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Function Field Sieve
In mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic subexponential complexity. Leonard Adleman developed it in 1994 and then elaborated it together with M. D. Huang in 1999.L. Adleman, M.D. Huang. "Function Field Sieve Method for Discrete Logarithms over Finite Fields". In: Inf. Comput. 151 (May 1999), pp. 5-16. DOI: 10.1006/inco.1998.2761. Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two. The discrete logarithm problem in a finite field consists of solving the equation a^x = b for a,b \in \mathbb_ , p a prime number and n an integer. The function f: \mathbb_ \to \mathbb_, x \mapsto a^x for a fixed a \in \mathbb_ is a one-way function used in cryptography. Several cryptographic methods are based on the DLP such as the Diffie-Hellman key exchange, the El Gamal cryptosystem and the Digital Signature Algorithm. Num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (algebra)
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : which satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(''a'') ≠ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form :\exp\left( \left(\sqrt + o(1)\right)(\ln n)^(\ln \ln n)^\right) =L_n\left .html"_;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in_L-notation.html" ;"title="">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation">">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation), where is the natural logarithm. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots). The pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Function Field
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebraic function field of ''n'' variables over ''k'' may be defined as a finite field extension of the field ''K'' = ''k''(''x''1,...,''x''''n'') of rational functions in ''n'' variables over ''k''. Example As an example, in the polynomial ring ''k'' 'X'',''Y''consider the ideal generated by the irreducible polynomial ''Y''2 − ''X''3 and form the field of fractions of the quotient ring ''k'' 'X'',''Y''(''Y''2 − ''X''3). This is a function field of one variable over ''k''; it can also be written as k(X)(\sqrt) (with degree 2 over k(X)) or as k(Y)(\sqrt (with degree 3 over k(Y)). We see that the degree of an algebraic function field is not a well-defined notion. Category structure The algebraic function fields over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of \mathbb_q(T), the field of rational functions in one variable over the finite field with q=p^n elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s. Formal definitions A ''global field'' is one of the following: ;An algebraic number field An algebraic number field ''F'' is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus ''F'' is a field that contains Q and has finite dimension when considered as a vector space over Q. ;The function field of an algebraic curve over a finite field A function field of a variety is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sub-exponential Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithm Problem
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b''''k'' can be defined for all integers ''k'', and the discrete logarithm log''b'' ''a'' is an integer ''k'' such that . In number theory, the more commonly used term is index: we can write ''x'' = ind''r'' ''a'' (mod ''m'') (read "the index of ''a'' to the base ''r'' modulo ''m''") for ''r''''x'' ≡ ''a'' (mod ''m'') if ''r'' is a primitive root of ''m'' and gcd(''a'',''m'') = 1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. Definition Let ''G'' be any group. Denote its group operation by mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-notation
''L''-notation is an asymptotic notation analogous to big-O notation, denoted as L_n alpha,c/math> for a bound variable n tending to infinity. Like big-O notation, it is usually used to roughly convey the rate of growth of a function, such as the computational complexity of a particular algorithm. Definition It is defined as :L_n alpha,ce^ where ''c'' is a positive constant, and \alpha is a constant 0 \leq \alpha \leq 1. L-notation is used mostly in computational number theory, to express the complexity of algorithms for difficult number theory problems, e.g. sieves for integer factorization and methods for solving discrete logarithms. The benefit of this notation is that it simplifies the analysis of these algorithms. The e^ expresses the dominant term, and the e^ takes care of everything smaller. When \alpha is 0, then :L_n alpha,c= L_n , c= e^ = (\ln n)^\, is a polynomial function of ln ''n''; When \alpha is 1 then :L_n alpha,c= L_n , c= e^ = n^\, is a full ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subexponential Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the Worst-case complexity, worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coppersmith Method
The Coppersmith method, proposed by Don Coppersmith, is a method to find small integer zeroes of univariate or bivariate polynomials modulo a given integer. The method uses the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) to find a polynomial that has the same zeroes as the target polynomial but smaller coefficients. In cryptography, the Coppersmith method is mainly used in attacks on RSA when parts of the secret key are known and forms a base for Coppersmith's attack. Approach Coppersmith's approach is a reduction of solving modular polynomial equations to solving polynomials over the integers. Let F(x) = x^n+a_x^+\ldots +a_1x+a_0 and assume that F(x_0)\equiv 0 \pmod for some integer , x_0, < M^. Coppersmith’s algorithm can be used to find this integer solution . Finding roots over is easy using, e.g., |
Index Calculus Algorithm
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms. Dedicated to the discrete logarithm in (\mathbb/q\mathbb)^* where q is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves. The algorithm collects relations among the discrete logarithms of small primes, computes them by a linear algebra procedure and finally expresses the desired discrete logarithm with respect to the discrete logarithms of small primes. Description Roughly speaking, the discrete log problem asks us to find an ''x'' such that g^x \equiv h \pmod, where ''g'', ''h'', and the modulus ''n'' are given. The algorithm (described in detail below) applies to the group (\mathbb/q\mathbb)^* where ''q'' is prime. It requires a ''factor base'' as input. This ''factor base'' is usually chosen to be the number −1 and the first ''r'' primes starting with 2. From the point of view of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field Sieve
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form :\exp\left( \left(\sqrt + o(1)\right)(\ln n)^(\ln \ln n)^\right) =L_n\left .html"_;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in_L-notation.html" ;"title="">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation">">frac,\sqrt[3right.html" ;"title=".html" ;"title="frac,\sqrt[3">frac,\sqrt[3right">.html" ;"title="frac,\sqrt[3">frac,\sqrt[3right/math> (in L-notation), where is the natural logarithm. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots). The pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
