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FourQ
In cryptography, FourQ is an elliptic curve developed by Microsoft Research. It is designed for key agreements schemes (elliptic-curve Diffie–Hellman) and digital signatures ( Schnorr), and offers about 128 bits of security. It is equipped with a reference implementation made by the authors of the original paper. The open source implementation is called ''FourQlib'' and runs on Windows and Linux and is available for x86, x64, and ARM. It is licensed under the MIT License and the source code is available on GitHub. Its name is derived from the four dimensional Gallant–Lambert–Vanstone scalar multiplication, which allows high performance calculations. The curve is defined over a two dimensional extension of the prime field defined by the Mersenne prime 2^ - 1. History The curve was published in 2015 by Craig Costello and Patrick Longa from Microsoft Research on ePrint. The paper was presented in Asiacrypt in 2015 in Auckland, New Zealand, and consequently a reference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic-curve Cryptography
Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security.Commercial National Security Algorithm Suite and Quantum Computing FAQ U.S. National Security Agency, January 2016. Elliptic curves are applicable for key agreement, s, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microsoft Research
Microsoft Research (MSR) is the research subsidiary of Microsoft. It was created in 1991 by Richard Rashid, Bill Gates and Nathan Myhrvold with the intent to advance state-of-the-art computing and solve difficult world problems through technological innovation in collaboration with academic, government, and industry researchers. The Microsoft Research team has more than 1,000 computer scientists, physicists, engineers, and mathematicians, including Turing Award winners, Fields Medal winners, MacArthur Fellows, and Dijkstra Prize winners. Between 2010 and 2018, 154,000 AI patents were filed worldwide, with Microsoft having by far the largest percentage of those patents, at 20%.Louis Columbus, January 6, 201Microsoft Leads The AI Patent Race Going Into 2019 ''Forbes'' According to estimates in trade publications, Microsoft spent about $6 billion annually in research initiatives from 2002-2010 and has spent from $10–14 billion annually since 2010. Microsoft Research has made signi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GitHub
GitHub, Inc. () is an Internet hosting service for software development and version control using Git. It provides the distributed version control of Git plus access control, bug tracking, software feature requests, task management, continuous integration, and wikis for every project. Headquartered in California, it has been a subsidiary of Microsoft since 2018. It is commonly used to host open source software development projects. As of June 2022, GitHub reported having over 83 million developers and more than 200 million repositories, including at least 28 million public repositories. It is the largest source code host . History GitHub.com Development of the GitHub.com platform began on October 19, 2007. The site was launched in April 2008 by Tom Preston-Werner, Chris Wanstrath, P. J. Hyett and Scott Chacon after it had been made available for a few months prior as a beta release. GitHub has an annual keynote called GitHub Universe. Org ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (group Theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Subgroup
In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or periodic group) if every element of ''A'' has finite order and is called torsion-free if every element of ''A'' except the identity is of infinite order. The proof that ''AT'' is closed under the group operation relies on the commutativity of the operation (see examples section). If ''A'' is abelian, then the torsion subgroup ''T'' is a fully characteristic subgroup of ''A'' and the factor group ''A''/''T'' is torsion-free. There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup. There is another covariant functor from the category of abelian groups to the category of torsion-free g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Subgroup Confinement Attack
In cryptography, a subgroup confinement attack, or small subgroup confinement attack, on a cryptographic method that operates in a large finite group is where an attacker attempts to compromise the method by forcing a key to be confined to an unexpectedly small subgroup of the desired group. Several methods have been found to be vulnerable to subgroup confinement attack, including some forms or applications of Diffie–Hellman key exchange Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exc ... and DH-EKE. References * * * Cryptographic attacks Finite groups {{crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Edwards Curve
In algebraic geometry, the twisted Edwards curves are plane models of elliptic curves, a generalisation of Edwards curves introduced by Bernstein, Birkner, Joye, Lange and Peters in 2008. The curve set is named after mathematician Harold M. Edwards. Elliptic curves are important in public key cryptography and twisted Edwards curves are at the heart of an electronic signature scheme called EdDSA that offers high performance while avoiding security problems that have surfaced in other digital signature schemes. Definition Each twisted Edwards curve is a twist of an Edwards curve. A twisted Edwards curve E_ over a field \mathbb with \operatorname(\mathbb) \neq 2 is an affine plane curve defined by the equation: : E_: a x^2+y^2= 1+dx^2y^2 where a, d are distinct non-zero elements of \mathbb. The special case a = 1 is ''untwisted'', because the curve reduces to an ordinary Edwards curve. Every twisted Edwards curve is birationally equivalent to an elliptic curve in Montgomery fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IETF
The Internet Engineering Task Force (IETF) is a standards organization for the Internet and is responsible for the technical standards that make up the Internet protocol suite (TCP/IP). It has no formal membership roster or requirements and all its participants are volunteers. Their work is usually funded by employers or other sponsors. The IETF was initially supported by the federal government of the United States but since 1993 has operated under the auspices of the Internet Society, an international non-profit organization. Organization The IETF is organized into a large number of working groups and birds of a feather informal discussion groups, each dealing with a specific topic. The IETF operates in a bottom-up task creation mode, largely driven by these working groups. Each working group has an appointed chairperson (or sometimes several co-chairs); a charter that describes its focus; and what it is expected to produce, and when. It is open to all who want to partic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microsoft
Microsoft Corporation is an American multinational corporation, multinational technology company, technology corporation producing Software, computer software, consumer electronics, personal computers, and related services headquartered at the Microsoft Redmond campus located in Redmond, Washington, United States. Its best-known software products are the Microsoft Windows, Windows line of operating systems, the Microsoft Office Productivity software#Office suite, suite, and the Internet Explorer and Microsoft Edge, Edge web browsers. Its flagship hardware products are the Xbox video game consoles and the Microsoft Surface lineup of touchscreen personal computers. Microsoft ranked No. 21 in the 2020 Fortune 500 rankings of the largest United States corporations by total revenue; it was the world's List of the largest software companies, largest software maker by revenue as of 2019. It is one of the Big Tech, Big Five American information technology companies, alongside Alphabet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auckland
Auckland (pronounced ) ( mi, Tāmaki Makaurau) is a large metropolitan city in the North Island of New Zealand. The most populous urban area in the country and the fifth largest city in Oceania, Auckland has an urban population of about It is located in the greater Auckland Region—the area governed by Auckland Council—which includes outlying rural areas and the islands of the Hauraki Gulf, and which has a total population of . While Europeans continue to make up the plurality of Auckland's population, the city became multicultural and cosmopolitan in the late-20th century, with Asians accounting for 31% of the city's population in 2018. Auckland has the fourth largest foreign-born population in the world, with 39% of its residents born overseas. With its large population of Pasifika New Zealanders, the city is also home to the biggest ethnic Polynesian population in the world. The Māori-language name for Auckland is ', meaning "Tāmaki desired by many", in ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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