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Non-commutative Cryptography
Non-commutative cryptography is the area of cryptology where the cryptographic primitives, methods and systems are based on algebraic structures like semigroups, Group (mathematics), groups and Ring (mathematics), rings which are non-commutative. One of the earliest applications of a non-commutative algebraic structure for cryptographic purposes was the use of braid groups to develop cryptographic protocols. Later several other non-commutative structures like Thompson groups, polycyclic groups, Grigorchuk groups, and matrix groups have been identified as potential candidates for cryptographic applications. In contrast to non-commutative cryptography, the currently widely used public-key cryptosystems like RSA (cryptosystem), RSA cryptosystem, Diffie–Hellman key exchange and elliptic curve cryptography are based on number theory and hence depend on commutative algebraic structures. Non-commutative cryptographic protocols have been developed for solving various cryptographic problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptology
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 synonymous wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encryption
In cryptography, encryption is the process of encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can decipher a ciphertext back to plaintext and access the original information. Encryption does not itself prevent interference but denies the intelligible content to a would-be interceptor. For technical reasons, an encryption scheme usually uses a pseudo-random encryption key generated by an algorithm. It is possible to decrypt the message without possessing the key but, for a well-designed encryption scheme, considerable computational resources and skills are required. An authorized recipient can easily decrypt the message with the key provided by the originator to recipients but not to unauthorized users. Historically, various forms of encryption have been used to aid in cryptography. Early encryption techniques were often used in military ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-way Function
In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient for a function to be called one-way (see Theoretical definition, below). The existence of such one-way functions is still an open conjecture. Their existence would prove that the complexity classes P and NP are not equal, thus resolving the foremost unsolved question of theoretical computer science.Oded Goldreich (2001). Foundations of Cryptography: Volume 1, Basic Tools,draft availablefrom author's site). Cambridge University Press. . (see als The converse is not known to be true, i.e. the existence of a proof that P≠NP would not directly imply the existence of one-way functions. In applied contexts, the terms "easy" and "hard" are usu ... [...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] |
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] |
Decryption
In cryptography, encryption is the process of encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can decipher a ciphertext back to plaintext and access the original information. Encryption does not itself prevent interference but denies the intelligible content to a would-be interceptor. For technical reasons, an encryption scheme usually uses a pseudo-random encryption key generated by an algorithm. It is possible to decrypt the message without possessing the key but, for a well-designed encryption scheme, considerable computational resources and skills are required. An authorized recipient can easily decrypt the message with the key provided by the originator to recipients but not to unauthorized users. Historically, various forms of encryption have been used to aid in cryptography. Early encryption techniques were often used in military ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word (group Theory)
In group theory, a word is any written product of group elements and their inverses. For example, if ''x'', ''y'' and ''z'' are elements of a group ''G'', then ''xy'', ''z''−1''xzz'' and ''y''−1''zxx''−1''yz''−1 are words in the set . Two different words may evaluate to the same value in ''G'', or even in every group. Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory. Definitions Let ''G'' be a group, and let ''S'' be a subset of ''G''. A word in ''S'' is any expression of the form :s_1^ s_2^ \cdots s_n^ where ''s''1,...,''sn'' are elements of ''S'', called generators, and each ''εi'' is ±1. The number ''n'' is known as the length of the word. Each word in ''S'' represents an element of ''G'', namely the product of the expression. By convention, the unique Uniqueness of identity element and inverses identity element can be represented by the empty word, which is the unique word ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice And Bob
Alice and Bob are fictional characters commonly used as placeholders in discussions about cryptographic systems and protocols, and in other science and engineering literature where there are several participants in a thought experiment. The Alice and Bob characters were invented 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 to aid comprehension. For example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
