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Elliptic Curve Primality Proving
In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and , in 1993. The concept of using elliptic curves in factorization had been developed by H. W. Lenstra in 1985, and the implications for its use in primality testing (and proving) followed quickly. Primality testing is a field that has been around since the time of Fermat, in whose time most algorithms were based on factoring, which become unwieldy with large input; modern algorithms treat the problems of determining whether a number is prime and what its factors are separately. It became of practical importance with the advent of modern cryptography. Although many current ... [...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] |
Recursion (computer Science)
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure) do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and . Repeatedly calling a function from within itself may cause the call stack to have a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. There is also the higher-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cornacchia's Algorithm
In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation x^2+dy^2=m, where 1\le d The algorithm First, find any solution to (perhaps by using an algorithm listed ); if no such exist, there can be no primitive solution to the original equation. Without loss of generality, we can assume that (if not, then replace with , which will still be a root of ). Then use the |
Ideal Class Group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of . The order of the group, which is finite, is called the class number of . The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. History and origin of the ideal class group Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue (''non-residue'') is −1. Its value at zero is 0. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol. Definition Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a perfect square modulo p and is a quadratic nonresidue modulo p otherwise. The Legendre symbol is a function of a and p defined as :\left(\frac\right) = \begi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial ax^2+bx+c is :b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. There is also the higher-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive Algorithm
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves from within their own code. The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science. Most computer programming languages support recursion by allowing a function to call itself from within its own code. Some functional programming languages (for instance, Clojure) do not define any looping constructs but rely solely on recursion to repeatedly call code. It is proved in computability theory that these recursive-only languages are Turing complete; this means that they are as powerful (they can be used to solve the same problems) as imperative languages based on control structures such as and . Repeatedly calling a function from within itself may cause the call stack to have a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoof's Algorithm
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography where it is important to know the number of points to judge the difficulty of solving the discrete logarithm problem in the group of points on an elliptic curve. The algorithm was published by René Schoof in 1985 and it was a theoretical breakthrough, as it was the first deterministic polynomial time algorithm for counting points on elliptic curves. Before Schoof's algorithm, approaches to counting points on elliptic curves such as the naive and baby-step giant-step algorithms were, for the most part, tedious and had an exponential running time. This article explains Schoof's approach, laying emphasis on the mathematical ideas underlying the structure of the algorithm. Introduction Let E be an elliptic curve defined over the finite field \mathbb_, where q=p^n for p a prime and n an integer \geq 1. Over a field of charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse's Theorem On Elliptic Curves
Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below. If ''N'' is the number of points on the elliptic curve ''E'' over a finite field with ''q'' elements, then Hasse's result states that :, N - (q+1), \le 2 \sqrt. The reason is that ''N'' differs from ''q'' + 1, the number of points of the projective line over the same field, by an 'error term' that is the sum of two complex numbers, each of absolute value . This result had originally been conjectured by Emil Artin in his thesis. It was proven by Hasse in 1933, with the proof published in a series of papers in 1936. Hasse's theorem is equivalent to the determination of the absolute value of the roots of the local zeta-function of ''E''. In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated with the elliptic curve. Hasse-Wei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
