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Baby-step Giant-step
In group theory, a branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm or order of an element in a finite abelian group by Daniel Shanks. The discrete log problem is of fundamental importance to the area of public key cryptography. Many of the most commonly used cryptography systems are based on the assumption that the discrete log is extremely difficult to compute; the more difficult it is, the more security it provides a data transfer. One way to increase the difficulty of the discrete log problem is to base the cryptosystem on a larger group. Theory The algorithm is based on a space–time tradeoff. It is a fairly simple modification of trial multiplication, the naive method of finding discrete logarithms. Given a cyclic group G of order n, a generator \alpha of the group and a group element \beta, the problem is to find an integer x such that : \alpha^x = \beta\,. The baby-step giant-step algorithm is based o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hash Table
In computing, a hash table, also known as hash map, is a data structure that implements an associative array or dictionary. It is an abstract data type that maps keys to values. A hash table uses a hash function to compute an ''index'', also called a ''hash code'', into an array of ''buckets'' or ''slots'', from which the desired value can be found. During lookup, the key is hashed and the resulting hash indicates where the corresponding value is stored. Ideally, the hash function will assign each key to a unique bucket, but most hash table designs employ an imperfect hash function, which might cause hash ''collisions'' where the hash function generates the same index for more than one key. Such collisions are typically accommodated in some way. In a well-dimensioned hash table, the average time complexity for each lookup is independent of the number of elements stored in the table. Many hash table designs also allow arbitrary insertions and deletions of key–value pairs, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Sutherland (mathematician)
Andrew Victor Sutherland is an American mathematician and Principal Research Scientist at the Massachusetts Institute of Technology. His research focuses on computational aspects of number theory and arithmetic geometry. He is known for his contributions to several projects involving large scale computations, including the Polymath project on bounded gaps between primes, the L-functions and Modular Forms Database, the sums of three cubes project, and the computation and classification of Sato-Tate conjecture, Sato-Tate distributions. Education and career Sutherland earned a bachelor's degree in mathematics from MIT in 1990. Following an entrepreneurial career in the software industry he returned to MIT and completed his doctoral degree in mathematics in 2007 under the supervision of Michael Sipser and Ronald Rivest, winning the George M. Sprowls prize for this thesis. He joined the MIT mathematics department as a Research Scientist in 2009, and was promoted to Principal Research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Cohen (number Theorist)
Henri Cohen (born 8 June 1947) is a number theorist, and a professor at the University of Bordeaux. He is best known for leading the team that created the PARI/GP computer algebra system. He introduced the Rankin–Cohen bracket and has written several textbooks in computational and algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob .... Selected publications * ; 2nd correct. print 19951st printing 1993ref> * * * * References External links Personal web page* Number theorists École Normale Supérieure alumni 20th-century French mathematicians 21st-century French mathematicians 1947 births Living people University of Bordeaux faculty {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Gelfond
Alexander Osipovich Gelfond (russian: Алекса́ндр О́сипович Ге́льфонд; 24 October 1906 – 7 November 1968) was a Soviet Union, Soviet mathematician. Gelfond–Schneider theorem, Gelfond's theorem, also known as the Gelfond-Schneider theorem is named after him. Biography Alexander Gelfond was born in Saint Petersburg, Russian Empire, the son of a professional physician and amateur philosopher Osip Gelfond. He entered the Moscow State University in 1924, started his postgraduate studies there in 1927 and obtained his Doctor of Philosophy, PhD in 1930. His advisors were Aleksandr Khinchin and Vyacheslav Stepanov. In 1930 he stayed for five months in Germany (in Berlin and Göttingen) where he worked with Edmund Landau, Carl Ludwig Siegel and David Hilbert. In 1931 he started teaching as a Professor at the Moscow State University and worked there until the last day of his life. Since 1933 he also worked at the Steklov Institute of Mathematics. In 1939 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollard's Rho Algorithm For Logarithms
Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's rho algorithm to solve the integer factorization problem. The goal is to compute \gamma such that \alpha ^ \gamma = \beta, where \beta belongs to a cyclic group G generated by \alpha. The algorithm computes integers a, b, A, and B such that \alpha^a \beta^b = \alpha^A \beta^B. If the underlying group is cyclic of order n, by substituting \beta as a^ and noting that two powers are equal if and only if the exponents are equivalent modulo the order of the base, in this case modulo n, we get that \gamma is one of the solutions of the equation (B-b) \gamma = (a-A) \pmod n. Solutions to this equation are easily obtained using the extended Euclidean algorithm. To find the needed a, b, A, and B the algorithm uses Floyd's cycle-finding algorithm to find a cycle in the sequence x_i = \alpha^ \beta^, where the function f: x_i \mapst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pohlig–Hellman Algorithm
In group theory, the Pohlig–Hellman algorithm, sometimes credited as the Silver–Pohlig–Hellman algorithm, Mollin 2006, pg. 344 is a special-purpose algorithm for computing discrete logarithms in a finite abelian group whose order is a smooth integer. The algorithm was introduced by Roland Silver, but first published by Stephen Pohlig and Martin Hellman (independent of Silver). Groups of prime-power order As an important special case, which is used as a subroutine in the general algorithm (see below), the Pohlig–Hellman algorithm applies to groups whose order is a prime power. The basic idea of this algorithm is to iteratively compute the p-adic digits of the logarithm by repeatedly "shifting out" all but one unknown digit in the exponent, and computing that digit by elementary methods. (Note that for readability, the algorithm is stated for cyclic groups — in general, G must be replaced by the subgroup \langle g\rangle generated by g, which is always cyclic.) :Input. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 exchange * Diffie–Hellman protocol * Diffie–Hellman handshake is a mathematical method of securely exchanging cryptographic keys over a public channel and was one of the first public-key protocols as conceived by Ralph Merkle and named after Whitfield Diffie and Martin Hellman. DH is one of the earliest practical examples of public key exchange implemented within the field of cryptography. Published in 1976 by Diffie and Hellman, this is the earliest publicly known work that proposed the idea of a private key and a corresponding public key. Traditionally, secure encrypted communication between two parties required that they first exchange keys by some secure physical means, such as paper key lists transported by a trusted courier. The Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In other words, if ''S'' is a subset of a group ''G'', then , the ''subgroup generated by S'', is the smallest subgroup of ''G'' containing every element of ''S'', which is equal to the intersection over all subgroups containing the elements of ''S''; equivalently, is the subgroup of all elements of ''G'' that can be expressed as the finite product of elements in ''S'' and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If ''G'' = , then we say that ''S'' ''generates'' ''G'', and the elements in ''S'' are called ''generators'' or ''group generators''. If ''S'' is the empty set, then is the trivial group , since we consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
