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Multivariate Cryptography
Multivariate cryptography is the generic term for asymmetric cryptographic primitives based on multivariate polynomials over a finite field F. In certain cases those polynomials could be defined over both a ground and an extension field. If the polynomials have the degree two, we talk about multivariate quadratics. Solving systems of multivariate polynomial equations is proven to be NP-complete. That's why those schemes are often considered to be good candidates for post-quantum cryptography. Multivariate cryptography has been very productive in terms of design and cryptanalysis. Overall, the situation is now more stable and the strongest schemes have withstood the test of time. It is commonly admitted that Multivariate cryptography turned out to be more successful as an approach to build signature schemes primarily because multivariate schemes provide the shortest signature among post-quantum algorithms. History presented their so-called C* scheme at the Eurocrypt con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security ( data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. Cryptography prior to the modern age was effectively synonymo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gröbner Basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases were introduced in 1965, together with an algorithm to compute them (Buchberger's algorithm), by Bruno Buchberger in his Ph.D. thesis. He named them after h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolas Courtois
Nicolas Tadeusz Courtois (born 14 November 1971) is a cryptographer and senior lecturer in computer science at University College London. Courtois was one of the co-authors of both the XSL attack against block ciphers, such as the Advanced Encryption Standard, and the XL system for solving systems of algebraic equations used in the attack. Other cryptographic results of Courtois include algebraic attacks on stream ciphers, attacks on the KeeLoq and Hitag 2 systems used for remote keyless automobile entry systems,. and an analysis of cryptographic weaknesses in public transit smart cards including the London Underground Oyster card and the Dutch OV-chipkaart The OV-chipkaart (short for ''openbaar vervoer chipkaart'', meaning ''public transport chipcard'') is a contactless smart card system used for all public transport in the Netherlands. First introduced in the Rotterdam Metro in April 2005, it has .... More recently, he has written about cryptocurrency. Courtois graduate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bart Preneel
Bart Preneel (born 15 October 1963 in Leuven, Belgium) is a Flemish cryptographer and cryptanalyst. He is a professor at Katholieke Universiteit Leuven, in the COSIC group. He was the president of the International Association for Cryptologic Research in 2008-2013 and project manager of ECRYPT. Education In 1987, Preneel received an electrical engineering degree in applied science from the Katholieke Universiteit, Leuven. In 1993, Preneel received a PhD from the Katholieke Universiteit Leuven. His dissertation in computer science, entitled ''Analysis and Design of Cryptographic Hash Functions'', was advised by Joos (Joseph) P. L. Vandewalle and René J. M. Govaerts. Career Along with Shoji Miyaguchi, he independently invented the Miyaguchi–Preneel scheme, a complex structure used in the hash function Whirlpool. He is one of the authors of the RIPEMD-160 hash function. He was also a co-inventor of the stream cipher MUGI which would later become a Japanese standard, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Charles Faugère
Jean-Charles Faugère is the head of the POLSYS project-team (Solvers for Algebraic Systems and Applications) of the Laboratoire d'Informatique de Paris 6 (LIP6) and Paris–Rocquencourt center of INRIA, in Paris. The team was formerly known as SPIRAL and SALSA. Faugère obtained his Ph.D. in mathematics in 1994 at the University of Paris VI, with the dissertation ''"Résolution des systemes d’équations algébriques"'' (Solving systems of algebraic equations), under the supervision of Daniel Lazard. He works on Gröbner bases and their applications, in particular, in cryptology. With his collaborators, he has devised the FGLM algorithm for computing Gröbner bases; he has also introduced the F4 and F5 algorithms for calculating Gröbner bases. In particular, his F5 algorithm allowed him to solve various problems in cryptography such as HFE; he also introduced a new type of cryptanalysis Cryptanalysis (from the Greek ''kryptós'', "hidden", and ''analýein'', "to analyze" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QUAD (cipher)
In cryptography, the QUAD, cipher is a relatively new stream cipher, which was designed with provable security arguments in mind. Description QUAD relies on the iteration of a randomly chosen multivariate quadratic system S=(Q1, ..., Qm) of m=kn equations in n unknowns over a finite field GF(q). The keystream generation process simply consists in iterating the three following steps in order to produce (k -1) n GF(q) keystream values at each iteration. *Compute the kn-tuple of GF(q) values S(x) = (Q1(x),..., Qkn(x)) where x is the current value of the internal state; *Output the sequence (Qn+1(x),..., Qkn(x)) of (k-1)n GF(q) keystream values *Update the internal state x with the sequence of n GF(q) first generated values (Q1(x),..., Qn(x)) QUAD is a modern stream cipher, i.e. it uses a key and an initialisation value (IV) to produce a keystream sequence. A Key and IV setup is also defined which also rely on multivariate quadratic system. Security The security of the keystream ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NESSIE
NESSIE (New European Schemes for Signatures, Integrity and Encryption) was a European research project funded from 2000 to 2003 to identify secure cryptographic primitives. The project was comparable to the NIST AES process and the Japanese Government-sponsored CRYPTREC project, but with notable differences from both. In particular, there is both overlap and disagreement between the selections and recommendations from NESSIE and CRYPTREC (as of the August 2003 draft report). The NESSIE participants include some of the foremost active cryptographers in the world, as does the CRYPTREC project. NESSIE was intended to identify and evaluate quality cryptographic designs in several categories, and to that end issued a public call for submissions in March 2000. Forty-two were received, and in February 2003 twelve of the submissions were selected. In addition, five algorithms already publicly known, but not explicitly submitted to the project, were chosen as "selectees". The project has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hash Function
A hash function is any function that can be used to map data of arbitrary size to fixed-size values. The values returned by a hash function are called ''hash values'', ''hash codes'', ''digests'', or simply ''hashes''. The values are usually used to index a fixed-size table called a ''hash table''. Use of a hash function to index a hash table is called ''hashing'' or ''scatter storage addressing''. Hash functions and their associated hash tables are used in data storage and retrieval applications to access data in a small and nearly constant time per retrieval. They require an amount of storage space only fractionally greater than the total space required for the data or records themselves. Hashing is a computationally and storage space-efficient form of data access that avoids the non-constant access time of ordered and unordered lists and structured trees, and the often exponential storage requirements of direct access of state spaces of large or variable-length keys. Use of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be added toge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can be repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unbalanced Oil And Vinegar
In cryptography, the unbalanced oil and vinegar (UOV) scheme is a modified version of the oil and vinegar scheme designed by J. Patarin. Both are digital signature protocols. They are forms of multivariate cryptography. The security of this signature scheme is based on an NP-hard mathematical problem. To create and validate signatures, a minimal quadratic equation system must be solved. Solving equations with variables is NP-hard. While the problem is easy if is either much much larger or much much smaller than , importantly for cryptographic purposes, the problem is thought to be difficult in the average case when and are nearly equal, even when using a quantum computer. Multiple signature schemes have been devised based on multivariate equations with the goal of achieving quantum resistance. A significant drawback with UOV is that the key size can be large. Typically , the number of variables, is chosen to be double , the number of equations. Encoding the coefficients of all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hidden Field Equations
Hidden Fields Equations (HFE), also known as HFE trapdoor function, is a public key cryptosystem which was introduced at Eurocrypt in 1996 and proposed by Jacques Patarin following the idea of the Matsumoto and Imai system. It is based on polynomials over finite fields \mathbb_q of different size to disguise the relationship between the private key and public key. HFE is in fact a family which consists of basic HFE and combinatorial versions of HFE. The HFE family of cryptosystems is based on the hardness of the problem of finding solutions to a system of multivariate quadratic equations (the so-called MQ problem) since it uses private affine transformations to hide the extension field and the private polynomials. Hidden Field Equations also have been used to construct digital signature schemes, e.g. Quartz and Sflash. Mathematical background One of the central notions to understand how Hidden Field Equations work is to see that for two extension fields \mathbb_ \mathbb_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
