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Number-Theoretic Algorithms

Number-Theoretic Algorithms This paper is a report on algorithms to solve problems in number theory. I believe the most interesting such problems to be those from elementary number theory whose complexity is still unknown. For this reason, I concentrate on methods to test primality, to find the prime factors of numbers, and to solve equations in various finite groups, rings, and fields. These problems have the attractive feature that they are easily stated and frequently can be solved by algorithms that are easy to implement. However, the intuition behind these algorithms and the methods used to analyze them are often anything but elementary; for this reason I describe not only the algorithms but also the underlying mathematics. The present review is not an exhaustive survey of the area, and neces­ sarily reflects my own biases and interests. For example, the inclusion of algorithms for polynomials over finite fields may seem inappropriate. However, the polynomials in one variable over a finite field have a well­ studied and attractive arithmetic, and it seemed interesting to contrast algorithms for this domain with algorithms for the integers. At other points, the subject impinges on algebraic number theory, but I omit the details of such http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annual Review of Computer Science Annual Reviews

Number-Theoretic Algorithms

Annual Review of Computer Science , Volume 4 (1) – Jun 1, 1990

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References (110)

Publisher
Annual Reviews
Copyright
Copyright 1990 Annual Reviews. All rights reserved
Subject
Review Articles
ISSN
8756-7016
DOI
10.1146/annurev.cs.04.060190.001003
Publisher site
See Article on Publisher Site

Abstract

This paper is a report on algorithms to solve problems in number theory. I believe the most interesting such problems to be those from elementary number theory whose complexity is still unknown. For this reason, I concentrate on methods to test primality, to find the prime factors of numbers, and to solve equations in various finite groups, rings, and fields. These problems have the attractive feature that they are easily stated and frequently can be solved by algorithms that are easy to implement. However, the intuition behind these algorithms and the methods used to analyze them are often anything but elementary; for this reason I describe not only the algorithms but also the underlying mathematics. The present review is not an exhaustive survey of the area, and neces­ sarily reflects my own biases and interests. For example, the inclusion of algorithms for polynomials over finite fields may seem inappropriate. However, the polynomials in one variable over a finite field have a well­ studied and attractive arithmetic, and it seemed interesting to contrast algorithms for this domain with algorithms for the integers. At other points, the subject impinges on algebraic number theory, but I omit the details of such

Journal

Annual Review of Computer ScienceAnnual Reviews

Published: Jun 1, 1990

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