Kunerth's Algorithm

	Kunerth's Algorithm Kunerth's algorithm is an algorithm for computing the modular square root of a given number.Leonard Eugene Dickson, "History of Numbers", vol 2, pp. 382–384. The algorithm does not require the factorization of the modulus, and relies on modular operations that is often easy when the given number is prime. Algorithm To find ''y'' from a given value :B = y^ \bmod, it takes the following steps: # find the modular square root of r\equiv \sqrt \pmod. This step is quite easy, irrespectively of how big ''N'' when B is a prime. # solve a quadratic equation associated with the modular square root of w^=A\cdot z^+B\cdot z+C. Most of Kunerth's examples in his original paper solve this equation by having ''C'' be a integer square and thus setting ''z'' to zero. #:Expand out the following equation to obtain the quadratic #::((B\cdot z + r)^2 \pm N)/B = w^. #:One can always make sure that the quadratic can be solved by adjusting the modulus ''N'' in the above equation. Thus #:::((B\cdot z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Methods Of Computing Square Roots Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted \sqrt, \sqrt /math>, or S^) of a real number. Arithmetically, it means given S, a procedure for finding a number which when multiplied by itself, yields S; algebraically, it means a procedure for finding the non-negative root of the equation x^2-S=0; geometrically, it means given two line segments, a procedure for constructing their geometric mean. Every real number except zero has two square roots. In addition to the principal square root, there is a negative square root equal in magnitude but opposite in sign to the principal square root, except for zero, which has double square roots of zero. The principal square root of most numbers is an irrational number with an infinite decimal expansion. As a result, the decimal expansion of any such square root can only be computed to some finite-precision approximation. However, even if we ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Articles Needing Expert Attention 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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Algorithms In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of space and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]