Ilya Sutskever (Hebrew: איליה סוצקבר; born 1986) is a computer scientist who specializes in machine learning. He has made several major contributions to the field of deep learning, including sequence-to-sequence learning, reasoning models, GPT models, and contributions to CLIP, DALL-E, and AlphaGo. With Alex Krizhevsky and Geoffrey Hinton, he co-created AlexNet, a convolutional neural network. One of the most highly cited computer scientists in history, he has won the NeurIPS Test of Time Award for his lasting impact on AI research three times in a row (2022–2024) and received the National Academy of Sciences Award for the Industrial Application of Science in 2026. Sutskever co-founded and was chief scientist at OpenAI, where he oversaw the research breakthroughs that led to large language models and to the launch of ChatGPT. He also led the research that led to reasoning models such as o1. In 2023, he was one of the members of OpenAI's board that ousted Sam Altman as its CEO; Altman was reinstated a week later, and Sutskever stepped down from the board. In June 2024, Sutskever co-founded the company Safe Superintelligence Inc., alongside Daniel Gross and Daniel Levy. Within a year, the company was valued at more than $30 billion. == Early life and education == Sutskever was born in 1986 into a Jewish family in Nizhny Novgorod, Russia (then Gorky, Russian SFSR, Soviet Union). At the age of 5, he immigrated to Israel with his family and grew up in Jerusalem. Sutskever proved to be a good student in school, and in eighth grade started taking classes at the Open University of Israel. At 16, he moved with his family to Canada, where he attended high school for a month before being admitted to the University of Toronto in Ontario as a third-year undergraduate student. At the University of Toronto, Sutskever received a bachelor's degree in mathematics in 2005, a master's degree in computer science in 2007, and a PhD in computer science in 2013. His doctoral advisor was Geoffrey Hinton. In 2012, Sutskever built AlexNet in collaboration with Geoffrey Hinton and Alex Krizhevsky. == Career and research == In 2012, Sutskever spent about two months as a postdoc with Andrew Ng at Stanford University. He then returned to the University of Toronto and joined Hinton's new research company DNNResearch, a spinoff of Hinton's research group. In 2013, Google acquired DNNResearch and hired Sutskever as a research scientist at Google Brain. At Google Brain, Sutskever worked with Oriol Vinyals and Quoc Viet Le to create the sequence-to-sequence learning algorithm, and worked on TensorFlow. He is also one of the AlphaGo paper's many co-authors. At the end of 2015, Sutskever left Google to become cofounder and chief scientist of the newly founded organization OpenAI. In 2022, Sutskever tweeted, "it may be that today's large neural networks are slightly conscious", which triggered debates about AI consciousness. He is considered to have played a key role in the development of ChatGPT, and later in leading the research that led to reasoning models. He is credited with establishing OpenAI’s scaling ethos. In 2023, he announced that he would co-lead OpenAI's new "Superalignment" project, which was trying to solve the alignment of superintelligences within four years. He wrote that even if superintelligence seems far off, it could happen this decade. Sutskever was formerly one of the six board members of the nonprofit entity that controlled OpenAI. In November 2023, the board fired Sam Altman, saying that "he was not consistently candid in his communications with the board". He authored a 52-page memo that relied heavily on information from Mira Murati, accusing Altman of lying, manipulating executives, and fostering internal division. Sutskever submitted the memo to the board after months of tension and dissatisfaction with Altman's leadership style, and ultimately joined the board in voting for Altman's termination. In an all-hands company meeting shortly after the board meeting, Sutskever said that firing Altman was "the board doing its duty", but the next week, he expressed regret at having participated in Altman's ouster. Altman's firing and OpenAI's co-founder Greg Brockman's resignation led three senior researchers to resign from OpenAI. After that, Sutskever stepped down from the OpenAI board and was absent from OpenAI's office. Some sources suggested he was leading the team remotely, while others said he no longer had access to the team's work. In May 2024, Sutskever announced his departure from OpenAI to focus on a new project that was "very personally meaningful" to him. His decision followed a turbulent period at OpenAI marked by leadership crises and internal debates about the direction of AI development and alignment protocols. Jan Leike, the other leader of the superalignment project, announced his departure hours later, citing an erosion of safety and trust in OpenAI's leadership. In June 2024, Sutskever announced Safe Superintelligence Inc., a new company he founded with Daniel Gross and Daniel Levy with offices in Palo Alto and Tel Aviv. In contrast to OpenAI, which releases revenue-generating products, Sutskever said the new company's "first product will be the safe superintelligence, and it will not do anything else up until then". In September 2024, the company announced that it had raised $1 billion from venture capital firms including Andreessen Horowitz, Sequoia Capital, DST Global, and SV Angel. In March 2025, Safe Superintelligence Inc. raised $2 billion more and reportedly reached a $32 billion valuation, notably due to Sutskever's reputation. In June 2025, SSI rejected an offer from Meta Platforms to buy the company. Sutskever became CEO of SSI shortly thereafter, after co-founder and CEO Gross left for Meta. In an October 2024 interview after winning the Nobel Prize in Physics, Geoffrey Hinton expressed support for Sutskever's decision to fire Altman, emphasizing concerns about AI safety. During the Musk v. Altman trial in 2026, Sutskever confirmed he had a $7 billion stake in OpenAI. === Awards and honors === In 2015, Sutskever was named in MIT Technology Review's 35 Innovators Under 35. In 2018, he was the keynote speaker at Nvidia Ntech 2018 and AI Frontiers Conference 2018. In 2022, he was elected a Fellow of the Royal Society (FRS). In 2023 and 2024, included in Time's list of the 100 most influential people in AI In 2022, 2023, and 2024, he won Neural Information Processing Systems’ Test of Time award, which recognizes papers that significantly shaped the AI field over at least ten years. In 2025, he received an honorary doctorate from his alma mater, the University of Toronto In 2026, he received the National Academy of Sciences Award for the Industrial Application of Science, presented for the first time in artificial intelligence.
Pocket (service)
Pocket, formerly known as Read It Later, was a social bookmarking service for storing, sharing and discovering web bookmarks, first released in 2007. Mozilla, the developer of Pocket, announced in May 2025 that it was discontinuing the service and would shut it down in July of that year. == History == Pocket was introduced in August 2007 as a Mozilla Firefox browser extension named Read It Later by Nathan (Nate) Weiner. Once his product was used by millions of people, he moved his office to Silicon Valley and four other people joined the Read It Later team. Weiner's intention was for the application to be like a TiVo directory for web content and to give users access to that content on any device. Read It Later obtained venture capital investments of US$2.5 million in 2011 and $5.0 million in 2012. The 2011 funding came from Foundation Capital, Baseline Ventures, Google Ventures, Founder Collective and unnamed angel investors. The company rejected an acquisition offer by Evernote after showing concerns that Evernote intended to shut down the Read It Later service and amalgamate its functionality into Evernote's main service. Initially, the Read It Later app was available in a free version and a paid version that included additional features. After the rebranding to Pocket, all paid features were made available in a free and advertisement-free app. In May 2014, a paid subscription service called Pocket Premium was introduced, adding server-side storage of articles and more powerful search tools. In June 2015, Pocket was included in Firefox, via a toolbar button and link to a user's Pocket list in the bookmark's menu. The integration was controversial, as users displayed concerns for the direct integration of a proprietary service into an open source application, and that it could not be completely disabled without editing advanced settings, unlike other third-party extensions. A Mozilla spokesperson stated that the feature was meant to leverage the service's popularity among Firefox users and clarified that all code related to the integration was open source. The spokesperson added that "[Mozilla had] gotten lots of positive feedback about the integration from users". On February 27, 2017, Pocket announced that it had been acquired by Mozilla Corporation, the commercial arm of Firefox's non-profit development group. Mozilla staff stated that Pocket would continue to operate as an independent subsidiary but that it would be leveraged as part of an ongoing "Context Graph" project. There were plans to open-source the server-side code of Pocket, though only parts of the project had been open-sourced as of 2024. On May 22, 2025, Mozilla announced that it would shut down Pocket on July 8, 2025. Exports of user data would be available until October 8, 2025, when accounts would be deleted. The email newsletter Pocket Hits was rebranded as Ten Tabs on June 12 as part of the closure, with it being changed to release only on weekdays. == Functions == The application allows the user to save an article or web page to remote servers for later reading. The article is sent to the user's Pocket list (synced to all of their devices) for offline reading. Pocket makes the article more readable by removing clutter and enabling the user to add tags and adjust text settings. == User base == The application had 17 million users and 1 billion saves, as of September 2015. Pocket was listed among Time magazine's 50 Best Android Applications for 2013. == Reception == Kent German of CNET said that "Read It Later is oh so incredibly useful for saving all the articles and news stories I find while commuting or waiting in line." Erez Zukerman of PC World said that supporting the developer is enough reason to buy what he deemed a "handy app". Bill Barol of Forbes said that although Read It Later works less well than Instapaper, "it makes my beloved Instapaper look and feel a little stodgy." In 2015, Pocket was awarded a Material Design Award for Adaptive Layout by Google for their Android application.
Coreu
COREU (French: Correspondance Européenne – Telex network of European correspondents, also EUKOR-Netzwerk in Austria) is a communication network of the European Union for the communication of the Council of the European Union, the European correspondents of the foreign ministries of the EU member states, permanent representatives of member states in Brussels, the European Commission, and the General Secretariat of the Council of the European Union. The European Parliament is not among the participants. COREU is the European equivalent of the American Secret Internet Protocol Router Network (SIPRNet, also known as Intelink-S). COREU's official aim is fast communication in case of crisis. The network enables a closer cooperation in matters regarding foreign affairs. In actuality the system's function exceeds that of mere communication, it also enables decision-making. COREU's first goal is to enable the exchange of information before and after decisions. Relaying upfront negotiations in preparation of meetings is the second goal. In addition, the system also allows the editing of documents and the decision-making, especially if there is little time. While the first two goals are preparatory measures for a shared foreign policy, the third is a methodical variant marked by practise that is defining for the image of the Common Foreign and Security Policy. == Members == (The following information dates from 2013): There is one representative in each of the capital cities in the EU.(since 1973) In Germany for example, this is the European correspondent (EU-KOR) from the Foreign Office. In Austria it is the European correspondent from the Referat II.1.a in the Federal Ministry for Europe, Integration and Foreign Affairs They are the correspondents (since 1982) for the European Commission They comprise the secretariat for the European Council They also make up the European External Action Service (EEAS) (responsible for foreign policy issues, since 1987) == Data volume and technical details == COREU functions as a spoke-hub distribution paradigm system with the hub in Brussels. The network is operated by the European Union Intelligence and Situation Centre (formerly Joint Situation Center, JSC). The technical infrastructure is located in a building of the European Council. COREU may be described as an advanced telex system with encrypted messages via dedicated terminals. Once a message has reached the destination, it is then redistributed via the local media. In contrast, messages of governments are transmitted via local media to the correspondents and from there delivered point-to-point to Brussels via COREU. In 2010, approximately 8500 communications had been distributed over this network. == History == A telex-based communication system under the name COREU was established in 1973. Originally, only the ministries of Foreign Affairs in the European capitals were connected to it. This telex system was replaced in 1997 by the mail system CORTESY (COREU Terminal Equipment System). The name was retained despite the technical innovation. COREU was reportedly compromised by hackers working for the People's Liberation Army Strategic Support Force, allowing for the theft of thousands of low-classified documents and diplomatic cables.
Application delivery network
An application delivery network (ADN) is a suite of technologies that, when deployed together, provide availability, security, visibility, and acceleration for Internet applications such as websites. ADN components provide supporting functionality that enables website content to be delivered to visitors and other users of that website, in a fast, secure, and reliable way. Gartner defines application delivery networking as the combination of WAN optimization controllers (WOCs) and application delivery controllers (ADCs). At the data center end of an ADN is the ADC, an advanced traffic management device that is often also referred to as a web switch, content switch, or multilayer switch, the purpose of which is to distribute traffic among a number of servers or geographically dislocated sites based on application specific criteria. In the branch office portion of an ADN is the WAN optimization controller, which works to reduce the number of bits that flow over the network using caching and compression, and shapes TCP traffic using prioritization and other optimization techniques. Some WOC components are installed on PCs or mobile clients, and there is typically a portion of the WOC installed in the data center. Application delivery networks are also offered by some CDN vendors. The ADC, one component of an ADN, evolved from layer 4-7 switches in the late 1990s when it became apparent that traditional load balancing techniques were not robust enough to handle the increasingly complex mix of application traffic being delivered over a wider variety of network connectivity options. == Application delivery techniques == The Internet was designed according to the end-to-end principle. This principle keeps the core network relatively simple and moves the intelligence as much as possible to the network end-points: the hosts and clients. An Application Delivery Network (ADN) enhances the delivery of applications across the Internet by employing a number of optimization techniques. Many of these techniques are based on established best-practices employed to efficiently route traffic at the network layer including redundancy and load balancing In theory, an Application Delivery Network (ADN) is closely related to a content delivery network. The difference between the two delivery networks lies in the intelligence of the ADN to understand and optimize applications, usually referred to as application fluency. Application Fluent Network (AFN) is based on the concept of Application Fluency to refer to WAN optimization techniques applied at Layer Four to Layer Seven of the OSI model for networks. Application Fluency implies that the network is fluent or intelligent in understanding and being able to optimize delivery of each application. Application Fluent Network is an addition of SDN capabilities. The acronym 'AFN' is used by Alcatel-Lucent Enterprise to refer to an Application Fluent Network. Application delivery uses one or more layer 4–7 switches, also known as a web switch, content switch, or multilayer switch to intelligently distribute traffic to a pool, also known as a cluster or farm, of servers. The application delivery controller (ADC) is assigned a single virtual IP address (VIP) that represents the pool of servers. Traffic arriving at the ADC is then directed to one of the servers in the pool (cluster, farm) based on a number of factors including application specific data values, application transport protocol, availability of servers, current performance metrics, and client-specific parameters. An ADN provides the advantages of load distribution, increase in capacity of servers, improved scalability, security, and increased reliability through application specific health checks. Increasingly the ADN comprises a redundant pair of ADC on which is integrated a number of different feature sets designed to provide security, availability, reliability, and acceleration functions. In some cases these devices are still separate entities, deployed together as a network of devices through which application traffic is delivered, each providing specific functionality that enhances the delivery of the application. == ADN optimization techniques == === TCP multiplexing === TCP Multiplexing is loosely based on established connection pooling techniques utilized by application server platforms to optimize the execution of database queries from within applications. An ADC establishes a number of connections to the servers in its pool and keeps the connections open. When a request is received by the ADC from the client, the request is evaluated and then directed to a server over an existing connection. This has the effect of reducing the overhead imposed by establishing and tearing down the TCP connection with the server, improving the responsiveness of the application. Some ADN implementations take this technique one step further and also multiplex HTTP and application requests. This has the benefit of executing requests in parallel, which enhances the performance of the application. === TCP optimization === There are a number of Request for Comments (RFCs) which describe mechanisms for improving the performance of TCP. Many ADN implement these RFCs in order to provide enhanced delivery of applications through more efficient use of TCP. The RFCs most commonly implemented are: Delayed Acknowledgements Nagle Algorithm Selective Acknowledgements Explicit Congestion Notification ECN Limited and Fast Retransmits Adaptive Initial Congestion Windows === Data compression and caching === ADNs also provide optimization of application data through caching and compression techniques. There are two types of compression used by ADNs today: industry standard HTTP compression and proprietary data reduction algorithms. It is important to note that the cost in CPU cycles to compress data when traversing a LAN can result in a negative performance impact and therefore best practices are to only utilize compression when delivering applications via a WAN or particularly congested high-speed data link. HTTP compression is asymmetric and transparent to the client. Support for HTTP compression is built into web servers and web browsers. All commercial ADN products currently support HTTP compression. A second compression technique is achieved through data reduction algorithms. Because these algorithms are proprietary and modify the application traffic, they are symmetric and require a device to reassemble the application traffic before the client can receive it. A separate class of devices known as WAN Optimization Controllers (WOC) provide this functionality, but the technology has been slowly added to the ADN portfolio over the past few years as this class of device continues to become more application aware, providing additional features for specific applications such as CIFS and SMB. == ADN reliability and availability techniques == === Advanced health checking === Advanced health checking is the ability of an ADN to determine not only the state of the server on which an application is hosted, but the status of the application it is delivering. Advanced health checking techniques allow the ADC to intelligently determine whether or not the content being returned by the server is correct and should be delivered to the client. This feature enables other reliability features in the ADN, such as resending a request to a different server if the content returned by the original server is found to be erroneous. === Load balancing algorithms === The load balancing algorithms found in today's ADN are far more advanced than the simplistic round-robin and least connections algorithms used in the early 1990s. These algorithms were originally loosely based on operating systems' scheduling algorithms, but have since evolved to factor in conditions peculiar to networking and application environments. It is more accurate to describe today's "load balancing" algorithms as application routing algorithms, as most ADN employ application awareness to determine whether an application is available to respond to a request. This includes the ability of the ADN to determine not only whether the application is available, but whether or not the application can respond to the request within specified parameters, often referred to as a service level agreement. Typical industry standard load balancing algorithms available today include: Round Robin Least Connections Fastest Response Time Weighted Round Robin Weighted Least Connections Custom values assigned to individual servers in a pool based on SNMP or other communication mechanism === Fault tolerance === The ADN provides fault tolerance at the server level, within pools or farms. This is accomplished by designating specific servers as a 'backup' that is activated automatically by the ADN in the event that the primary server(s) in the pool fail. The ADN also ensures application availability and reliability through its ability to seamlessly "failover"
Sentiment analysis
Sentiment analysis (also known as opinion mining) is the use of natural language processing, text analysis, computational linguistics, and biometrics to systematically identify, extract, quantify, and study affective states and subjective information. Sentiment analysis is widely applied to voice of the customer materials such as reviews and survey responses, online and social media, and healthcare materials for applications that range from marketing to customer service to clinical medicine. With the rise of deep language models, such as RoBERTa, more difficult data domains can be analyzed, e.g., news texts where authors typically express their opinion/sentiment less explicitly. == Types == A basic task in sentiment analysis is classifying the polarity of a given text at the document, sentence, or feature/aspect level—whether the expressed opinion in a document, a sentence or an entity feature/aspect is positive, negative, or neutral. Advanced, "beyond polarity" sentiment classification looks, for instance, at emotional states such as enjoyment, anger, disgust, sadness, fear, and surprise. Precursors to sentimental analysis include the General Inquirer, which provided hints toward quantifying patterns in text and, separately, psychological research that examined a person's psychological state based on analysis of their verbal behavior. Subsequently, the method described in a patent by Volcani and Fogel, looked specifically at sentiment and identified individual words and phrases in text with respect to different emotional scales. A current system based on their work, called EffectCheck, presents synonyms that can be used to increase or decrease the level of evoked emotion in each scale. Many other subsequent efforts were less sophisticated, using a mere polar view of sentiment, from positive to negative, such as work by Turney, and Pang who applied different methods for detecting the polarity of product reviews and movie reviews respectively. This work is at the document level. One can also classify a document's polarity on a multi-way scale, which was attempted by Pang and Snyder among others: Pang and Lee expanded the basic task of classifying a movie review as either positive or negative to predict star ratings on either a 3- or a 4-star scale, while Snyder performed an in-depth analysis of restaurant reviews, predicting ratings for various aspects of the given restaurant, such as the food and atmosphere (on a five-star scale). First steps to bringing together various approaches—learning, lexical, knowledge-based, etc.—were taken in the 2004 AAAI Spring Symposium where linguists, computer scientists, and other interested researchers first aligned interests and proposed shared tasks and benchmark data sets for the systematic computational research on affect, appeal, subjectivity, and sentiment in text. Even though in most statistical classification methods, the neutral class is ignored under the assumption that neutral texts lie near the boundary of the binary classifier, several researchers suggest that, as in every polarity problem, three categories must be identified. Moreover, it can be proven that specific classifiers such as the Max Entropy and SVMs can benefit from the introduction of a neutral class and improve the overall accuracy of the classification. There are in principle two ways for operating with a neutral class. Either, the algorithm proceeds by first identifying the neutral language, filtering it out and then assessing the rest in terms of positive and negative sentiments, or it builds a three-way classification in one step. This second approach often involves estimating a probability distribution over all categories (e.g. naive Bayes classifiers as implemented by the NLTK). Whether and how to use a neutral class depends on the nature of the data: if the data is clearly clustered into neutral, negative and positive language, it makes sense to filter the neutral language out and focus on the polarity between positive and negative sentiments. If, in contrast, the data are mostly neutral with small deviations towards positive and negative affect, this strategy would make it harder to clearly distinguish between the two poles. A different method for determining sentiment is the use of a scaling system whereby words commonly associated with having a negative, neutral, or positive sentiment are given an associated number on a −10 to +10 scale (most negative up to most positive) or simply from 0 to a positive upper limit such as +4. This makes it possible to adjust the sentiment of a given term relative to its environment (usually on the level of the sentence). When a piece of unstructured text is analyzed using natural language processing, each concept in the specified environment is given a score based on the way sentiment words relate to the concept and its associated score. This allows movement to a more sophisticated understanding of sentiment, because it is now possible to adjust the sentiment value of a concept relative to modifications that may surround it. Words, for example, that intensify, relax or negate the sentiment expressed by the concept can affect its score. Alternatively, texts can be given a positive and negative sentiment strength score if the goal is to determine the sentiment in a text rather than the overall polarity and strength of the text. There are various other types of sentiment analysis, such as aspect-based sentiment analysis, grading sentiment analysis (positive, negative, neutral), multilingual sentiment analysis and detection of emotions. === Subjectivity/objectivity identification === This task is commonly defined as classifying a given text (usually a sentence) into one of two classes: objective or subjective. This problem can sometimes be more difficult than polarity classification. The subjectivity of words and phrases may depend on their context and an objective document may contain subjective sentences (e.g., a news article quoting people's opinions). Moreover, as mentioned by Su, results are largely dependent on the definition of subjectivity used when annotating texts. However, Pang showed that removing objective sentences from a document before classifying its polarity helped improve performance. Subjective and objective identification, emerging subtasks of sentiment analysis to use syntactic, semantic features, and machine learning knowledge to identify if a sentence or document contains facts or opinions. Awareness of recognizing factual and opinions is not recent, having possibly first presented by Carbonell at Yale University in 1979. The term objective refers to the incident carrying factual information. Example of an objective sentence: 'To be elected president of the United States, a candidate must be at least thirty-five years of age.' The term subjective describes the incident contains non-factual information in various forms, such as personal opinions, judgment, and predictions, also known as 'private states'. In the example down below, it reflects a private states 'We Americans'. Moreover, the target entity commented by the opinions can take several forms from tangible product to intangible topic matters stated in Liu (2010). Furthermore, three types of attitudes were observed by Liu (2010), 1) positive opinions, 2) neutral opinions, and 3) negative opinions. Example of a subjective sentence: 'We Americans need to elect a president who is mature and who is able to make wise decisions.' This analysis is a classification problem. Each class's collections of words or phrase indicators are defined for to locate desirable patterns on unannotated text. For subjective expression, a different word list has been created. Lists of subjective indicators in words or phrases have been developed by multiple researchers in the linguist and natural language processing field states in Riloff et al. (2003). A dictionary of extraction rules has to be created for measuring given expressions. Over the years, in subjective detection, the features extraction progression from curating features by hand to automated features learning. At the moment, automated learning methods can further separate into supervised and unsupervised machine learning. Patterns extraction with machine learning process annotated and unannotated text have been explored extensively by academic researchers. However, researchers recognized several challenges in developing fixed sets of rules for expressions respectably. Much of the challenges in rule development stems from the nature of textual information. Six challenges have been recognized by several researchers: 1) metaphorical expressions, 2) discrepancies in writings, 3) context-sensitive, 4) represented words with fewer usages, 5) time-sensitive, and 6) ever-growing volume. Metaphorical expressions. The text contains metaphoric expression may impact on the performance on the extraction. Besides, metaphors take in different forms, which may have been contribu
CLEVER score
The CLEVER (Cross Lipschitz Extreme Value for nEtwork Robustness) score is a way of measuring the robustness of an artificial neural network towards adversarial attacks. It was developed by a team at the MIT-IBM Watson AI Lab in IBM Research and first presented at the 2018 International Conference on Learning Representations. It was mentioned and reviewed by Ian Goodfellow as well. It was adopted into an educational game Fool The Bank by Narendra Nath Joshi, Abhishek Bhandwaldar and Casey Dugan
Cryptographic multilinear map
A cryptographic n {\displaystyle n} -multilinear map is a kind of multilinear map, that is, a function e : G 1 × ⋯ × G n → G T {\displaystyle e:G_{1}\times \cdots \times G_{n}\rightarrow G_{T}} such that for any integers a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} and elements g i ∈ G i {\displaystyle g_{i}\in G_{i}} , e ( g 1 a 1 , … , g n a n ) = e ( g 1 , … , g n ) ∏ i = 1 n a i {\displaystyle e(g_{1}^{a_{1}},\ldots ,g_{n}^{a_{n}})=e(g_{1},\ldots ,g_{n})^{\prod _{i=1}^{n}a_{i}}} , and which in addition is efficiently computable and satisfies some security properties. It has several applications on cryptography, as key exchange protocols, identity-based encryption, and broadcast encryption. There exist constructions of cryptographic 2-multilinear maps, known as bilinear maps, however, the problem of constructing such multilinear maps for n > 2 {\displaystyle n>2} seems much more difficult and the security of the proposed candidates is still unclear. == Definition == === For n = 2 === In this case, multilinear maps are mostly known as bilinear maps or pairings, and they are usually defined as follows: Let G 1 , G 2 {\displaystyle G_{1},G_{2}} be two additive cyclic groups of prime order q {\displaystyle q} , and G T {\displaystyle G_{T}} another cyclic group of order q {\displaystyle q} written multiplicatively. A pairing is a map: e : G 1 × G 2 → G T {\displaystyle e:G_{1}\times G_{2}\rightarrow G_{T}} , which satisfies the following properties: Bilinearity ∀ a , b ∈ F q ∗ , ∀ P ∈ G 1 , Q ∈ G 2 : e ( a P , b Q ) = e ( P , Q ) a b {\displaystyle \forall a,b\in F_{q}^{},\ \forall P\in G_{1},Q\in G_{2}:\ e(aP,bQ)=e(P,Q)^{ab}} Non-degeneracy If g 1 {\displaystyle g_{1}} and g 2 {\displaystyle g_{2}} are generators of G 1 {\displaystyle G_{1}} and G 2 {\displaystyle G_{2}} , respectively, then e ( g 1 , g 2 ) {\displaystyle e(g_{1},g_{2})} is a generator of G T {\displaystyle G_{T}} . Computability There exists an efficient algorithm to compute e {\displaystyle e} . In addition, for security purposes, the discrete logarithm problem is required to be hard in both G 1 {\displaystyle G_{1}} and G 2 {\displaystyle G_{2}} . === General case (for any n) === We say that a map e : G 1 × ⋯ × G n → G T {\displaystyle e:G_{1}\times \cdots \times G_{n}\rightarrow G_{T}} is an n {\displaystyle n} -multilinear map if it satisfies the following properties: All G i {\displaystyle G_{i}} (for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} ) and G T {\displaystyle G_{T}} are groups of same order; if a 1 , … , a n ∈ Z {\displaystyle a_{1},\ldots ,a_{n}\in \mathbb {Z} } and ( g 1 , … , g n ) ∈ G 1 × ⋯ × G n {\displaystyle (g_{1},\ldots ,g_{n})\in G_{1}\times \cdots \times G_{n}} , then e ( g 1 a 1 , … , g n a n ) = e ( g 1 , … , g n ) ∏ i = 1 n a i {\displaystyle e(g_{1}^{a_{1}},\ldots ,g_{n}^{a_{n}})=e(g_{1},\ldots ,g_{n})^{\prod _{i=1}^{n}a_{i}}} ; the map is non-degenerate in the sense that if g 1 , … , g n {\displaystyle g_{1},\ldots ,g_{n}} are generators of G 1 , … , G n {\displaystyle G_{1},\ldots ,G_{n}} , respectively, then e ( g 1 , … , g n ) {\displaystyle e(g_{1},\ldots ,g_{n})} is a generator of G T {\displaystyle G_{T}} There exists an efficient algorithm to compute e {\displaystyle e} . In addition, for security purposes, the discrete logarithm problem is required to be hard in G 1 , … , G n {\displaystyle G_{1},\ldots ,G_{n}} . === Candidates === All the candidates multilinear maps are actually slightly generalizations of multilinear maps known as graded-encoding systems, since they allow the map e {\displaystyle e} to be applied partially: instead of being applied in all the n {\displaystyle n} values at once, which would produce a value in the target set G T {\displaystyle G_{T}} , it is possible to apply e {\displaystyle e} to some values, which generates values in intermediate target sets. For example, for n = 3 {\displaystyle n=3} , it is possible to do y = e ( g 2 , g 3 ) ∈ G T 2 {\displaystyle y=e(g_{2},g_{3})\in G_{T_{2}}} then e ( g 1 , y ) ∈ G T {\displaystyle e(g_{1},y)\in G_{T}} . The three main candidates are GGH13, which is based on ideals of polynomial rings; CLT13, which is based approximate GCD problem and works over integers, hence, it is supposed to be easier to understand than GGH13 multilinear map; and GGH15, which is based on graphs.