Sir Demis Hassabis (/ˈdɛ.mɪs/ DE-mis /hɑːˈsɑː.bis/ hah-SAH-bees; born Dimitrios Hassapis, Greek: Δημήτριος Χασάπης, 27 July 1976) is a British artificial intelligence (AI) researcher and entrepreneur. He is the chief executive officer and co-founder of Google DeepMind and Isomorphic Labs, and a UK Government AI Adviser. In 2024, Hassabis and John M. Jumper were jointly awarded the Nobel Prize in Chemistry for their AI research contributions to protein structure prediction. Hassabis is a Fellow of the Royal Society and has won awards for his research efforts, including the Breakthrough Prize, the Canada Gairdner International Award and the Lasker Award. He was appointed a CBE in 2017, and knighted in 2024 for his work on AI. He was also listed among the Time 100 most influential people in the world in 2017 and 2025, and was one of the "Architects of AI" collectively chosen as Time's 2025 Person of the Year. == Early life and education == Hassabis was born to Costas and Angela Hassapis. His father is a Greek Cypriot and his mother is a Chinese Singaporean. Demis grew up in North London. His original surname was "Hassapis" (Greek: Χασάπης), meaning "butcher" in Greek, but he later, according to Ingo Althöfer, "executed a point mutation by changing ‘p’ to ‘b’". One of his younger brothers still carries the original surname. In his early career, he was a video game AI programmer and designer, and an expert board games player. A child prodigy in chess from the age of four, when he first learnt chess by watching his father playing against his uncle, Hassabis reached master standard at the age of 13 with an Elo rating of 2300 and captained many of the England junior chess teams. He represented the University of Cambridge in the Oxford–Cambridge varsity chess matches of 1995, 1996 and 1997, winning a half blue. He first got interested in technology after buying his first computer in 1984, a ZX Spectrum 48K, funded from chess winnings. He taught himself how to program from books. He subsequently wrote his first AI program on a Commodore Amiga to play the reversi board game. Between 1988 and 1990, Hassabis was educated at Queen Elizabeth's School, Barnet, a boys' grammar school in North London. He was subsequently home-schooled by his parents for a year, before studying at the comprehensive school of Christ's College in East Finchley. He completed his A-level exams two years early at 16. === Bullfrog Productions === Asked by Cambridge University to take a gap year owing to his young age, Hassabis began his computer games career at Bullfrog Productions after entering an Amiga Power "Win-a-job-at-Bullfrog" competition. He began by playtesting on Syndicate and then at 17 co-designing and lead-programming on the 1994 game Theme Park, with the game's designer Peter Molyneux. Theme Park, a simulation video game, sold several million copies and inspired a whole genre of simulation sandbox games. Despite being offered a seven-figure sum to remain in the games industry, he turned it down. He earned enough from his gap year to pay his own way through university. === University of Cambridge === Hassabis left Bullfrog to study at Queens' College of the University of Cambridge, where he completed the Computer Science Tripos and graduated in 1997 with a double first. == Career and research == === Lionhead === After graduating from Cambridge, Hassabis worked at Lionhead Studios. Games designer Peter Molyneux, with whom Hassabis had worked at Bullfrog Productions, had recently founded the company. At Lionhead, Hassabis worked as lead AI programmer on the 2001 god game Black & White. === Elixir Studios === Hassabis left Lionhead in 1998 to found Elixir Studios, a London-based independent games developer, signing publishing deals with Eidos Interactive, Vivendi Universal and Microsoft. In addition to managing the company, Hassabis served as executive designer of the games Republic: The Revolution and Evil Genius. Each received BAFTA nominations for their interactive music scores, created by James Hannigan. The release of Elixir's first game, Republic: The Revolution, a highly ambitious and unusual political simulation game, was delayed due to its huge scope, which involved an AI simulation of the workings of an entire fictional country. The final game was reduced from its original vision and greeted with lukewarm reviews, receiving a Metacritic score of 62/100. Evil Genius, a tongue-in-cheek Austin Powers parody, fared much better with a score of 75/100. In April 2005 the intellectual property and technology rights were sold to various publishers and the studio was closed. === Neuroscience research === Following Elixir Studios, Hassabis returned to academia to obtain his PhD in cognitive neuroscience from UCL Queen Square Institute of Neurology in 2009 supervised by Eleanor Maguire. He sought to find inspiration in the human brain for new AI algorithms. He continued his neuroscience and artificial intelligence research as a visiting scientist jointly at Massachusetts Institute of Technology (MIT), in the lab of Tomaso Poggio, and Harvard University, before earning a Henry Wellcome postdoctoral research fellowship to the Gatsby Computational Neuroscience Unit at UCL in 2009 working with Peter Dayan. Working in the field of imagination, memory, and amnesia, he co-authored several influential papers published in Nature, Science, Neuron, and PNAS. His very first academic work, published in PNAS, was a landmark paper that showed systematically for the first time that patients with damage to their hippocampus, known to cause amnesia, were also unable to imagine themselves in new experiences. The finding established a link between the constructive process of imagination and the reconstructive process of episodic memory recall. Based on this work and a follow-up functional magnetic resonance imaging (fMRI) study, Hassabis developed a new theoretical account of the episodic memory system identifying scene construction, the generation and online maintenance of a complex and coherent scene, as a key process underlying both memory recall and imagination. This work received widespread coverage in the mainstream media and was listed in the top 10 scientific breakthroughs of the year by the journal Science. He later generalised these ideas to advance the notion of a 'simulation engine of the mind' whose role it was to imagine events and scenarios to aid with better planning. === DeepMind === Hassabis is the CEO and co-founder of DeepMind, a machine learning AI startup, founded in London in 2010 with Shane Legg and Mustafa Suleyman. Hassabis met Legg when both were postdocs at the Gatsby Computational Neuroscience Unit, and he and Suleyman had been friends through family. Hassabis also recruited his university friend and Elixir partner David Silver. DeepMind's mission is to "solve intelligence" and then use intelligence "to solve everything else". More concretely, DeepMind aims to combine insights from systems neuroscience with new developments in machine learning and computing hardware to unlock increasingly powerful general-purpose learning algorithms that will work towards the creation of an artificial general intelligence (AGI). The company has focused on training learning algorithms to master games, and in December 2013 it announced that it had made a pioneering breakthrough by training an algorithm called a Deep Q-Network (DQN) to play Atari games at a superhuman level by using only the raw pixels on the screen as inputs. DeepMind's early investors included several high-profile tech entrepreneurs. In 2014, Google purchased DeepMind for £400 million. Although most of the company has remained an independent entity based in London, DeepMind Health has since been directly incorporated into Google Health. Since the Google acquisition, the company has notched up a number of significant achievements, perhaps the most notable being the creation of AlphaGo, a program that defeated world champion Lee Sedol at the complex game of Go. Go had been considered a holy grail of AI, for its high number of possible board positions and resistance to existing programming techniques. However, AlphaGo beat European champion Fan Hui 5–0 in October 2015 before winning 4–1 against former world champion Lee Sedol in March 2016 and winning 3–0 against the world's top-ranked player Ke Jie in 2017. Additional DeepMind accomplishments include creating a neural Turing machine, reducing the energy used by the cooling systems in Google's data centres by 40%, and advancing research on AI safety. DeepMind has also been responsible for technical advances in machine learning, having produced a number of award-winning papers. In particular, the company has made significant advances in deep learning and reinforcement learning, and pioneered the field of deep reinforcement learning which combines these two methods. Hassabis has predicted that artificial intelligence will be "one of the most beneficial techn
Microsoft Whiteboard
Microsoft Whiteboard is a free multi-platform application, as well as an online service and a feature in Microsoft Teams, which simulates a virtual whiteboard and enables real-time collaboration between users. == Overview and features == Microsoft Whiteboard allows users to draw on a virtual whiteboard using input methods such as a stylus pen or a mouse and keyboard, and write down notes, draw connections between shareable ideas, and interact in real time. Microsoft Whiteboard is available to download on the following platforms and devices: Microsoft Windows (on Windows 10 or above) Android Apple iOS Surface Hub devices It is also available on the web and as a feature in Microsoft Teams. Microsoft Whiteboard allows users with Microsoft accounts to view, edit, and share whiteboards using the provided tools and options. The feature set includes tools for drawing, shapes, and media. Drawing in Microsoft Whiteboard is called inking. It works both on mobile devices and computers. The inking toolbar has customizable pencils, a ruler, a highlighter, an eraser, and an object selector. Whiteboard can recognize shapes drawn by hand and straighten them. Holding the Shift key on a computer while inking draws straight lines. Microsoft Whiteboard has keyboard shortcuts for some functions. Additional features include inserting sticky notes, text boxes, stickers, as well as images. Grid lines and colors are adjustable. Different templates can be inserted into the whiteboard. Users can also share their reactions. A feature limited to boards created in Microsoft Teams, is the ability to make them read-only; other participants from the meeting cannot edit them. == Reviews == PC Magazine gave Microsoft Whiteboard a score of 3.5 out of 5, praising the app's free availability and plentiful templates. It compared it to other, paid whiteboarding solutions, and concluded that Microsoft offers the best free one. Some of the cons, described by PCMag, include the inability to view boards without a Microsoft account and the inability to create custom templates.
Batch cryptography
Batch cryptography is a field of cryptology focused on the design of cryptographic protocols that perform operations—such as encryption, decryption, key exchange, and authentication—on multiple inputs simultaneously, rather than processing each input individually. Batching cryptographic operations can significantly reduce the marginal cost of handling individual inputs—a principle that was first introduced by Amos Fiat in 1989.
HashClash
HashClash was a volunteer computing project running on the Berkeley Open Infrastructure for Network Computing (BOINC) software platform to find collisions in the MD5 hash algorithm. It was based at Department of Mathematics and Computer Science at the Eindhoven University of Technology, and Marc Stevens initiated the project as part of his master's degree thesis. The project ended after Stevens defended his M.Sc. thesis in June 2007. However, SHA1 was added later, and the code repository was ported to git in 2017. The project was used to create a rogue certificate authority certificate in 2009.
NRENum.net
The NRENum.net service is an end-user ENUM service run by TERENA and the participating national research and education networking organisations (NRENs), primarily for academia. NRENum.net is considered as a complementary service and a valid alternative to the Golden ENUM tree. The domain nrenum.net is being populated in order to provide the infrastructure in DNS for storage of E.164 numbers. The NRENum.net service includes the operation of the Tier-0 root Domain Name Server(s) and the delegation of county codes to NRENum.net Registries. NRENum.net is a registered community trademark of TERENA. == Service description == E.164 Telephone Number Mapping (ENUM) is a standard protocol that is the result of work of the Internet Engineering Task Force's Telephone Number Mapping working group. ENUM translates a telephone number into a domain name. This allows users to continue to use the existing phone number formats they are familiar with, while allowing the call to be routed using DNS. This makes ENUM a quick, stable and cheap link between telecommunications systems and the Internet. RFC 3761 discusses the use of the Domain Name System for storage of E.164 numbers. More specifically, how DNS can be used for identifying available services connected to one E.164 number. The RIPE NCC provides DNS operations for e164.arpa (known as Golden ENUM tree) in accordance with the instructions from the Internet Architecture Board. The NRENum.net service is an end-user ENUM service run by TERENA and the participating NRENs primarily for academia. NRENum.net is considered as a complementary service and a valid alternative to the Golden ENUM tree. The domain nrenum.net is being populated in order to provide the infrastructure in DNS for storage of E.164 numbers. The NRENum.net service includes the operation of the Tier-0 root Domain Name Servers and the delegation of county codes to NRENum.net Registries. NRENum.net is a registered community trademark of TERENA. NRENum.net facilitates services such as Voice over IP and videoconferencing. NRENum.net tree refers to the tree structure where: Tier-0 root Domain Name Servers (technically one master and several secondary servers ensuring resilience) are run by the hosting organisations and coordinated by the NRENum.net Operations Team. Tier-1 Domain Name Servers are run by the NRENum.net (national or regional) Registries responsible for the country code(s) delegated. Tier-2 and lower DNS sub-delegations may be implemented, regulated by the national service policies. An NRENum.net Registry is an entity that is authorised by the NRENum.net Operations Team to operate the national or regional Tier-1 Domain Name Server and be responsible for the county code(s) delegated. In many countries there is a National Research and Education Networking organisation (NREN) that acts as the Registry of the country. An NRENum.net Registrar is responsible for the number/block registration in the Tier-1 DNS and a Number Validation Entity is responsible for the validation of the E.164 telephone numbers to be registered. The NREN may at the same time have the role of the NRENum.net Registry, Registrar and Validation Entity for the country code(s) delegated. A Registrant (end user) is an E.164 telephone number holder. Holders of E.164 numbers who want to be listed in the service must contact the appropriate NRENum.net Registrar. Number (block) delegation is the technical process of assigning country codes to national registries, or number blocks under country codes to end users. Number (block) registration is the technical process of configuring DNS and populating it with the appropriate ENUM records (i.e., adding NAPTR records to DNS) via registrars. The ITU-T strictly regulates the number structure of valid E.164 telephone numbers and assigns number blocks to national authorities (telecom regulators) or recently to global entities directly. The national authorities can further delegate the number ranges to local operators within the country or region. A virtual number has either a non-valid E.164 number structure (e.g., longer than 15 digits) or has a valid structure but is not assigned to any national authorities or operators. The number Validation Entity is responsible for checking the numbers to be registered to NRENum.net. == History == The idea for the NRENum.net service was conceived in 2006. NRENum.net became operational in August 2006, and was run by Bernie Höneisen, a staff member of SWITCH, and Kewin Stöckigt, a staff member of AARNet, as a private service, with technical support from SWITCH and the participants in the TERENA Task Force on Enhanced Communication Services (TF-ECS). When that task force completed its activities in 2008, TERENA agreed to take over the coordination of the NRENum.net service. By that time, nine NRENs had joined NRENum.net. The service continued to grow during the next years, and in March 2012 NRENum.net went global when RNP from Brazil joined the service as its 14th partificpant and the first outside Europe. In 2011, the participants decided to migrate the operation of the service's master Domain Name Server to NIIF and the operation of the two secondary DNSs to CARNET and SWITCH. In 2013, Internet2, AARNet and NORDUnet set up additional secondary Domain Name Servers for their regions, thereby completing the global distribution of DNS slaves and bringing the resilience of the NRENum.net infrastructure to a high level. == Governance == TERENA has established a lightweight global governance structure. The Global NRENum.net Governance Committee (GNGC) is the highest-level strategic body responsible for overall NRENum.net service definition, sustainability and long-term strategy. This includes formulating and recommending service governance principles and policies. Its members are nominated by the NRENum.net Registries in the various world regions, and are appointed by TERENA. The GNGC is composed of two members representing Europe, two representing the Asia-Pacific region, and two representing the Americas. The NRENum.net Operations Team is responsible for the day-to-day operations of the Tier-0 root DNSs and the handling of country code delegation requests. It may escalate technical or policy issues to the GNGC for discussion. TERENA is responsible for ensuring the correct and secure operations of the NRENum.net service performed by the NRENum.net Operations Team and governance by the GNGC. TERENA also supports the development of technical improvements to the NRENum.net service and promotes the deployment of NRENum.net worldwide. == Geographical deployment == Thirty-two county codes are delegated in the NRENum.net service. Below these are listed per world region. === Europe === === Asia-Pacific === === North America === +1 United States (Internet2) === Latin America === === Caribbean === === Africa === +262 Réunion, Mayotte (RENATER)
Application software
Application software is software that is intended for end-user use – not operating, administering or programming a computer. It includes programs such as word processors, web browsers, media players, and mobile applications used in daily tasks. An application (app, application program, software application) is any program that can be categorized as application software. Application is a subjective classification that is often used to differentiate from system and utility software. Application software represents the user-facing layer of computing systems, designed to translate complex system capabilities into task-oriented, goal-driven workflows. Unlike system software, which focuses on hardware orchestration and resource management, application software is centered on problem abstraction, user interaction, and domain-specific functionality. The abbreviation app became popular with the 2008 introduction of the iOS App Store, to refer to applications for mobile devices such as smartphones and tablets. Later, with the release of the Mac App Store in 2010 and the Windows Store in 2011, it began to be used to refer to end-user software in general, regardless of platform. Applications may be bundled with the computer and its system software or published separately. Applications may be proprietary or open-source. == Terminology == === Meaning program and software === When used as an adjective, application can have a broader meaning than that described in this article. For example, concepts such as application programming interface (API), application server, application virtualization, application lifecycle management and portable application refer to programs and software in general. === Distinction between system and application software === The distinction between system and application software is subjective and has been the subject of controversy. For example, one of the key questions in the United States v. Microsoft Corp. antitrust trial was whether Microsoft's Internet Explorer web browser was part of its Windows operating system or a separate piece of application software. As another example, the GNU/Linux naming controversy is, in part, due to disagreement about the relationship between the Linux kernel and the operating systems built over this kernel. In some types of embedded systems, the application software and the operating system software may be indistinguishable by the user, as in the case of software used to control a VCR, DVD player, or microwave oven. The above definitions may exclude some applications that may exist on some computers in large organizations. For an alternative definition of an app: see Application Portfolio Management. === Killer application === A killer application (killer app, coined in the late 1980s) is an application that is so popular that it causes demand for its host platform to increase. For example, VisiCalc was the first modern spreadsheet software for the Apple II and helped sell the then-new personal computers into offices. For the BlackBerry, it was its email software. === Software suite === As software suite consists of multiple applications bundled together. They usually have related functions, features, and user interfaces, and may be able to interact with each other, e.g. open each other's files. Business applications often come in suites, e.g. Microsoft Office, LibreOffice and iWork, which bundle together a word processor, a spreadsheet, etc.; but suites exist for other purposes, e.g. graphics or music. == Ways to classify == As there so many applications and since their attributes vary so dramatically, there are many different ways to classify them. === By legal aspects === Proprietary software is protected under an exclusive copyright, and a software license grants limited usage rights. Such applications may allow add-ons from third parties. Free and open-source software (FOSS) can be run, distributed, sold, and extended for any purpose. FOSS software released under a free license may be perpetual and also royalty-free. Perhaps, the owner, the holder or third-party enforcer of any right (copyright, trademark, patent, or ius in re aliena) are entitled to add exceptions, limitations, time decays or expiring dates to the license terms of use. Public-domain software is a type of FOSS that is royalty-free and can be run, distributed, modified, reversed, republished, or created in derivative works without any copyright attribution and therefore revocation. It can even be sold, but without transferring the public domain property to other single subjects. Public-domain software can be released under a (un)licensing legal statement, which enforces those terms and conditions for an indefinite duration (for a lifetime, or forever). === By platform === An application can be categorized by the host platform on which it runs. Notable platforms include operating system (native), web browser, cloud computing and mobile. For example a web application runs in a web browser whereas a more traditional, native application runs in the environment of a computer's operating system. There has been a contentious debate regarding web applications replacing native applications for many purposes, especially on mobile devices such as smartphones and tablets. Web apps have indeed greatly increased in popularity for some uses, but the advantages of applications make them unlikely to disappear soon, if ever. Furthermore, the two can be complementary, and even integrated. === Horizontal vs. vertical === Application software can be seen as either horizontal or vertical. Horizontal applications are more popular and widespread, because they are general purpose, for example word processors or databases. Vertical applications are niche products, designed for a particular type of industry or business, or department within an organization. Integrated suites of software will try to handle every specific aspect possible of, for example, manufacturing or banking worker, accounting, or customer service. === By purpose === There are many types of application software: Enterprise Addresses the needs of an entire organization's processes and data flows, across several departments, often in a large distributed environment. Examples include enterprise resource planning systems, customer relationship management (CRM) systems, data replication engines, and supply chain management software. Departmental Software is a sub-type of enterprise software with a focus on smaller organizations or groups within a large organization. (Examples include travel expense management and IT Helpdesk.) Enterprise infrastructure Provides common capabilities needed to support enterprise software systems. (Examples include databases, email servers, and systems for managing networks and security.) Application platform as a service (aPaaS) A cloud computing service that offers development and deployment environments for application services. Knowledge worker Lets users create and manage information, often for and individual media editors may aid in multiple information worker tasks. Content access Used primarily to access content without editing, but may include software that allows for content editing. Such software addresses the needs of individuals and groups to consume digital entertainment and published digital content. (Examples include media players, web browsers, and help browsers.) Educational Related to content access software, but has the content or features adapted for use by educators or students. For example, it may deliver evaluations (tests), track progress through material, or include collaborative capabilities. Simulation Simulates physical or abstract systems for either research, training, or entertainment purposes. Media development Generates print and electronic media for others to consume, most often in a commercial or educational setting. This includes graphic-art software, desktop publishing software, multimedia development software, HTML editors, digital-animation editors, digital audio and video composition, and many others. Engineering Used in developing hardware and software products. This includes computer-aided design (CAD), computer-aided engineering (CAE), computer language editing and compiling tools, integrated development environments, and application programmer interfaces. Entertainment Refers to video games, screen savers, programs to display motion pictures or play recorded music, and other forms of entertainment which can be experienced through the use of a computing device. == Taxonomy == This section is a taxonomy of kinds of applications. This organization is but one of many different ways to organize them. A kind is included in only one category even if it logically fits in multiple. === General-purpose === Calculator Spreadsheet Web browser Web mapping E-commerce Social media === Communication === Chat Email Presentation software Phone Messages Networking software Web conferencing === Documentation === Desktop
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.