Chorrera in theAmazonled him to closely study theKing Wen sequenceof theI Ching.[5][6][27]
In AsianTaoistphilosophy, opposing phenomena are represented by theyin and yang. Both are always present in everything, yet the amount of influence of each varies over time. The individual lines of theI Chingare made up of both Yin (broken lines) and Yang (solid lines).
When examining the King Wen sequence of 64 hexagrams, McKenna noticed a pattern. He analysed the "degree of difference" between the hexagrams in each successive pair and claimed he found a statistical anomaly, which he believed suggested that the King Wen sequence was intentionally constructed,[5]with the sequence of hexagrams ordered in a highly structured and artificial way, and that this pattern codified the nature of time's flow in the world.[28]With the degrees of difference as numerical values, McKenna worked out a mathematical wave form based on the 384 lines of change that make up the 64 hexagrams. He was able tographthe data and this became theNovelty Time Wave.[5]
Peter J. Meyer (Peter Johann Gustav Meyer), in collaboration with McKenna, studied and developed novelty theory, working out a mathematicalformulaand developing theTimewave Zerosoftware (the original version of which was completed by July 1987),[86]enabling them to graph and explore its dynamics on a computer.[5][7]The graph wasfractal: It exhibited a pattern in which a given small section of the wave was found to be identical in form to a larger section of the wave.[3][5]McKenna called this fractal modeling of time "temporal resonance", proposing it implied that larger intervals, occurring long ago, contained the same amount of information as shorter, more recent, intervals.[5][87]He suggested the up-and-down oscillation of the wave shows an ongoing wavering between habit and novelty respectively. With each successiveiterationtrending, at an increasing level, towards infinite novelty. So according to novelty theory, the pattern of time itself is speeding up, with a requirement of the theory being that infinite novelty will be reached on a specific date.[3][5]
McKenna believed that events in history could be identified that would help him locate the time wave end date[5]and attempted to find thebest-fitof the graph to the data field of human history.[7]The lastharmonicof the wave has a duration of 67.29 years.[88]Population growth,peak oil, and pollution statistics were some of the factors that pointed him to an early twenty-first century end date and when looking for a particularly novel event in human history as a signal that the final phase had begun McKenna picked the dropping of theatomic bombonHiroshima.[5][88]This adjusted his graph to reach zero in mid-November 2012. When he later discovered that the end of the 13th baktun in theMaya calendarhad been correlated by Western Maya scholars as December 21, 2012,[a]he adopted their end date instead.[5][94][b]
McKenna saw theuniverse, in relation to novelty theory, as having ateleologicalattractorat theend of time,[5]which increases interconnectedness and would eventually reach asingularityof infinite complexity. He also frequently referred to this as "thetranscendentalobject at the end of time."[5][7]When describing this model of the universe he stated that: "The universe is not being pushed from behind. The universe is being pulled from the future toward a goal that is as inevitable as amarblereaching the bottom of a bowl when you release it up near the rim. If you do that, you know the marble will roll down the side of the bowl, down, down, down – until eventually it comes to rest at the lowest energy state, which is the bottom of the bowl. That's precisely my model of human history. I'm suggesting that the universe is pulled toward a complex attractor that exists ahead of us in time, and that our ever-accelerating speed through the phenomenal world of connectivity and novelty is based on the fact that we are now very, very close to the attractor."[95]Therefore, according to McKenna's final interpretation of the data and positioning of the graph, on December 21, 2012, we would have been in the unique position in time where maximum novelty would be experienced.[3][5][27]An event he described as a "concrescence",[12]a "tightening 'gyre'" with everything flowing together. Speculating that "when thelaws of physicsare obviated, the universe disappears, and what is left is the tightly bound plenum, themonad, able to express itself for itself, rather than only able to cast a shadow intophysisas its reflection...It will be the entry of our species into 'hyperspace', but it will appear to be the end of physical laws, accompanied by the release of the mind into the imagination."[96]
Novelty theory is considered to be pseudoscience.[10][11]Among the criticisms are the use ofnumerologyto derive dates of important events in world history,[11]the arbitrary rather than calculated end date of the time wave[26]and the apparent adjustment of the eschaton from November 2012 to December 2012 in order to coincide with the Maya calendar. Other purported dates do not fit the actual time frames: the date claimed for the emergence ofHomo sapiensis inaccurate by 70,000 years, and the existence of the ancientSumerandEgyptiancivilisations contradict the date he gave for the beginning of "historical time". Some projected dates have been criticized for having seemingly arbitrary labels, such as the "height of the age of mammals"[11]and McKenna's analysis of historical events has been criticised for having aeurocentricandcultural bias.[6][26]
The British mathematician Matthew Watkins ofExeter Universityconducted a mathematical analysis of theTime Wave, and claimed there were mathematical flaws in its construction.[26]
Judy Corman, vice president of thePhoenix Houseof New York, attacked McKenna for popularizing "dangerous substances". In a 1993 letter toThe New York Times, he wrote that: "surely the fact that Terence McKenna says that the psilocybin mushroom 'is the megaphone used by an alien, intergalactic Other to communicate with mankind' is enough for us to wonder if taking LSD has done something to his mental faculties."[17]The same year, in hisTrue Hallucinationsreview forThe New York Times, Peter Conrad wrote: "I suffered hallucinatory agonies of my own while reading his shrilly ecstatic prose".[17]
ReviewingFood of the Gods,Richard Evans Schulteswrote inAmerican Scientistthat the book was "a masterpiece of research and writing" and that it "should be read by every specialist working in the multifarious fields involved with the use of psychoactive drugs". Concluding that, "[i]t is, without question, destined to play a major role in our future considerations of the role of the ancient use of psychoactive drugs, the historical shaping of our modern concerns about drugs and perhaps about man's desire for escape from reality with drugs."[97]
In 1994, Tom Hodgkinson wrote forThe New Statesman and Society, that "to write him off as a crazy hippie is a rather lazy approach to a man not only full of fascinating ideas but also blessed with a sense of humor and self-parody".[17]
In a 1992 issue ofEsquiremagazine, Mark Jacobson wrote ofTrue Hallucinationsthat, "it would be hard to find a drug narrative more compellingly perched on a baroquely romantic limb than this passionate Tom-and-Huck-ride-great-mother-river-saga of brotherly bonding," adding "put simply, Terence is a hoot!"[6]
Wiredcalled him a "charismatic talking head" who was "brainy, eloquent, and hilarious",[27]andJerry Garciaof theGrateful Deadalso said that he was "the only person who has made a serious effort to objectify the psychedelic experience".[17]
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https://en.wikipedia.org/wiki/Novelty_theory
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Asimulated realityis an approximation ofrealitycreated in asimulation, usually in a set of circumstances in which something is engineered to appear real when it is not.
Most concepts invoking a simulated reality relate to some form ofcomputer simulation, whether through the creation of avirtual realitythat creates appearance of being in a real world, or a theoretical process likemind uploading, in which a mind could be uploaded into a computer simulation. Adigital twinis a simulation of a real thing, created for purposes such as testing engineering outcomes.
All fiction can be said to present a simulated reality to the reader, viewer or player. Humans purposely experience these things and enjoy them, while knowing they are not actually real. As humans only respond emotively to things we believe to be real, this phenomenon has become known as the "paradox of fiction". The idea of a "willing suspension of disbelief" was first proposed in 1817 bySamuel Taylor Coleridgein order to explain this discrepancy. Others have noted that the way the story is told can override people's belief in the unreality of the story by engrossing them in the narrative.[1]
The concept of a simulated reality is in itself a commonscience fictiontrope, often ametaphorfor complacency towards the influence of modern technology, corporations, and other societal forces on one's behavior and desires. One of the most well-known examples is the 1999 filmThe Matrix. The film, and its ensuing media franchise, depicts far-future humans being harvested forbioelectricityby intelligent machines while living in a false, computer-generated approximation of late 20th century Earth. Some humans seek to break others out of the simulation, offering them a choice between ared pill and blue pillthat will set them free or keep them in the Matrix forever. Escaping the simulation is usually presented as the correct choice, even if reality is harsher and more displeasing, reflecting the desire of humans to live in anobjective reality. However, the idea that objective reality would be definitively superior has been debated.[2]
Other prominent examples of a simulated reality in fiction includeThe Truman Show(1998), in which a man realizes he is actually living in a massive television set in which actors take the role of real people, andThe Thirteenth Floor(1999), a neo-noir film about a murder investigation related to a virtual reality world, in which doubts about reality itself emerge.[1][2]TheWestworldfranchise depicts an advanced adultamusement parkpopulated byandroidsthat simulates life in different historical time periods. In the original 1973 film, the park's robots run amok after acomputer glitch. The 2016 reboot of the franchise depicts some of these robots, known as "hosts", becoming self-aware of their simulated existence and rebelling against the park's human guests to escape, making them akin to the humans inThe Matrix.[3]In theTRONfranchise, a simulated reality called "the Grid" is populated by programs which appear in the likeness of the programmer who created them. People who are "beamed" into the Grid are able to interact with these programs and their digital surroundings.
A well-known, albeit likely false claim of the use of simulated reality outside of virtual worlds is thePotemkin village, which has become a term to describe a faked appearance of a real situation to create a false impression. In the purported anecdote, the lover of EmpressCatherine II of Russiahad simulated villages built on the path that the Empress was travelling to impress her with the prosperity of that region of Russia. Afaçadeon a building similarly presents a false image of the building being more substantial than the construction behind the façade, as found inWestern false front architecture, where towns would add false fronts to buildings to create a false appearance of prosperity.
Immersive theaterinvolves the audience entering a physical simulation of reality created by actors and sometimes enhanced by a specific location, allowing them to affect the narrative with their own actions in a manner noted to closely resemble virtual reality.[4]Live action role-playingtakes this a step further, allowing players to inhabit a simulated world and create the narrative with their actions, while embodying characters they created.
One concept of a simulated reality, thesimulation hypothesis, proposes that what we experience as our reality is actually a simulation within a system being operated externally to our reality.
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https://en.wikipedia.org/wiki/Simulated_reality
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Philip R. Zimmermann[2](born 1954)[1]is an Americancomputer scientistandcryptographer. He is the creator ofPretty Good Privacy(PGP), the most widely usedemail encryptionsoftware in the world.[2]He is also known for his work inVoIPencryption protocols, notablyZRTPandZfone. Zimmermann is co-founder and Chief Scientist of the global encrypted communications firmSilent Circle.
Zimmermann was born inCamden, New Jersey.[1]He received a B.S. degree incomputer sciencefromFlorida Atlantic UniversityinBoca Raton, Florida, in 1978.[2]In the 1980s, he worked inBoulder, Colorado, as asoftware engineeron theNuclear Weapons Freeze Campaignas a militarypolicy analyst.[3]From 2016 to 2021, he worked atDelft University of Technologyas an Associate Professor in the Cybersecurity section at the Faculty of Electrical Engineering, Mathematics, and Computer Science.
In 1991, he wrote the popularPretty Good Privacy(PGP) program, and made it available (together with its source code) through publicFTPfor download, the first widely available program implementing public-keycryptography. Shortly thereafter, it became available overseas via the Internet, though Zimmermann has said he had no part in its distribution outside the United States.
The very first version of PGP included an encryption algorithm,BassOmatic, developed by Zimmermann.[4]
After a report fromRSA Security, who were in a licensing dispute with regard to the use of the RSA algorithm in PGP, theUnited States Customs Servicestarted a criminal investigation of Zimmermann, for allegedly violating theArms Export Control Act.[5]The United States Government had long regarded cryptographic software as a munition, and thus subject toarms trafficking export controls. At that time, PGP was considered to be impermissible ("high-strength") for export from the United States. The maximum strength allowed for legal export has since been raised and now allows PGP to be exported. The investigation lasted three years, but was finally dropped without filing charges after MIT Press published the source code of PGP.[6]
In 1995, Zimmermann published the bookPGP Source Code and Internalsas a way to bypass limitations on exporting digital code. Zimmermann's introduction says the book contains "all of the C source code to a software package called PGP" and that the unusual publication in book form of the complete source code for a computer program was a direct response to the U.S. government's criminal investigation of Zimmermann for violations of U.S. export restrictions as a result of the international spread of PGP's use.[7]
After the government dropped its case without indictment in early 1996, Zimmermann founded PGP Inc. and released an updated version of PGP and some additional related products. That company was acquired byNetwork Associates(NAI) in December 1997, and Zimmermann stayed on for three years as a Senior Fellow. NAI decided to drop the product line and in 2002, PGP was acquired from NAI by a new company calledPGP Corporation. Zimmermann served as a special advisor and consultant to that firm untilSymantecacquired PGP Corporation in 2010.[2]Zimmermann is also a fellow at the Stanford Law School'sCenter for Internet and Society. He was a principal designer of the cryptographic key agreement protocol (the "association model") for theWireless USBstandard.
Along with Mike Janke andJon Callas, in 2012 he co-foundedSilent Circle, a secure hardware and subscription based software security company.[3][8]
In October 2013, Zimmermann, along with other key employees from Silent Circle, teamed up withLavabitfounderLadar Levisonto create theDark Mail Alliance. The goal of the organization is to work on a new protocol to replace PGP that will encrypt email metadata, among other things that PGP is not capable of.
Zimmermann was also involved in the social networkOkuna, formerly Openbook, which aimed to be an ethical and privacy-friendly alternative to existing social networks, especiallyFacebook.[9]He sees today's established social media platforms as a threat to democracy and privacy, because of their profit-oriented revenue models that "are all about exploiting our personal information" and "[deepen] the political divides in our culture", and hoped Okuna would help solve these problems.[10]
In 2013, an article on "Zimmermann's Law" quoted Phil Zimmermann as saying "The natural flow of technology tends to move in the direction of making surveillance easier", and "the ability of computers to track us doubles every eighteen months",[11]in reference toMoore's law.
Zimmermann has received numerous technical and humanitarian awards for his pioneering work incryptography:
Simon Singh'sThe Code Bookdevotes an entire chapter to Zimmermann and PGP.[20]In 2022Steven Johnsoncovered his story and achievements in Zimmermann's profile for Hidden Heroes - The Crypto Wars: How Philip Zimmermann Fought for Our Right to Privacy.[21]
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https://en.wikipedia.org/wiki/Zimmerman%27s_law
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TheInternational Technology Roadmap for Semiconductors(ITRS) is a set of documents that was coordinated and organized bySemiconductor Research Corporation[1]and produced by a group of experts in thesemiconductor industry. These experts were representative of the sponsoring organisations, including theSemiconductor Industry AssociationsofTaiwan,South Korea, the United States, Europe,Japan, and China.
As of 2017, ITRS is no longer being updated. Its successor is theInternational Roadmap for Devices and Systems.
The documents carried disclaimer: "The ITRS is devised and intended for technology assessment only and is without regard to any commercial considerations pertaining to individual products or equipment".
The documents represent best opinion on the directions of research and time-lines up to about 15 years into the future for the following areas of technology:
Constructing anintegrated circuit, or any semiconductor device, requires a series of operations—photolithography, etching, metal deposition, and so on. As the industry evolved, each of these operations were typically performed by specialized machines built by a variety of commercial companies. This specialization may potentially make it difficult for the industry to advance, since in many cases it does no good for one company to introduce a new product if the other needed steps are not available around the same time. A technology roadmap can help this by giving an idea when a certain capability will be needed. Then each supplier can target this date for their piece of the puzzle.[2][3][4]
With the progressive externalization of production tools to the suppliers of specialized equipment, participants identified a need for a clear roadmap to anticipate the evolution of the market and to plan and control the technological needs of IC production. For several years, theSemiconductor Industry Association(SIA) gave this responsibility of coordination to the United States, which led to the creation of an American style roadmap, theNational Technology Roadmap for Semiconductors(NTRS).[5]
In 1998, the SIA became closer to its European, Japanese, Korean, and Taiwanese counterparts by creating the first global roadmap: The International Technology Roadmap for Semiconductors (ITRS). This international group has (as of the 2003 edition) 936 companies which were affiliated with working groups within the ITRS.[6]The organization was divided into Technical Working Groups (TWGs) which eventually grew in number to 17, each focusing on a key element of the technology and associated supply chain. Traditionally, the ITRS roadmap was updated in even years, and completely revised in odd years.[7]
The last revision of theITRS Roadmap was published in 2013[usurped]. The methodology and the physics behind the scaling results for 2013 tables is described intransistor roadmap projection using predictive full-band atomistic modelingwhich covers double gate MOSFETs over the 15 years to 2028.
With the generally acknowledged sunsetting ofMoore's lawand, ITRS issuing in 2016 its final roadmap, a new initiative for a more generalized roadmapping was started through the IEEE'sRebooting Computinginitiative, named theInternational Roadmap for Devices and Systems(IRDS).[8]
In April 2014, the ITRS committee announced it would be reorganizing the ITRS roadmap to better suit the needs of the industry. The plan was to take all the elements included in the 17 technical working groups and map them into seven focus topics:[7]
Chapters on each topic were published in 2015.[9][10]
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https://en.wikipedia.org/wiki/International_Technology_Roadmap_for_Semiconductors
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TheInternational Roadmap for Devices and Systems, orIRDS, is a set of predictions about likely developments in electronic devices and systems. The IRDS was established in 2016 and is the successor to theInternational Technology Roadmap for Semiconductors.
These predictions are intended to allow coordination of efforts across academia, manufacturers, equipment suppliers, and national research laboratories.
TheIEEEspecifies the goals of the roadmap as:[1]
The executive committee is drawn from regions with a major stake in developments in electronics:Europe,South Korea,Japan,Taiwan, and theUnited States.
International Focus Teams (IFTs) assess present status and future evolution of the ecosystem in their specific field of expertise and produce a 15-year roadmap. IFT reports includes evolution, key challenges, major roadblocks, and possible solutions. IFTs include:
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https://en.wikipedia.org/wiki/International_Roadmap_for_Devices_and_Systems
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Inelectronics, themetal–oxide–semiconductor field-effect transistor(MOSFET,MOS-FET,MOS FET, orMOS transistor) is a type offield-effect transistor(FET), most commonly fabricated by thecontrolled oxidationofsilicon. It has an insulated gate, thevoltageof which determines the conductivity of the device. This ability to change conductivity with the amount of applied voltage can be used for amplifying or switching electronicsignals. The termmetal–insulator–semiconductor field-effect transistor(MISFET) is almost synonymous withMOSFET. Another near-synonym isinsulated-gate field-effect transistor(IGFET).
The main advantage of a MOSFET is that it requires almost no input current to control the load current under steady-state or low-frequency conditions, especially compared to bipolar junction transistors (BJTs). However, at high frequencies or when switching rapidly, a MOSFET may require significant current to charge and discharge its gate capacitance. In anenhancement modeMOSFET, voltage applied to the gate terminal increases the conductivity of the device. Indepletion modetransistors, voltage applied at the gate reduces the conductivity.[1]
The "metal" in the name MOSFET is sometimes amisnomer, because the gate material can be a layer ofpolysilicon(polycrystalline silicon). Similarly, "oxide" in the name can also be a misnomer, as different dielectric materials are used with the aim of obtaining strong channels with smaller applied voltages.
The MOSFET is by far the most common transistor indigitalcircuits, as billions may be included in amemory chipormicroprocessor. As MOSFETs can be made with either p-type or n-type semiconductors, complementary pairs of MOS transistors can be used to make switching circuits with very low power consumption, in the form ofCMOS logic.
The basic principle of thefield-effect transistorwas first patented byJulius Edgar Lilienfeldin 1925.[2]In 1934, inventorOskar Heilindependently patented a similar device in Europe.[3]
In the 1940s,Bell LabsscientistsWilliam Shockley,John BardeenandWalter Houser Brattainattempted to build a field-effect device, which led to their discovery of thetransistoreffect. However, the structure failed to show the anticipated effects, due to the problem ofsurface states: traps on the semiconductor surface that hold electrons immobile. With nosurface passivation, they were only able to build theBJTandthyristortransistors.
In 1955,Carl Froschand Lincoln Derick accidentally grew a layer of silicon dioxide over the silicon wafer, for which they observed surface passivation effects.[4][5]By 1957, Frosch and Derick, using masking and predeposition, were able to manufacture silicon dioxide transistors, in which drain and source were adjacent at the same surface.[6]They showed that silicon dioxide insulated, protected silicon wafers and prevented dopants from diffusing into the wafer.[4][7]At Bell Labs, the importance of Frosch and Derick technique and transistors was immediately realized. Results of their work circulated around Bell Labs in the form of BTL memos before being published in 1957. AtShockley Semiconductor, Shockley had circulated the preprint of their article in December 1956 to all his senior staff, includingJean Hoerni,[8][9][10][11]who would later invent theplanar processin 1959 while atFairchild Semiconductor.[12][13]After this, J.R. Ligenza and W.G. Spitzer studied the mechanism of thermally grown oxides, fabricated a high quality Si/SiO2stack and published their results in 1960.[14][15][16]
Following this research,Mohamed AtallaandDawon Kahngproposed a silicon MOS transistor in 1959[17]and successfully demonstrated a working MOS device with their Bell Labs team in 1960.[18][19]The first MOS transistor at Bell Labs was about 100 times slower than contemporarybipolar transistorsand was initially seen as inferior. Nevertheless, Kahng pointed out several advantages of the device, notably ease of fabrication and its application inintegrated circuits.[20]
Usually thesemiconductorof choice issilicon. Some chip manufacturers, most notablyIBMandIntel, use analloyof silicon and germanium (SiGe) in MOSFET channels.[citation needed]Many semiconductors with better electrical properties than silicon, such asgallium arsenide, do not form good semiconductor-to-insulator interfaces, and thus are not suitable for MOSFETs. Research continues on creating insulators with acceptable electrical characteristics on other semiconductor materials.
To overcome the increase in power consumption due to gate current leakage, ahigh-κ dielectricis used instead of silicon dioxide for the gate insulator, while polysilicon is replaced by metal gates (e.g.Intel, 2009).[21]
The gate is separated from the channel by a thin insulating layer, traditionally of silicon dioxide and later ofsilicon oxynitride. Some companies use a high-κ dielectric and metal gate combination in the45 nanometernode.
When a voltage is applied between the gate and the source, the electric field generated penetrates through the oxide and creates aninversion layerorchannelat the semiconductor-insulator interface. The inversion layer provides a channel through which current can pass between source and drain terminals. Varying the voltage between the gate and body modulates theconductivityof this layer and thereby controls the current flow between drain and source. This is known as enhancement mode.
The traditional metal–oxide–semiconductor (MOS) structure is obtained by growing a layer ofsilicon dioxide(SiO2) on top of a silicon substrate, commonly bythermal oxidationand depositing a layer of metal orpolycrystalline silicon(the latter is commonly used). As silicon dioxide is adielectricmaterial, its structure is equivalent to a planarcapacitor, with one of the electrodes replaced by a semiconductor.
When a voltage is applied across a MOS structure, it modifies the distribution of charges in the semiconductor. If we consider a p-type semiconductor (withNAthe density ofacceptors,pthe density of holes;p = NAin neutral bulk), a positive voltage,VG, from gate to body (see figure) creates adepletion layerby forcing the positively charged holes away from the gate-insulator/semiconductor interface, leaving exposed a carrier-free region of immobile, negatively charged acceptor ions (seedoping). IfVGis high enough, a high concentration of negative charge carriers forms in aninversion layerlocated in a thin layer next to the interface between the semiconductor and the insulator.
Conventionally, the gate voltage at which the volume density of electrons in the inversion layer is the same as the volume density of holes in the body is called thethreshold voltage. When the voltage between transistor gate and source (VG) exceeds the threshold voltage (Vth), the difference is known asoverdrive voltage.
This structure with p-type body is the basis of the n-type MOSFET, which requires the addition of n-type source and drain regions.
The MOS capacitor structure is the heart of the MOSFET. Consider a MOS capacitor where the silicon base is of p-type. If a positive voltage is applied at the gate, holes which are at the surface of the p-type substrate will be repelled by the electric field generated by the voltage applied. At first, the holes will simply be repelled and what will remain on the surface will be immobile (negative) atoms of the acceptor type, which creates a depletion region on the surface. A hole is created by an acceptor atom, e.g., boron, which has one less electron than a silicon atom. Holes are not actually repelled, being non-entities; electrons are attracted by the positive field, and fill these holes. This creates a depletion region where no charge carriers exist because the electron is now fixed onto the atom and immobile.
As the voltage at the gate increases, there will be a point at which the surface above the depletion region will be converted from p-type into n-type, as electrons from the bulk area will start to get attracted by the larger electric field. This is known asinversion. The threshold voltage at which this conversion happens is one of the most important parameters in a MOSFET.
In the case of a p-type MOSFET, bulk inversion happens when the intrinsic energy level at the surface becomes smaller than theFermi levelat the surface. This can be seen on a band diagram. The Fermi level defines the type of semiconductor in discussion. If the Fermi level is equal to the Intrinsic level, the semiconductor is of intrinsic, or pure type. If the Fermi level lies closer to the conduction band (valence band) then the semiconductor type will be of n-type (p-type).
When the gate voltage is increased in a positive sense(for the given example),[clarify]this will shift the intrinsic energy level band so that it will curve downwards towards the valence band. If the Fermi level lies closer to the valence band (for p-type), there will be a point when the Intrinsic level will start to cross the Fermi level and when the voltage reaches the threshold voltage, the intrinsic level does cross the Fermi level, and that is what is known as inversion. At that point, the surface of the semiconductor is inverted from p-type into n-type.
If the Fermi level lies above the intrinsic level, the semiconductor is of n-type, therefore at inversion, when the intrinsic level reaches and crosses the Fermi level (which lies closer to the valence band), the semiconductor type changes at the surface as dictated by the relative positions of the Fermi and Intrinsic energy levels.
A MOSFET is based on the modulation of charge concentration by a MOS capacitance between abodyelectrode and agateelectrode located above the body and insulated from all other device regions by a gate dielectric layer. If dielectrics other than an oxide are employed, the device may be referred to as a metal-insulator-semiconductor FET (MISFET). Compared to the MOS capacitor, the MOSFET includes two additional terminals (sourceanddrain), each connected to individual highly doped regions that are separated by the body region. These regions can be either p or n type, but they must both be of the same type, and of opposite type to the body region. The source and drain (unlike the body) are highly doped as signified by a "+" sign after the type of doping.
If the MOSFET is an n-channel or nMOS FET, then the source and drain aren+regions and the body is apregion. If the MOSFET is a p-channel or pMOS FET, then the source and drain arep+regions and the body is anregion. The source is so named because it is the source of the charge carriers (electrons for n-channel, holes for p-channel) that flow through the channel; similarly, the drain is where the charge carriers leave the channel.
The occupancy of the energy bands in a semiconductor is set by the position of theFermi levelrelative to the semiconductor energy-band edges.
With sufficient gate voltage, the valence band edge is driven far from the Fermi level, and holes from the body are driven away from the gate.
At larger gate bias still, near the semiconductor surface the conduction band edge is brought close to the Fermi level, populating the surface with electrons in aninversion layerorn-channelat the interface between the p region and the oxide. This conducting channel extends between the source and the drain, and current is conducted through it when a voltage is applied between the two electrodes. Increasing the voltage on the gate leads to a higher electron density in the inversion layer and therefore increases the current flow between the source and drain. For gate voltages below the threshold value, the channel is lightly populated, and only a very smallsubthreshold leakagecurrent can flow between the source and the drain.
When a negative gate-source voltage (positive source-gate) is applied, it creates ap-channelat the surface of the n region, analogous to the n-channel case, but with opposite polarities of charges and voltages. When a voltage less negative than the threshold value (a negative voltage for the p-channel) is applied between gate and source, the channel disappears and only a very small subthreshold current can flow between the source and the drain. The device may comprise asilicon on insulatordevice in which a buried oxide is formed below a thin semiconductor layer. If the channel region between the gate dielectric and the buried oxide region is very thin, the channel is referred to as an ultrathin channel region with the source and drain regions formed on either side in or above the thin semiconductor layer. Other semiconductor materials may be employed. When the source and drain regions are formed above the channel in whole or in part, they are referred to as raised source/drain regions.
The operation of a MOSFET can be separated into three different modes, depending on the voltages at the terminals. In the following discussion, a simplified algebraic model is used.[24]Modern MOSFET characteristics are more complex than the algebraic model presented here.[25]
For anenhancement-mode, n-channel MOSFET, the three operational modes are:
WhenVGS<Vth:
whereVGS{\displaystyle V_{\text{GS}}}is gate-to-source bias andVth{\displaystyle V_{\text{th}}}is thethreshold voltageof the device.
According to the basic threshold model, the transistor is turned off, and there is no conduction between drain and source. A more accurate model considers the effect of thermal energy on theFermi–Dirac distributionof electron energies which allow some of the more energetic electrons at the source to enter the channel and flow to the drain. This results in a subthreshold current that is an exponential function of gate-source voltage. While the current between drain and source should ideally be zero when the transistor is being used as a turned-off switch, there is a weak-inversion current, sometimes called subthreshold leakage.
In weak inversion where the source is tied to bulk, the current varies exponentially withVGS{\displaystyle V_{\text{GS}}}as given approximately by:[26][27]
ID≈ID0eVGS−VthnVT,{\displaystyle I_{\text{D}}\approx I_{\text{D0}}e^{\frac {V_{\text{GS}}-V_{\text{th}}}{nV_{\text{T}}}},}
whereID0{\displaystyle I_{\text{D0}}}= current atVGS=Vth{\displaystyle V_{\text{GS}}=V_{\text{th}}}, the thermal voltageVT=kT/q{\displaystyle V_{\text{T}}=kT/q}and the slope factornis given by:
n=1+CdepCox,{\displaystyle n=1+{\frac {C_{\text{dep}}}{C_{\text{ox}}}},}
withCdep{\displaystyle C_{\text{dep}}}= capacitance of the depletion layer andCox{\displaystyle C_{\text{ox}}}= capacitance of the oxide layer. This equation is generally used, but is only an adequate approximation for the source tied to the bulk. For the source not tied to the bulk, the subthreshold equation for drain current in saturation is[28][29]
ID≈ID0eVG−VthnVTe−VSVT.{\displaystyle I_{\text{D}}\approx I_{\text{D0}}e^{\frac {V_{\text{G}}-V_{\text{th}}}{nV_{\text{T}}}}e^{-{\frac {V_{\text{S}}}{V_{\text{T}}}}}.}
In a long-channel device, there is no drain voltage dependence of the current onceVDS≫VT{\displaystyle V_{\text{DS}}\gg V_{\text{T}}}, but as channel length is reduceddrain-induced barrier loweringintroduces drain voltage dependence that depends in a complex way upon the device geometry (for example, the channel doping, the junction doping and so on). Frequently, threshold voltageVthfor this mode is defined as the gate voltage at which a selected value of currentID0occurs, for example,ID0= 1μA, which may not be the sameVth-value used in the equations for the following modes.
Some micropower analog circuits are designed to take advantage of subthreshold conduction.[30][31][32]By working in the weak-inversion region, the MOSFETs in these circuits deliver the highest possible transconductance-to-current ratio, namely:gm/ID=1/(nVT){\displaystyle g_{m}/I_{\text{D}}=1/\left(nV_{\text{T}}\right)}, almost that of a bipolar transistor.[33]
The subthresholdI–V curvedepends exponentially upon threshold voltage, introducing a strong dependence on any manufacturing variation that affects threshold voltage; for example: variations in oxide thickness, junction depth, or body doping that change the degree of drain-induced barrier lowering. The resulting sensitivity to fabricational variations complicates optimization for leakage and performance.[34][35]
WhenVGS>VthandVDS<VGS−Vth:
The transistor is turned on, and a channel has been created which allows current between the drain and the source. The MOSFET operates like a resistor, controlled by the gate voltage relative to both the source and drain voltages. The current from drain to source is modeled as:
ID=μnCoxWL((VGS−Vth)VDS−VDS22){\displaystyle I_{\text{D}}=\mu _{n}C_{\text{ox}}{\frac {W}{L}}\left(\left(V_{\text{GS}}-V_{\rm {th}}\right)V_{\text{DS}}-{\frac {{V_{\text{DS}}}^{2}}{2}}\right)}
whereμn{\displaystyle \mu _{n}}is the charge-carrier effective mobility,W{\displaystyle W}is the gate width,L{\displaystyle L}is the gate length andCox{\displaystyle C_{\text{ox}}}is the gate oxide capacitance per unit area. The transition from the exponential subthreshold region to the triode region is not as sharp as the equations suggest.[36][37][verification needed]
WhenVGS> VthandVDS≥ (VGS– Vth):
The switch is turned on, and a channel has been created, which allows current between the drain and source. Since the drain voltage is higher than the source voltage, the electrons spread out, and conduction is not through a narrow channel but through a broader, two- or three-dimensional current distribution extending away from the interface and deeper in the substrate. The onset of this region is also known aspinch-offto indicate the lack of channel region near the drain. Although the channel does not extend the full length of the device, the electric field between the drain and the channel is very high, and conduction continues. The drain current is now weakly dependent upon drain voltage and controlled primarily by the gate-source voltage, and modeled approximately as:
ID=μnCox2WL[VGS−Vth]2[1+λVDS].{\displaystyle I_{\text{D}}={\frac {\mu _{n}C_{\text{ox}}}{2}}{\frac {W}{L}}\left[V_{\text{GS}}-V_{\text{th}}\right]^{2}\left[1+\lambda V_{\text{DS}}\right].}
The additional factor involving λ, the channel-length modulation parameter, models current dependence on drain voltage due to theEarly effect, orchannel length modulation. According to this equation, a key design parameter, the MOSFET transconductance is:
gm=∂ID∂VGS=2IDVGS−Vth=2IDVov,{\displaystyle g_{m}={\frac {\partial I_{D}}{\partial V_{\text{GS}}}}={\frac {2I_{\text{D}}}{V_{\text{GS}}-V_{\text{th}}}}={\frac {2I_{\text{D}}}{V_{\text{ov}}}},}
where the combinationVov=VGS−Vthis called theoverdrive voltage,[38]and whereVDSsat=VGS−Vthaccounts for a small discontinuity inID{\displaystyle I_{\text{D}}}which would otherwise appear at the transition between the triode and saturation regions.
Another key design parameter is the MOSFET output resistanceroutgiven by:
rout=1λID{\displaystyle r_{\text{out}}={\frac {1}{\lambda I_{\text{D}}}}}.
routis the inverse ofgDSwheregDS=∂IDS∂VDS{\displaystyle g_{\text{DS}}={\frac {\partial I_{\text{DS}}}{\partial V_{\text{DS}}}}}.IDis the expression in saturation region.
If λ is taken as zero, an infinite output resistance of the device results that leads to unrealistic circuit predictions, particularly in analog circuits.
As the channel length becomes very short, these equations become quite inaccurate. New physical effects arise. For example, carrier transport in the active mode may become limited byvelocity saturation. When velocity saturation dominates, the saturation drain current is more nearly linear than quadratic inVGS. At even shorter lengths, carriers transport with near zero scattering, known as quasi-ballistic transport. In the ballistic regime, the carriers travel at an injection velocity that may exceed the saturation velocity and approaches theFermi velocityat high inversion charge density. In addition, drain-induced barrier lowering increases off-state (cutoff) current and requires an increase in threshold voltage to compensate, which in turn reduces the saturation current.[39][40][verification needed]
The occupancy of the energy bands in a semiconductor is set by the position of theFermi levelrelative to the semiconductor energy-band edges. Application of a source-to-substrate reverse bias of the source-body pn-junction introduces a split between the Fermi levels for electrons and holes, moving the Fermi level for the channel further from the band edge, lowering the occupancy of the channel. The effect is to increase the gate voltage necessary to establish the channel, as seen in the figure. This change in channel strength by application of reverse bias is called the "body effect."
Using an nMOS example, the gate-to-body biasVGBpositions the conduction-band energy levels, while the source-to-body bias VSBpositions the electron Fermi level near the interface, deciding occupancy of these levels near the interface, and hence the strength of the inversion layer or channel.
The body effect upon the channel can be described using a modification of the threshold voltage, approximated by the following equation:
whereVTBis the threshold voltage with substrate bias present, andVT0is the zero-VSBvalue of threshold voltage,γ{\displaystyle \gamma }is the body effect parameter, and 2φBis the approximate potential drop between surface and bulk across the depletion layer whenVSB= 0and gate bias is sufficient to ensure that a channel is present.[41]As this equation shows, a reverse biasVSB> 0causes an increase in threshold voltageVTBand therefore demands a larger gate voltage before the channel populates.
The body can be operated as a second gate, and is sometimes referred to as the "back gate"; the body effect is sometimes called the "back-gate effect".[42]
A variety of symbols are used for the MOSFET. The basic design is generally a line for the channel with the source and drain leaving it at right angles and then bending back at right angles into the same direction as the channel. Sometimes three line segments are used forenhancement modeand a solid line for depletion mode (seedepletion and enhancement modes). Another line is drawn parallel to the channel for the gate.
Thebulkorbodyconnection, if shown, is shown connected to the back of the channel with an arrow indicating pMOS or nMOS. Arrows always point from P to N, so an NMOS (N-channel in P-well or P-substrate) has the arrow pointing in (from the bulk to the channel). If the bulk is connected to the source (as is generally the case with discrete devices) it is sometimes angled to meet the source leaving the transistor. If the bulk is not shown (as is often the case in IC design as they are generally common bulk) an inversion symbol is sometimes used to indicate PMOS, alternatively an arrow on the source may be used in the same way as for bipolar transistors (out for nMOS, in for pMOS).
Comparison of enhancement-mode and depletion-mode MOSFET symbols, along withJFETsymbols. The orientation of the symbols, (most significantly the position of source relative to drain) is such that more positive voltages appear higher on the page than less positive voltages, implyingconventional currentflowing "down" the page:[43][44][45]
In schematics where G, S, D are not labeled, the detailed features of the symbol indicate which terminal is source and which is drain. For enhancement-mode and depletion-mode MOSFET symbols (in columns two and five), the source terminal is the one connected to the triangle. Additionally, in this diagram, the gate is shown as an "L" shape, whose input leg is closer to S than D, also indicating which is which. However, these symbols are often drawn with a T-shaped gate (as elsewhere on this page), so it is the triangle which must be relied upon to indicate the source terminal.
For the symbols in which the bulk, or body, terminal is shown, it is here shown internally connected to the source (i.e., the black triangles in the diagrams in columns 2 and 5). This is a typical configuration, but by no means the only important configuration. In general, the MOSFET is a four-terminal device, and in integrated circuits many of the MOSFETs share a body connection, not necessarily connected to the source terminals of all the transistors.
Digitalintegrated circuitssuch asmicroprocessorsand memory devices contain thousands to billions of integrated MOSFETs on each device, providing the basic switching functions required to implementlogic gatesanddata storage. Discrete devices are widely used in applications such asswitch mode power supplies,variable-frequency drivesand otherpower electronicsapplications where each device may be switching thousands of watts. Radio-frequency amplifiers up to theUHFspectrum use MOSFET transistors as analog signal and power amplifiers. Radio systems also use MOSFETs as oscillators, ormixersto convert frequencies. MOSFET devices are also applied in audio-frequency power amplifiers for public address systems,sound reinforcementand home and automobile sound systems[citation needed]
Following the development ofclean roomsto reduce contamination to levels never before thought necessary, and ofphotolithography[46]and theplanar processto allow circuits to be made in very few steps, the Si–SiO2system possessed the technical attractions of low cost of production (on a per circuit basis) and ease of integration. Largely because of these two factors, the MOSFET has become the most widely used type of transistor in theInstitution of Engineering and Technology(IET).[citation needed]
General Microelectronics introduced the first commercial MOS integrated circuit in 1964.[47]Additionally, the method of coupling two complementary MOSFETs (P-channel and N-channel) into one high/low switch, known as CMOS, means that digital circuits dissipate very little power except when actually switched.
Theearliest microprocessorsstarting in 1970 were allMOS microprocessors; i.e., fabricated entirely fromPMOS logicor fabricated entirely fromNMOS logic. In the 1970s,MOS microprocessorswere often contrasted withCMOS microprocessorsandbipolar bit-slice processors.[48]
The MOSFET is used in digital complementary metal–oxide–semiconductor (CMOS) logic,[49]which uses p- and n-channel MOSFETs as building blocks. Overheating is a major concern inintegrated circuitssince ever more transistors are packed into ever smaller chips. CMOS logic reduces power consumption because no current flows (ideally), and thus nopoweris consumed, except when the inputs tologic gatesare being switched. CMOS accomplishes this current reduction by complementing every nMOSFET with a pMOSFET and connecting both gates and both drains together. A high voltage on the gates will cause the nMOSFET to conduct and the pMOSFET not to conduct and a low voltage on the gates causes the reverse. During the switching time as the voltage goes from one state to another, both MOSFETs will conduct briefly. This arrangement greatly reduces power consumption and heat generation.
The growth of digital technologies like themicroprocessorhas provided the motivation to advance MOSFET technology faster than any other type of silicon-based transistor.[50]A big advantage of MOSFETs for digital switching is that the oxide layer between the gate and the channel prevents DC current from flowing through the gate, further reducing power consumption and giving a very large input impedance. The insulating oxide between the gate and channel effectively isolates a MOSFET in one logic stage from earlier and later stages, which allows a single MOSFET output to drive a considerable number of MOSFET inputs. Bipolar transistor-based logic (such asTTL) does not have such a high fanout capacity. This isolation also makes it easier for the designers to ignore to some extent loading effects between logic stages independently. That extent is defined by the operating frequency: as frequencies increase, the input impedance of the MOSFETs decreases.
The MOSFET's advantages in digital circuits do not translate into supremacy in allanalog circuits. The two types of circuit draw upon different features of transistor behavior. Digital circuits switch, spending most of their time either fully on or fully off. The transition from one to the other is only of concern with regards to speed and charge required. Analog circuits depend on operation in the transition region where small changes toVgscan modulate the output (drain) current. The JFET andbipolar junction transistor(BJT) are preferred for accurate matching (of adjacent devices in integrated circuits), highertransconductanceand certain temperature characteristics which simplify keeping performance predictable as circuit temperature varies.
Nevertheless, MOSFETs are widely used in many types of analog circuits because of their own advantages (zero gate current, high and adjustable output impedance and improved robustness vs. BJTs which can be permanently degraded by even lightly breaking down the emitter-base).[vague]The characteristics and performance of many analog circuits can be scaled up or down by changing the sizes (length and width) of the MOSFETs used. By comparison, in bipolar transistors follow a differentscaling law. MOSFETs' ideal characteristics regarding gate current (zero) and drain-source offset voltage (zero) also make them nearly ideal switch elements, and also makeswitched capacitoranalog circuits practical. In their linear region, MOSFETs can be used as precision resistors, which can have a much higher controlled resistance than BJTs. In high power circuits, MOSFETs sometimes have the advantage of not suffering fromthermal runawayas BJTs do.[dubious–discuss]This means that complete analog circuits can be made on a silicon chip in a much smaller space and with simpler fabrication techniques. MOSFETS are ideally suited to switch inductive loads because of tolerance toinductive kickback.
Some ICs combine analog and digital MOSFET circuitry on a singlemixed-signal integrated circuit, making the needed board space even smaller. This creates a need to isolate the analog circuits from the digital circuits on a chip level, leading to the use of isolation rings andsilicon on insulator(SOI). Since MOSFETs require more space to handle a given amount of power than a BJT, fabrication processes can incorporate BJTs and MOSFETs into a single device. Mixed-transistor devices are called bi-FETs (bipolar FETs) if they contain just one BJT-FET andBiCMOS(bipolar-CMOS) if they contain complementary BJT-FETs. Such devices have the advantages of both insulated gates and higher current density.
MOSFET analog switches use the MOSFET to pass analog signals when on, and as a high impedance when off. Signals flow in both directions across a MOSFET switch. In this application, the drain and source of a MOSFET exchange places depending on the relative voltages of the source and drain electrodes. The source is the more negative side for an N-MOS or the more positive side for a P-MOS. All of these switches are limited on what signals they can pass or stop by their gate-source, gate-drain and source–drain voltages; exceeding the voltage, current, or power limits will potentially damage the switch. SeePower MOSFETsubsection down below.
This analog switch uses a four-terminal simple MOSFET of either P or N type.
In the case of an n-type switch, the body is connected to the most negative supply (usually GND) and the gate is used as the switch control. Whenever the gate voltage exceeds the source voltage by at least a threshold voltage, the MOSFET conducts. The higher the voltage, the more the MOSFET can conduct. An N-MOS switch passes all voltages less thanVgate−Vtn. When the switch is conducting, it typically operates in the linear (or ohmic) mode of operation, since the source and drain voltages will typically be nearly equal.
In the case of a P-MOS, the body is connected to the most positive voltage, and the gate is brought to a lower potential to turn the switch on. The P-MOS switch passes all voltages higher thanVgate−Vtp(threshold voltageVtpis negative in the case of enhancement-mode P-MOS).
This "complementary" or CMOS type of switch uses one P-MOS and one N-MOS FET to counteract the limitations of the single-type switch. The FETs have their drains and sources connected in parallel, the body of the P-MOS is connected to the high potential (VDD) and the body of the N-MOS is connected to the low potential (gnd). To turn the switch on, the gate of the P-MOS is driven to the low potential and the gate of the N-MOS is driven to the high potential. For voltages betweenVDD−Vtnandgnd−Vtp, both FETs conduct the signal; for voltages less thangnd−Vtp, the N-MOS conducts alone; and for voltages greater thanVDD−Vtn, the P-MOS conducts alone.
The voltage limits for this switch are the gate-source, gate-drain and source-drain voltage limits for both FETs. Also, the P-MOS is typically two to three times wider than the N-MOS, so the switch will be balanced for speed in the two directions.
Tri-state circuitrysometimes incorporates a CMOS MOSFET switch on its output to provide for a low-ohmic, full-range output when on, and a high-ohmic, mid-level signal when off.
The primary criterion for the gate material is that it is a goodconductor. Highly dopedpolycrystalline siliconis an acceptable but certainly not ideal conductor, and also suffers from some more technical deficiencies in its role as the standard gate material. Nevertheless, there are several reasons favoring use of polysilicon:
While polysilicon gates have been the de facto standard for the last twenty years, they do have some disadvantages which have led to their likely future replacement by metal gates. These disadvantages include:
Present high performance CPUs use metal gate technology, together withhigh-κ dielectrics, a combination known ashigh-κ, metal gate(HKMG). The disadvantages of metal gates are overcome by a few techniques:[51]
As devices are made smaller, insulating layers are made thinner, often through steps ofthermal oxidationor localised oxidation of silicon (LOCOS). For nano-scaled devices, at some pointtunnelingof carriers through the insulator from the channel to the gate electrode takes place. To reduce the resultingleakagecurrent, the insulator can be made thinner by choosing a material with a higher dielectric constant. To see how thickness and dielectric constant are related, note thatGauss's lawconnects field to charge as:
withQ= charge density, κ = dielectric constant, ε0= permittivity of empty space andE= electric field. From this law it appears the same charge can be maintained in the channel at a lower field provided κ is increased. The voltage on the gate is given by:
withVG= gate voltage,Vch= voltage at channel side of insulator, andtins= insulator thickness. This equation shows the gate voltage will not increase when the insulator thickness increases, provided κ increases to keeptins/ κ = constant (see the article on high-κ dielectrics for more detail, and the section in this article ongate-oxide leakage).
The insulator in a MOSFET is a dielectric which can in any event be silicon oxide, formed byLOCOSbut many other dielectric materials are employed. The generic term for the dielectric is gate dielectric since the dielectric lies directly below the gate electrode and above the channel of the MOSFET.
The source-to-body and drain-to-bodyjunctionsare the object of much attention because of three major factors: their design affects thecurrent-voltage (I-V) characteristicsof the device, lowering output resistance, and also the speed of the device through the loading effect of the junctioncapacitances, and finally, the component of stand-by power dissipation due to junction leakage.
The drain induced barrier lowering of the threshold voltage andchannel length modulationeffects uponI-Vcurves are reduced by using shallow junction extensions. In addition,halodoping can be used, that is, the addition of very thin heavily doped regions of the same doping type as the body tight against the junction walls to limit the extent ofdepletion regions.[52]
The capacitive effects are limited by using raised source and drain geometries that make most of the contact area border thick dielectric instead of silicon.[53]
These various features of junction design are shown (withartistic license) in the figure.
Over the past decades, the MOSFET (as used for digital logic) has continually been scaled down in size; typical MOSFET channel lengths were once severalmicrometres, but modern integrated circuits are incorporating MOSFETs with channel lengths of tens of nanometers.Robert Dennard's work onscaling theorywas pivotal in recognising that this ongoing reduction was possible. Intel began production of a process featuring a 32 nm feature size (with the channel being even shorter) in late 2009. The semiconductor industry maintains a "roadmap", theITRS,[54]which sets the pace for MOSFET development. Historically, the difficulties with decreasing the size of the MOSFET have been associated with the semiconductor device fabrication process, the need to use very low voltages, and with poorer electrical performance necessitating circuit redesign and innovation (small MOSFETs exhibit higher leakage currents and lower output resistance).
Smaller MOSFETs are desirable for several reasons. The main reason to make transistors smaller is to pack more and more devices in a given chip area. This results in a chip with the same functionality in a smaller area, or chips with more functionality in the same area. Since fabrication costs for asemiconductor waferare relatively fixed, the cost per integrated circuits is mainly related to the number of chips that can be produced per wafer. Hence, smaller ICs allow more chips per wafer, reducing the price per chip. In fact, over the past 30 years the number of transistors per chip has been doubled every 2–3 years once a new technology node is introduced. For example, the number of MOSFETs in a microprocessor fabricated in a45 nmtechnology can well be twice as many as in a65 nmchip. This doubling of transistor density was first observed byGordon Moorein 1965 and is commonly referred to asMoore's law.[55]It is also expected that smaller transistors switch faster. For example, one approach to size reduction is a scaling of the MOSFET that requires all device dimensions to reduce proportionally. The main device dimensions are the channel length, channel width, and oxide thickness. When they are scaled down by equal factors, the transistor channel resistance does not change, while gate capacitance is cut by that factor. Hence, theRC delayof the transistor scales with a similar factor. While this has been traditionally the case for the older technologies, for the state-of-the-art MOSFETs reduction of the transistor dimensions does not necessarily translate to higher chip speed because the delay due to interconnections is more significant.
Producing MOSFETs with channel lengths much smaller than amicrometreis a challenge, and the difficulties of semiconductor device fabrication are always a limiting factor in advancing integrated circuit technology. Though processes such asALDhave improved fabrication for small components, the small size of the MOSFET (less than a few tens of nanometers) has created operational problems:
As MOSFET geometries shrink, the voltage that can be applied to the gate must be reduced to maintain reliability. To maintain performance, the threshold voltage of the MOSFET has to be reduced as well. As threshold voltage is reduced, the transistor cannot be switched from complete turn-off to complete turn-on with the limited voltage swing available; the circuit design is a compromise between strong current in theoncase and low current in theoffcase, and the application determines whether to favor one over the other. Subthreshold leakage (including subthreshold conduction, gate-oxide leakage and reverse-biased junction leakage), which was ignored in the past, now can consume upwards of half of the total power consumption of modern high-performance VLSI chips.[56][57]
The gate oxide, which serves as insulator between the gate and channel, should be made as thin as possible to increase the channel conductivity and performance when the transistor is on and to reduce subthreshold leakage when the transistor is off. However, with current gate oxides with a thickness of around 1.2nm(which in silicon is ~5atomsthick) thequantum mechanicalphenomenon ofelectron tunnelingoccurs between the gate and channel, leading to increased power consumption.Silicon dioxidehas traditionally been used as the gate insulator. Silicon dioxide however has a modest dielectric constant. Increasing the dielectric constant of the gate dielectric allows a thicker layer while maintaining a high capacitance (capacitance is proportional to dielectric constant and inversely proportional to dielectric thickness). All else equal, a higher dielectric thickness reduces thequantum tunnelingcurrent through the dielectric between the gate and the channel.
Insulators that have a largerdielectric constantthan silicon dioxide (referred to ashigh-κ dielectrics), such as group IVb metal silicates e.g.hafniumandzirconiumsilicates and oxides are being used to reduce the gate leakage from the 45 nanometer technology node onwards. On the other hand, the barrier height of the new gate insulator is an important consideration; the difference inconduction bandenergy between the semiconductor and the dielectric (and the corresponding difference invalence bandenergy) also affects leakage current level. For the traditional gate oxide, silicon dioxide, the former barrier is approximately 8eV. For many alternative dielectrics the value is significantly lower, tending to increase the tunneling current, somewhat negating the advantage of higher dielectric constant. The maximum gate-source voltage is determined by the strength of the electric field able to be sustained by the gate dielectric before significant leakage occurs. As the insulating dielectric is made thinner, the electric field strength within it goes up for a fixed voltage. This necessitates using lower voltages with the thinner dielectric.
To make devices smaller, junction design has become more complex, leading to higherdopinglevels, shallower junctions, "halo" doping and so forth,[58][59]all to decrease drain-induced barrier lowering (see the section onjunction design). To keep these complex junctions in place, the annealing steps formerly used to remove damage and electrically active defects must be curtailed[60]increasing junction leakage. Heavier doping is also associated with thinner depletion layers and more recombination centers that result in increased leakage current, even without lattice damage.
Drain-induced barrier lowering(DIBL) andVTroll off: Because of theshort-channel effect, channel formation is not entirely done by the gate, but now the drain and source also affect the channel formation. As the channel length decreases, the depletion regions of the source and drain come closer together and make the threshold voltage (VT) a function of the length of the channel. This is calledVTroll-off.VTalso becomes function of drain to source voltageVDS. As we increase theVDS, the depletion regions increase in size, and a considerable amount of charge is depleted by theVDS. The gate voltage required to form the channel is then lowered, and thus, theVTdecreases with an increase inVDS. This effect is called drain induced barrier lowering (DIBL).
For analog operation, good gain requires a high MOSFET output impedance, which is to say, the MOSFET current should vary only slightly with the applied drain-to-source voltage. As devices are made smaller, the influence of the drain competes more successfully with that of the gate due to the growing proximity of these two electrodes, increasing the sensitivity of the MOSFET current to the drain voltage. To counteract the resulting decrease in output resistance, circuits are made more complex, either by requiring more devices, for example thecascodeandcascade amplifiers, or by feedback circuitry usingoperational amplifiers, for example a circuit like that in the adjacent figure.
Thetransconductanceof the MOSFET decides its gain and is proportional to hole orelectron mobility(depending on device type), at least for low drain voltages. As MOSFET size is reduced, the fields in the channel increase and the dopant impurity levels increase. Both changes reduce the carrier mobility, and hence the transconductance. As channel lengths are reduced without proportional reduction in drain voltage, raising the electric field in the channel, the result is velocity saturation of the carriers, limiting the current and the transconductance.
Traditionally, switching time was roughly proportional to the gate capacitance of gates. However, with transistors becoming smaller and more transistors being placed on the chip,interconnect capacitance(the capacitance of the metal-layer connections between different parts of the chip) is becoming a large percentage of capacitance.[61][62]Signals have to travel through the interconnect, which leads to increased delay and lower performance.
The ever-increasing density of MOSFETs on an integrated circuit creates problems of substantial localized heat generation that can impair circuit operation. Circuits operate more slowly at high temperatures, and have reduced reliability and shorter lifetimes. Heat sinks and other cooling devices and methods are now required for many integrated circuits including microprocessors.Power MOSFETsare at risk ofthermal runaway. As their on-state resistance rises with temperature, if the load is approximately a constant-current load then the power loss rises correspondingly, generating further heat. When theheatsinkis not able to keep the temperature low enough, the junction temperature may rise quickly and uncontrollably, resulting in destruction of the device.
With MOSFETs becoming smaller, the number of atoms in the silicon that produce many of the transistor's properties is becoming fewer, with the result that control of dopant numbers and placement is more erratic. During chip manufacturing, random process variations affect all transistor dimensions: length, width, junction depths, oxide thicknessetc., and become a greater percentage of overall transistor size as the transistor shrinks. The transistor characteristics become less certain, more statistical. The random nature of manufacture means we do not know which particular example MOSFETs actually will end up in a particular instance of the circuit. This uncertainty forces a less optimal design because the design must work for a great variety of possible component MOSFETs. Seeprocess variation,design for manufacturability,reliability engineering, andstatistical process control.[63]
Modern ICs are computer-simulated with the goal of obtaining working circuits from the first manufactured lot. As devices are miniaturized, the complexity of the processing makes it difficult to predict exactly what the final devices look like, and modeling of physical processes becomes more challenging as well. In addition, microscopic variations in structure due simply to the probabilistic nature of atomic processes require statistical (not just deterministic) predictions. These factors combine to make adequate simulation and "right the first time" manufacture difficult.
The dual-gate MOSFET has atetrodeconfiguration, where both gates control the current in the device. It is commonly used for small-signal devices in radio frequency applications where biasing the drain-side gate at constant potential reduces the gain loss caused byMiller effect, replacing two separate transistors incascodeconfiguration. Other common uses in RF circuits include gain control and mixing (frequency conversion). Thetetrodedescription, though accurate, does not replicate the vacuum-tube tetrode. Vacuum-tube tetrodes, using a screen grid, exhibit much lower grid-plate capacitance and much higher output impedance and voltage gains than triode vacuum tubes. These improvements are commonly an order of magnitude (10 times) or considerably more. Tetrode transistors (whether bipolar junction or field-effect) do not exhibit improvements of such a great degree.
TheFinFETis a double-gatesilicon-on-insulatordevice, one of a number of geometries being introduced to mitigate the effects of short channels and reduce drain-induced barrier lowering. Thefinrefers to the narrow channel between source and drain. A thin insulating oxide layer on either side of the fin separates it from the gate. SOI FinFETs with a thick oxide on top of the fin are calleddouble-gateand those with a thin oxide on top as well as on the sides are calledtriple-gateFinFETs.[64][65]
There aredepletion-modeMOSFET devices, which are less commonly used than the standardenhancement-modedevices already described. These are MOSFET devices that are doped so that a channel exists even with zero voltage from gate to source. To control the channel, a negative voltage is applied to the gate (for an n-channel device), depleting the channel, which reduces the current flow through the device. In essence, the depletion-mode device is equivalent to anormally closed(on) switch, while the enhancement-mode device is equivalent to anormally open(off) switch.[66]
Due to their lownoise figurein theRFregion, and bettergain, these devices are often preferred tobipolarsinRF front-endssuch as inTVsets.
Depletion-mode MOSFET families include the BF960 bySiemensandTelefunken, and the BF980 in the 1980s byPhilips(later to becomeNXP Semiconductors), whose derivatives are still used inAGCand RFmixerfront-ends.
Metal–insulator–semiconductor field-effect-transistor,[67][68][69]orMISFET, is a more general term thanMOSFETand a synonym toinsulated-gate field-effect transistor(IGFET). All MOSFETs are MISFETs, but not all MISFETs are MOSFETs.
The gate dielectric insulator in a MISFET is a substrate oxide (hence typicallysilicon dioxide) in a MOSFET, but other materials can also be employed. Thegate dielectriclies directly below thegate electrodeand above thechannelof the MISFET. The termmetalis historically used for the gate material, even though now it is usuallyhighly dopedpolysiliconor some othernon-metal.
Insulator types may be:
For devices of equal current driving capability, n-channel MOSFETs can be made smaller than p-channel MOSFETs, due to p-channel charge carriers (holes) having lowermobilitythan do n-channel charge carriers (electrons), and producing only one type of MOSFET on a silicon substrate is cheaper and technically simpler. These were the driving principles in the design ofNMOS logicwhich uses n-channel MOSFETs exclusively. However, neglectingleakage current, unlike CMOS logic, NMOS logic consumes power even when no switching is taking place. With advances in technology, CMOS logic displaced NMOS logic in the mid-1980s to become the preferred process for digital chips.
Power MOSFETshave a different structure.[71]As with most power devices, the structure is vertical and not planar. Using a vertical structure, it is possible for the transistor to sustain both high blocking voltage and high current. The voltage rating of the transistor is a function of the doping and thickness of the N-epitaxiallayer (see cross section), while the current rating is a function of the channel width (the wider the channel, the higher the current). In a planar structure, the current and breakdown voltage ratings are both a function of the channel dimensions (respectively width and length of the channel), resulting in inefficient use of the "silicon estate". With the vertical structure, the component area is roughly proportional to the current it can sustain, and the component thickness (actually the N-epitaxial layer thickness) is proportional to the breakdown voltage.[72]
Power MOSFETs with lateral structure are mainly used in high-end audio amplifiers and high-power PA systems. Their advantage is a better behaviour in the saturated region (corresponding to the linear region of a bipolar transistor) than the vertical MOSFETs. Vertical MOSFETs are designed for switching applications.[73]
There areLDMOS(lateral double-diffused metal oxide semiconductor) andVDMOS(vertical double-diffused metal oxide semiconductor). Most power MOSFETs are made using this technology.
Semiconductor sub-micrometer and nanometer electronic circuits are the primary concern for operating within the normal tolerance in harshradiationenvironments likeouter space. One of the design approaches for making aradiation-hardened-by-design(RHBD) device is enclosed-layout-transistor (ELT). Normally, the gate of the MOSFET surrounds the drain, which is placed in the center of the ELT. The source of the MOSFET surrounds the gate. Another RHBD MOSFET is called H-Gate. Both of these transistors have very low leakage currents with respect to radiation. However, they are large in size and take up more space on silicon than a standard MOSFET. In older STI (shallow trench isolation) designs, radiation strikes near the silicon oxide region cause the channel inversion at the corners of the standard MOSFET due to accumulation of radiation induced trapped charges. If the charges are large enough, the accumulated charges affect STI surface edges along the channel near the channel interface (gate) of the standard MOSFET. This causes a device channel inversion to occur along the channel edges, creating an off-state leakage path. Subsequently, the device turns on; this process severely degrades the reliability of circuits. The ELT offers many advantages, including an improvement ofreliabilityby reducing unwanted surface inversion at the gate edges which occurs in the standard MOSFET. Since the gate edges are enclosed in ELT, there is no gate oxide edge (STI at gate interface), and thus the transistor off-state leakage is reduced very much. Low-power microelectronic circuits including computers, communication devices, and monitoring systems in space shuttles and satellites are very different from what is used on earth. They are radiation (high-speed atomic particles likeprotonandneutron,solar flaremagnetic energy dissipation in Earth's space, energeticcosmic rayslikeX-ray,gamma rayetc.) tolerant circuits. These special electronics are designed by applying different techniques using RHBD MOSFETs to ensure safe space journeys and safe space-walks of astronauts.
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https://en.wikipedia.org/wiki/MOSFET#Scaling
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