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Anshel–Anshel–Goldfeld Key Exchange
Anshel–Anshel–Goldfeld protocol, also known as a commutator key exchange, is a key-exchange protocol using nonabelian groups. It was invented by Drs. Michael Anshel, Iris Anshel, and Dorian Goldfeld. Unlike other group-based protocols, it does not employ any commuting or commutative subgroups of a given platform group and can use any nonabelian group with efficiently computable normal forms. It is often discussed specifically in application of braid groups, which notably are infinite (and the group elements can take variable quantities of space to represent). The computed shared secret is an element of the group, so in practice this scheme must be accompanied with a sufficiently secure compressive hash function to normalize the group element to a usable bitstring. Description Let G be a fixed nonabelian group called a ''platform group''. Alice's public/private information: * ''Alice's public key'' is a tuple of elements =(a_1,\ldots,a_n) in G. * ''Alice's private key'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonabelian Group
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ''b'' ≠ ''b'' ∗ ''a''. This class of groups contrasts with the abelian groups, where all pairs of group elements commute. Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory. See also * Ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dorian M
Dorian may refer to: Ancient Greece * Dorians, one of the main ethnic divisions of ancient Greeks * Doric Greek, or Dorian, the dialect spoken by the Dorians Art and entertainment Films * ''Dorian'' (film), the Canadian title of the 2004 film ''Pact with the Devil'' * '' Dorian Blues'', a 2004 film Literature * '' Dorian, an Imitation'', a 2002 novel by Will Self * ''Dorian'', a 1921 novel by Nephi Anderson * ''Dorian'', a 2022 play by Darryl Pinckney and Robert Wilson Music * Dorians (band), from Armenia * Dorian (Spanish band), a Spanish band * Dorian mode, various musical modes * Dorian Recordings, a label noted for early music recordings * Toccata and Fugue in D minor, BWV 538, or "Dorian", an organ piece by Johann Sebastian Bach * Ukrainian Dorian scale, a musical mode * "Dorian," a song by Demons and Wizards on their album '' Touched by the Crimson King'' People * Dorian (name), a given name (includes a list of people with the name) * Dorian (rapper) (born 1984 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Groups
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see ); and in monodromy invariants of algebraic geometry. Introduction In this introduction let ; the generalization to other values of will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected with an item of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hash Function
A hash function is any Function (mathematics), function that can be used to map data (computing), data of arbitrary size to fixed-size values, though there are some hash functions that support variable-length output. The values returned by a hash function are called ''hash values'', ''hash codes'', (''hash/message'') ''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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice And Bob
Alice and Bob are fictional characters commonly used as placeholders in discussions about cryptography, cryptographic systems and Cryptographic protocol, protocols, and in other science and engineering literature where there are several participants in a thought experiment. The Alice and Bob characters were created by Ron Rivest, Adi Shamir, and Leonard Adleman in their 1978 paper "A Method for Obtaining Digital Signatures and Public-key Cryptosystems". Subsequently, they have become common archetypes in many scientific and engineering fields, such as quantum cryptography, game theory and physics. As the use of Alice and Bob became more widespread, additional characters were added, sometimes each with a particular meaning. These characters do not have to refer to people; they refer to generic agents which might be different computers or even different programs running on a single computer. Overview Alice and Bob are the names of fictional characters used for convenience and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (that is, if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the '' commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many group theorists define the commutator as : . Using the first definition, this can be expressed as . Identities (group theory) Commutator identities are an important tool in group th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugacy Problem
In abstract algebra, the conjugacy problem for a group ''G'' with a given presentation is the decision problem of determining, given two words ''x'' and ''y'' in ''G'', whether or not they represent conjugate elements of ''G''. That is, the problem is to determine whether there exists an element ''z'' of ''G'' such that :y = zxz^.\,\! The conjugacy problem is also known as the transformation problem. The conjugacy problem was identified by Max Dehn in 1911 as one of the fundamental decision problems in group theory; the other two being the word problem and the isomorphism problem. The conjugacy problem contains the word problem as a special case: if ''x'' and ''y'' are words, deciding if they are the same word is equivalent to deciding if xy^ is the identity, which is the same as deciding if it's conjugate to the identity. In 1912 Dehn gave an algorithm that solves both the word and conjugacy problem for the fundamental groups of closed orientable two-dimensional manifolds of g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid Group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Emil Artin, Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see ); and in monodromy invariants of algebraic geometry. Introduction In this introduction let ; the generalization to other values of will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Eraser
Algebraic Eraser (AE)Also referred to as the colored Burau key agreement protocol (CBKAP), Anshel–Anshel–Goldfeld–Lemieux key agreement protocol, Algebraic Eraser key agreement protocol (AEKAP), and Algebraic Eraser Diffie–Hellman (AEDH). is an anonymous key agreement protocol that allows two parties, each having an AE public–private key pair, to establish a shared secret over an insecure channel. This shared secret may be directly used as a key, or to derive another key that can then be used to encrypt subsequent communications using a symmetric key cipher. Algebraic Eraser was developed by Iris Anshel, Michael Anshel, Dorian Goldfeld and Stephane Lemieux. SecureRF owns patents covering the protocol and unsuccessfully attempted (as of July 2019) to standardize the protocol as part of ISO/IEC 29167-20, a standard for securing radio-frequency identification devices and wireless sensor networks. Keyset parameters Before two parties can establish a key t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
