Statistical machine translation (SMT) is a machine translation approach where translations are generated on the basis of statistical models whose parameters are derived from the analysis of bilingual text corpora. The statistical approach contrasts with the rule-based approaches to machine translation as well as with example-based machine translation, that superseded the previous rule-based approach that required explicit description of each and every linguistic rule, which was costly, and which often did not generalize to other languages. The first ideas of statistical machine translation were introduced by Warren Weaver in 1949, including the ideas of applying Claude Shannon's information theory. Statistical machine translation was re-introduced in the late 1980s and early 1990s by researchers at IBM's Thomas J. Watson Research Center. Before the introduction of neural machine translation, it was by far the most widely studied machine translation method. == Basis == The idea behind statistical machine translation comes from information theory. A document is translated according to the probability distribution p ( e | f ) {\displaystyle p(e|f)} that a string e {\displaystyle e} in the target language (for example, English) is the translation of a string f {\displaystyle f} in the source language (for example, French). The problem of modeling the probability distribution p ( e | f ) {\displaystyle p(e|f)} has been approached in a number of ways. One approach which lends itself well to computer implementation is to apply Bayes' theorem, that is p ( e | f ) ∝ p ( f | e ) p ( e ) {\displaystyle p(e|f)\propto p(f|e)p(e)} , where the translation model p ( f | e ) {\displaystyle p(f|e)} is the probability that the source string is the translation of the target string, and the language model p ( e ) {\displaystyle p(e)} is the probability of seeing that target language string. This decomposition is attractive as it splits the problem into two subproblems. Finding the best translation e ~ {\displaystyle {\tilde {e}}} is done by picking up the one that gives the highest probability: e ~ = a r g max e ∈ e ∗ p ( e | f ) = a r g max e ∈ e ∗ p ( f | e ) p ( e ) {\displaystyle {\tilde {e}}=arg\max _{e\in e^{}}p(e|f)=arg\max _{e\in e^{}}p(f|e)p(e)} . For a rigorous implementation of this one would have to perform an exhaustive search by going through all strings e ∗ {\displaystyle e^{}} in the native language. Performing the search efficiently is the work of a machine translation decoder that uses the foreign string, heuristics and other methods to limit the search space and at the same time keeping acceptable quality. This trade-off between quality and time usage can also be found in speech recognition. As the translation systems are not able to store all native strings and their translations, a document is typically translated sentence by sentence. Language models are typically approximated by smoothed n-gram models, and similar approaches have been applied to translation models, but this introduces additional complexity due to different sentence lengths and word orders in the languages. Statistical translation models were initially word based (Models 1-5 from IBM Hidden Markov model from Stephan Vogel and Model 6 from Franz-Joseph Och), but significant advances were made with the introduction of phrase based models. Later work incorporated syntax or quasi-syntactic structures. == Benefits == The most frequently cited benefits of statistical machine translation (SMT) over rule-based approach are: More efficient use of human and data resources There are many parallel corpora in machine-readable format and even more monolingual data. Generally, SMT systems are not tailored to any specific pair of languages. More fluent translations owing to use of a language model == Shortcomings == Corpus creation can be costly. Specific errors are hard to predict and fix. Results may have superficial fluency that masks translation problems. Statistical machine translation usually works less well for language pairs with significantly different word order. The benefits obtained for translation between Western European languages are not representative of results for other language pairs, owing to smaller training corpora and greater grammatical differences. == Word-based translation == In word-based translation, the fundamental unit of translation is a word in some natural language. Typically, the number of words in translated sentences are different, because of compound words, morphology and idioms. The ratio of the lengths of sequences of translated words is called fertility, which tells how many foreign words each native word produces. Necessarily it is assumed by information theory that each covers the same concept. In practice this is not really true. For example, the English word corner can be translated in Spanish by either rincón or esquina, depending on whether it is to mean its internal or external angle. Simple word-based translation cannot translate between languages with different fertility. Word-based translation systems can relatively simply be made to cope with high fertility, such that they could map a single word to multiple words, but not the other way about. For example, if we were translating from English to French, each word in English could produce any number of French words— sometimes none at all. But there is no way to group two English words producing a single French word. An example of a word-based translation system is the freely available GIZA++ package (GPLed), which includes the training program for IBM models and HMM model and Model 6. The word-based translation is not widely used today; phrase-based systems are more common. Most phrase-based systems are still using GIZA++ to align the corpus. The alignments are used to extract phrases or deduce syntax rules. And matching words in bi-text is still a problem actively discussed in the community. Because of the predominance of GIZA++, there are now several distributed implementations of it online. == Phrase-based translation == In phrase-based translation, the aim is to reduce the restrictions of word-based translation by translating whole sequences of words, where the lengths may differ. The sequences of words are called blocks or phrases. These are typically not linguistic phrases, but phrasemes that were found using statistical methods from corpora. It has been shown that restricting the phrases to linguistic phrases (syntactically motivated groups of words, see syntactic categories) decreased the quality of translation. The chosen phrases are further mapped one-to-one based on a phrase translation table, and may be reordered. This table could be learnt based on word-alignment, or directly from a parallel corpus. The second model is trained using the expectation maximization algorithm, similarly to the word-based IBM model. == Syntax-based translation == Syntax-based translation is based on the idea of translating syntactic units, rather than single words or strings of words (as in phrase-based MT), i.e. (partial) parse trees of sentences/utterances. Until the 1990s, with advent of strong stochastic parsers, the statistical counterpart of the old idea of syntax-based translation did not take off. Examples of this approach include DOP-based MT and later synchronous context-free grammars. == Hierarchical phrase-based translation == Hierarchical phrase-based translation combines the phrase-based and syntax-based approaches to translation. It uses synchronous context-free grammar rules, but the grammars can be constructed by an extension of methods for phrase-based translation without reference to linguistically motivated syntactic constituents. This idea was first introduced in Chiang's Hiero system (2005). == Language models == A language model is an essential component of any statistical machine translation system, which aids in making the translation as fluent as possible. It is a function that takes a translated sentence and returns the probability of it being said by a native speaker. A good language model will for example assign a higher probability to the sentence "the house is small" than to "small the is house". Other than word order, language models may also help with word choice: if a foreign word has multiple possible translations, these functions may give better probabilities for certain translations in specific contexts in the target language. == Systems implementing statistical machine translation == Google Translate (started transition to neural machine translation in 2016) Microsoft Translator (started transition to neural machine translation in 2016) Yandex.Translate (switched to hybrid approach incorporating neural machine translation in 2017) == Challenges with statistical machine translation == Problems with statistical machine translation include: === Sentence alignment === Single sentences in one language can be found translated into several sentences in the o
Cone tracing
Cone tracing and beam tracing are a derivative of the ray tracing algorithm that replaces rays, which have no thickness, with thick rays. == Principles == In ray tracing, rays are often modeled as geometric ray with no thickness to perform efficient geometric queries such as a ray-triangle intersection. From a physics of light transport point of view, however, this is an inaccurate model provided the pixel on the sensor plane has non-zero area. In the simplified pinhole camera optics model, the energy reaching the pixel comes from the integral of radiance from the solid angle by which the sensor pixel sees the scene through the pinhole at the focal plane. This yields the key notion of pixel footprint on surfaces or in the texture space, which is the back projection of the pixel on to the scene. Note that this approach can also represent a lens-based camera and thus depth of field effects, using a cone whose cross-section decreases from the lens size to zero at the focal plane, and then increases. Real optical system do not focus on exact points because of diffraction and imperfections. This can be modeled with a point spread function (PSF) weighted within a solid angle larger than the pixel. From a signal processing point of view, ignoring the point spread function and approximating the integral of radiance with a single, central sample (through a ray with no thickness) can lead to strong aliasing because the "projected geometric signal" has very high frequencies exceeding the Nyquist-Shannon maximal frequency that can be represented using the uniform pixel sampling rate. The physically based image formation model can be approximated by the convolution with the point spread function assuming the function is shift-invariant and linear. In practice, techniques such as multisample anti-aliasing estimate this cone-based model by oversampling the signal and then performing a convolution (the reconstruction filter). The backprojected cone footprint onto the scene can also be used to directly pre-filter the geometry and textures of the scene. Note that contrary to intuition, the reconstruction filter should not be the pixel footprint (as the pinhole camera model would suggest), since a box filter has poor spectral properties. Conversely, the ideal sinc function is not practical, having infinite support with possibly negative values which often creates ringing artifacts due to the Gibbs phenomenon. A Gaussian or a Lanczos filter are considered good compromises. == Computer graphics models == Cone and Beam early papers rely on different simplifications: the first considers a circular section and treats the intersection with various possible shapes. The second treats an accurate pyramidal beam through the pixel and along a complex path, but it only works for polyhedrical shapes. Cone tracing solves certain problems related to sampling and aliasing, which can plague conventional ray tracing. However, cone tracing creates a host of problems of its own. For example, just intersecting a cone with scene geometry leads to an enormous variety of possible results. For this reason, cone tracing has remained mostly unpopular. In recent years, increases in computer speed have made Monte Carlo algorithms like distributed ray tracing - i.e. stochastic explicit integration of the pixel - much more used than cone tracing because the results are exact provided enough samples are used. But the convergence is so slow that even in the context of off-line rendering a huge amount of time can be required to avoid noise. Differential cone-tracing, considering a differential angular neighborhood around a ray, avoids the complexity of exact geometry intersection but requires a LOD representation of the geometry and appearance of the objects. MIPmapping is an approximation of it limited to the integration of the surface texture within a cone footprint. Differential ray-tracing extends it to textured surfaces viewed through complex paths of cones reflected or refracted by curved surfaces. Raymarching methods over signed distance fields (SDFs) naturally allow easy use of cone-like tracing, at zero additional cost to the tracing, and both speeds up tracing and improves quality. Voxel cone tracing is a real-time algorithm that uses a hierarchical voxel representation of scene geometry, such as a sparse voxel octree, to support fast cone tracing for indirect illumination. This approach allows for the approximation of effects like glossy reflections and ambient occlusion at interactive framerates without the need for precomputation.
Change data capture
In databases, change data capture (CDC) is a set of software design patterns used to determine and track the data that has changed (the "deltas") so that action can be taken using the changed data. The result is a delta-driven dataset. CDC is an approach to data integration that is based on the identification, capture and delivery of the changes made to enterprise data sources. For instance it can be used for incremental update of data loading. CDC occurs often in data warehouse environments since capturing and preserving the state of data across time is one of the core functions of a data warehouse, but CDC can be utilized in any database or data repository system. == Methodology == System developers can set up CDC mechanisms in a number of ways and in any one or a combination of system layers from application logic down to physical storage. In a simplified CDC context, one computer system has data believed to have changed from a previous point in time, and a second computer system needs to take action based on that changed data. The former is the source, the latter is the target. It is possible that the source and target are the same system physically, but that would not change the design pattern logically. Multiple CDC solutions can exist in a single system. === Timestamps on rows === Tables whose changes must be captured may have a column that represents the time of last change. Names such as LAST_UPDATE, LAST_MODIFIED, etc. are common. Any row in any table that has a timestamp in that column that is more recent than the last time data was captured is considered to have changed. Timestamps on rows are also frequently used for optimistic locking so this column is often available. === Version numbers on rows === Database designers give tables whose changes must be captured a column that contains a version number. Names such as VERSION_NUMBER, etc. are common. One technique is to mark each changed row with a version number. A current version is maintained for the table, or possibly a group of tables. This is stored in a supporting construct such as a reference table. When a change capture occurs, all data with the latest version number is considered to have changed. Once the change capture is complete, the reference table is updated with a new version number. (Do not confuse this technique with row-level versioning used for optimistic locking. For optimistic locking each row has an independent version number, typically a sequential counter. This allows a process to atomically update a row and increment its counter only if another process has not incremented the counter. But CDC cannot use row-level versions to find all changes unless it knows the original "starting" version of every row. This is impractical to maintain.) === Status indicators on rows === This technique can either supplement or complement timestamps and versioning. It can configure an alternative if, for example, a status column is set up on a table row indicating that the row has changed (e.g., a boolean column that, when set to true, indicates that the row has changed). Otherwise, it can act as a complement to the previous methods, indicating that a row, despite having a new version number or a later date, still shouldn't be updated on the target (for example, the data may require human validation). === Time/version/status on rows === This approach combines the three previously discussed methods. As noted, it is not uncommon to see multiple CDC solutions at work in a single system, however, the combination of time, version, and status provides a particularly powerful mechanism and programmers should utilize them as a trio where possible. The three elements are not redundant or superfluous. Using them together allows for such logic as, "Capture all data for version 2.1 that changed between 2005-06-01 00:00 and 2005-07-01 00:00 where the status code indicates it is ready for production." === Triggers on tables === May include a publish/subscribe pattern to communicate the changed data to multiple targets. In this approach, triggers log events that happen to the transactional table into another queue table that can later be "played back". For example, imagine an Accounts table, when transactions are taken against this table, triggers would fire that would then store a history of the event or even the deltas into a separate queue table. The queue table might have schema with the following fields: Id, TableName, RowId, Timestamp, Operation. The data inserted for our Account sample might be: 1, Accounts, 76, 2008-11-02 00:15, Update. More complicated designs might log the actual data that changed. This queue table could then be "played back" to replicate the data from the source system to a target. Data capture offers a challenge in that the structure, contents and use of a transaction log is specific to a database management system. Unlike data access, no standard exists for transaction logs. Most database management systems do not document the internal format of their transaction logs, although some provide programmatic interfaces to their transaction logs (for example: Oracle, DB2, SQL/MP, SQL/MX and SQL Server 2008). Other challenges in using transaction logs for change data capture include: Coordinating the reading of the transaction logs and the archiving of log files (database management software typically archives log files off-line on a regular basis). Translation between physical storage formats that are recorded in the transaction logs and the logical formats typically expected by database users (e.g., some transaction logs save only minimal buffer differences that are not directly useful for change consumers). Dealing with changes to the format of the transaction logs between versions of the database management system. Eliminating uncommitted changes that the database wrote to the transaction log and later rolled back. Dealing with changes to the metadata of tables in the database. CDC solutions based on transaction log files have distinct advantages that include: minimal impact on the database (even more so if one uses log shipping to process the logs on a dedicated host). no need for programmatic changes to the applications that use the database. low latency in acquiring changes. transactional integrity: log scanning can produce a change stream that replays the original transactions in the order they were committed. Such a change stream include changes made to all tables participating in the captured transaction. no need to change the database schema == Confounding factors == As often occurs in complex domains, the final solution to a CDC problem may have to balance many competing concerns. === Unsuitable source systems === Change data capture both increases in complexity and reduces in value if the source system saves metadata changes when the data itself is not modified. For example, some Data models track the user who last looked at but did not change the data in the same structure as the data. This results in noise in the Change Data Capture. === Tracking the capture === Actually tracking the changes depends on the data source. If the data is being persisted in a modern database then Change Data Capture is a simple matter of permissions. Two techniques are in common use: Tracking changes using database triggers Reading the transaction log as, or shortly after, it is written. If the data is not in a modern database, CDC becomes a programming challenge. === Push versus pull === Push: the source process creates a snapshot of changes within its own process and delivers rows downstream. The downstream process uses the snapshot, creates its own subset and delivers them to the next process. Pull: the target that is immediately downstream from the source, prepares a request for data from the source. The downstream target delivers the snapshot to the next target, as in the push model. === Alternatives === Sometimes the slowly changing dimension is used as an alternative method. CDC and SCD are similar in that both methods can detect changes in a data set. The most common forms of SCD are type 1 (overwrite), type 2 (maintain history) or 3 (only previous and current value). SCD 2 can be useful if history is needed in the target system. CDC overwrites in the target system (akin to SCD1), and is ideal when only the changed data needs to arrive at the target, i.e. a delta-driven dataset.
Social media use in education
Social media in education is the use of social media to enhance education. Social media are "a group of Internet-based applications...that allow the creation and exchange of user-generated content". It is also known as the read/write web. As time went on and technology evolved, social media has been an integral part of people's lives, including students, scholars, and teachers. However, social media are controversial because, in addition to providing new means of connection, critics claim that they damage self-esteem, shorten attention spans, and increase mental health issues. A 2016 dissertation presented surveys that focused on the impact of social media. It reported that 54.6% of students believed that social media affected their studies positively (38% agree, 16.6% strongly agree). About 40% disagreed, and 4.7% of students strongly disagreed. 53% of female students reported that social media negatively impacted their studies. Among male students, 40% agreed that social media had a negative impact on studies, while 59% disagreed. A 2023 article dives deep into the rewards system of the brain in response to social media. This study compares the social rewards system in our brain to those from social media. From ages 10-12, most are receiving a cell phone, social rewards in the brain start to feel more satisfying. Leading to adulthood, the effects of social rewards are less likely to feel reliant on feedback from peers. Equivalent to a more mature prefrontal cortex, this enables a better management of their emotional reaction to these social rewards, meaning a more balanced and controlled reaction. == History == A survey from Cambridge International of nearly 20,000 teachers and students (ages 12–19) from 100 countries found that 48% of students use a desktop computer in class, 42% uses phones, 33% use interactive whiteboards and 20% use tablets. Desktop computers are more used than tablets. Teachers were abandoning the "no phones at school" rule. A 2024 research survey through Common Sense Education reported 54% of age 8-12 and 69% of ages 13-18 social media is an extensive distraction from homework. === United States === The long-running technology boom accelerated after the millennium. As of 2018, 95% of US teenage students had access to a smartphone and 45% said they were online almost constantly. In the early days of social media, access to technology was a significant issue as many students did not own not compatible devices and school budgets were often insufficient to purchase devices for student use. Despite backlash, Missouri passed a law that prohibited teachers from communicating privately with students over social media in 2011. Supporters were concerned that online communication between underage students and faculty could lead to inappropriate relationships. Some schools adopted a "Bring Your Own Device" (BYOD) policy, allowing students to bring Internet-accessing devices, such as phones or tablets to class. During the pandemic, the federal government offered funds that allowed more schools to purchase devices. Over time, more students acquired phones with social media access. Personal devices increased student satisfaction, but reduced teachers' ability to control device use in their classrooms. A 2018 Pew Research study reported that 95% of teenagers had a phone and used social media consistently. === Canada === The Peel District School Board (PDSB) in Ontario accepted the use of social media in the classroom. In 2013, the PDSB introduced BYOD and unblocked many social media sites. That was later replaced by a policy that dealt specifically with social media. == Uses == === Classroom === In the classroom, social media offers a way to systematically distribute and gather information from students. Teachers can supply documents, and audio/video media to students for immediate or later use. One study on higher education reported that devices and social media: created opportunities for interaction provided occasions for collaboration sped up information access offered more ways to learn situated learning. Frustrations included anti-technology instructors, device challenges, and devices as a distraction. Social media in classrooms can have a negative effect. A Yale University publication reported that students who used laptops in class for non-academic reasons had poorer performance. Students spent most of their time on social media, shopping, and other personal activities. Social media has helped many educators mentor their students more effectively. === Outside of class === Social media offer a venue for video calls, stories, feeds, and game playing that can enhance the learning process. Teachers can utilize social media to communicate with their students. Social media can provide students with resources that they can utilize in essays, projects, and presentations. Students can easily access comments made by teachers and peers and offer feedback to teachers. Social media can offer students the opportunity to collaborate by sharing information without requiring face to face meetings. Social media can allow students to more easily connect with experts, to go beyond course materials. Instructors in a 2010 study reported that online technologies (social media) can help students become comfortable having discussions outside the classroom better than traditional means. Teachers may face some risk when using social media outside the classroom, without appropriate work rules. Studies explores how college students' engagement with social media platforms influences their communication preferences and habits, particularly in relation to using school email for academic purposes. === Professional development === Social media can aid professional development, as teachers become students, enhancing knowledge transfer, skill master, and collaboration. === Non-academic uses === Schools can use social media to make public announcements. Teachers and administrators can communicate other important information to parents and students and to receive feedback from them. Families can keep up with school events and policies. === Ecology education === The potential of using social media in ecological, nature and forest education include: virtual nature groups can help promote good habits in forest tourism and recreation (nature ethics), by entering general rules in the regulations by administrators, e.g. "DO NOT PICK UP PLANTS UNKNOWN TO US", which is to protects rare species from pointless picking. social media activity motivates people to learn about nature in the field, allows them to gain knowledge, dispels popular myths, enables contact with scientists and practitioners, promotes valuable literature, websites, and at the same time reveals distortions and substantive errors in popular news services. contact is not only virtual. Despite financial barriers and distance, Internet users organize nature conventions. Such meetings are an opportunity not only to make friends, but also to learn about nature together and have fun. the possibility of contact between scientists and nature lovers via Facebook has become a source of cooperation in species inventory, e.g. the online campaign of the NATRIX Herpetological Society, which consists not only of collecting reports of observations of the smooth snake by Internet users, but also of drawing attention to the biology and threats to this species. Social media has become a place where ecology education quickly reaches people of different ages and social statuses. The nature groups that have been created, in which nature lovers, biologists, foresters and scientists participate, can have a real impact on the state of knowledge and data collection through citizen science. == Apps and services == Social media can allow students to participate in their field by working with organizations outside the classroom. By offering easier access to peers outside the classroom, students can broaden their perspectives and find support resources. Social media aided learning outside of the classroom through collaboration and innovation. One specific study, "Exploring education-related use of social media," called this "audience connectors". Audience connectors bring students together while studying with WhatsApp and Facebook. This study reported that "60 percent [of students in the study] agreed that technology changes education for the better." While social media can promote a beneficial education platform, downsides exist. Students may become skilled at "lifting material from the internet" rather than enhancing their personal understanding. Another downside is student attention spans decline. A concern raised by the students of this study showed how many use spell-check as a crutch and will see a trend of points taken off when spell-check is not an option. Apps like X allowed teachers to make classroom accounts where students can learn about social media in a controlled context. Teachers can post assignments on th
Opinion Space
Developed at UC Berkeley, "Opinion Space" (also known as The Collective Discovery Engine) is a social media technology designed to help communities generate and exchange ideas about important issues and policies. Version 1.0 was launched on April 4, 2009, at UC Berkeley, and explored the question "Do you think legalizing marijuana is a good idea?" It has since undergone 4 different iterations, and been used in partnership with various organizations including The Occupy movement (Version 4.0, 5/24/2013) and the African Robots Network (Version 4.0, 5/25/2013). Opinion Space has also been used in collaboration with the United States State Department and the University of California's Berkeley Center for New Media (Version 2.0, 12/1/2009 and Version 3.0, 2/25/2012) to gain public perspective on foreign policy issues. Then U.S. Secretary of State Hillary Rodham Clinton explained, "Opinion Space will harness the power of connection technologies to provide a unique forum for international dialogue. This is...an opportunity to extend our engagement beyond the halls of government directly to the people of the world" (2010). The website uses data visualization and statistical analysis to present and develop public opinion and ideas. Opinion Space is a self-organizing system that uses an intuitive graphical "map" that displays patterns, trends, and insights as they emerge and employs the wisdom of crowds to identify and highlight the most insightful ideas. The system uses a game model that incorporates techniques from deliberative polling, collaborative filtering, and multidimensional visualization.
ShowDocument
ShowDocument is an online web application that allows multiple users to conduct web meetings, upload, share and review documents from remote locations. The service was developed by the HBR Labs company, established in 2007. == Features == Users can collaborate on and review documents in real time, with annotations and text being visible to all users and accessible for co-editing. The idea of every user being able to annotate can cause conflicts within the sessions, and so main navigation options are under the "presenter"'s control - which can be given to a different user as well. An earlier version of the application, by contrast, had allowed all users to navigate and edit at once, causing the system to drop all incomplete edits. It is possible to draw and write on a virtual whiteboard, and to stream a YouTube video to a group in full synchronization. A feature also exists for co-browsing of Google Maps. Entering an open session in the application can be done with a given code number, or by receiving a link through an Email message. Different file formats can be uploaded and saved either online or offline, such as PDF. A PDF file's text cannot be edited - text is edited through the separate text editor. Although the platform contains a text chat, it is not intended to replace instant messaging software, as there are no extensive messaging features. The application has a paid and free version, with the free version having a few limitations: audio and video options are disabled, number of participants is limited and sessions are time-limited. == Development == ShowDocument was first developed in 2007. On September 8, 2009, HBR labs released a new update which included features such as secure online document storage and mobile device support.
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.