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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] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monic Polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\cdots+c_2x^2+c_1x+c_0 Univariate polynomials If a polynomial has only one indeterminate (univariate polynomial), then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in ''x'' of degree ''n'' then takes the general form displayed above, where : ''c''''n'' ≠ 0, ''c''''n''−1, ..., ''c''2, ''c''1 and ''c''0 are constants, the coefficients of the polynomial. Here the term ''c''''n''''x''''n'' is called the ''leading term'', and its coefficient ''c''''n'' the ''leading coefficient''; if the leading coefficient , the univariate polynomial is called monic. Properties Multiplicatively closed The set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Wiedemann Algorithm
The block Wiedemann algorithm for computing kernel vectors of a matrix over a finite field is a generalization by Don Coppersmith of an algorithm due to Doug Wiedemann. Wiedemann's algorithm Let M be an n\times n square matrix over some finite field F, let x_ be a random vector of length n, and let x = M x_. Consider the sequence of vectors S = \left , Mx, M^2x, \ldots\right/math> obtained by repeatedly multiplying the vector by the matrix M; let y be any other vector of length n, and consider the sequence of finite-field elements S_y = \left \cdot x, y \cdot Mx, y \cdot M^2x \ldots\right/math> We know that the matrix M has a minimal polynomial; by the Cayley–Hamilton theorem we know that this polynomial is of degree (which we will call n_0) no more than n. Say \sum_^ p_rM^r = 0. Then \sum_^ y \cdot (p_r (M^r x)) = 0; so the minimal polynomial of the matrix annihilates the sequence S and hence S_y. But the Berlekamp–Massey algorithm allows us to calculate relatively effi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Lanczos Algorithm For Nullspace Of A Matrix Over A Finite Field
In computer science, the block Lanczos algorithm is an algorithm for finding the nullspace of a matrix over a finite field, using only multiplication of the matrix by long, thin matrices. Such matrices are considered as vectors of tuples of finite-field entries, and so tend to be called 'vectors' in descriptions of the algorithm. The block Lanczos algorithm is amongst the most efficient methods known for finding nullspaces, which is the final stage in integer factorization algorithms such as the quadratic sieve and number field sieve, and its development has been entirely driven by this application. It is based on, and bears a strong resemblance to, the Lanczos algorithm for finding eigenvalues of large sparse real matrices. Parallelization issues The algorithm is essentially not parallel: it is of course possible to distribute the matrix–'vector' multiplication, but the whole vector must be available for the combination step at the end of each iteration, so all the machines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the polynomial . That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number 1 + i is algebraic because it is a root of . All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers. The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental. Examples * All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer and a (non-zero) natural number , satisfies the above definition, because is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Norm
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K''. The field ''L'' is then a finite dimensional vector space over ''K''. Multiplication by α, an element of ''L'', :m_\alpha\colon L\to L :m_\alpha (x) = \alpha x, is a ''K''-linear transformation of this vector space into itself. The norm, N''L''/''K''(''α''), is defined as the determinant of this linear transformation. If ''L''/''K'' is a Galois extension, one may compute the norm of α ∈ ''L'' as the product of all the Galois conjugates of α: :\operatorname_(\alpha)=\prod_ \sigma(\alpha), where Gal(''L''/''K'') denotes the Galois group of ''L''/''K''. (Note that there may be a repetition in the terms of the product.) For a general field extension ''L''/''K'', and nonzero α in ''L'', let ''σ''(''α''), ..., σ(''α'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855) although some special cases of the method—albeit presented without proof—were known to Chinese mathematicians as early as circa 179 AD. To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. (subtraction can be achieved by multiplying one row with -1 and adding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Sieving
Lattice sieving is a technique for finding smooth number, smooth values of a bivariate polynomial f(a,b) over a large region. It is almost exclusively used in conjunction with the number field sieve. The original idea of the lattice sieve came from John Pollard (mathematician), John Pollard.Arjen K. Lenstra and H. W. Lenstra, Jr. (eds.). "The development of the number field sieve". Lecture Notes in Math. (1993) 1554. Springer-Verlag. The algorithm implicitly involves the ideal (ring theory), ideal structure of the number field of the polynomial; it takes advantage of the theorem that any prime ideal above some rational prime ''p'' can be written as p \mathbb Z[\alpha] + (u + v \alpha) \mathbb Z[\alpha]. One then picks many prime numbers ''q'' of an appropriate size, usually just above the factor base limit, and proceeds by :: For each ''q'', list the prime ideals above ''q'' by factorising the polynomial f(a,b) over GF(q) ::: For each of these prime ideals, which are called 'spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9. In any standard positional numeral system, a number is conventionally written as with ''x'' as the string of digits and ''y'' as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four. Etymology ''Radix'' is a Latin word for "root". ''Root'' can be considered a synonym for ''base,'' in the arithmetical sense. In numeral systems In the system with radix 13, for example, a string of digits such as 398 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
