Mathematical Sciences: Topics in Number Theory

数学科学：数论主题

• 批准号: 8803297
• 负责人: Carl Pomerance
• 金额: $ 7.95万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-15 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8803297&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Number; Theory

There are three foci of this research. The first one concerns Artin's conjecture for composite numbers. That is, what can one say about the elements of maximal order modulo a composite number. The second problem concerns finding efficient discrete logarithm algorithms and efficient algorithms for factoring large numbers. The principal investigator and his colleagues have built a special purpose computer for implementing his quadratic sieve algorithm for factoring large integers. He will study the practical aspects of implementing on this and other computers linear and cubic sieve algorithms which are promising practical methods and he will also attempt to find new methods, especially for the discrete logarithm problem. The third problem concerns elliptic curves over finite fields and their applications to primality testing. It is hoped that a polylog algorithm for constructing an elliptic curve with a prescribed order modulo a given prime can be found. This is known from a result of the principal investigator to give a polylog primality test. This research is in the area of computational number theory with a heavy concentration on algorithms for factoring integers and testing whether they are primes.

本研究有三个焦点。 第一个 关于Artin的合数猜想 一个什么 可以说模a的最大阶元素 合数 第二个问题是找到有效的 离散对数算法和有效算法 分解大量的数字。 主要研究者及其 同事们已经建立了一个特殊用途的计算机， 他的分解大整数的二次筛选算法。 他 将研究在这方面实施的实际问题， 其他计算机线性和立方筛算法， 有希望的实用方法，他也将试图找到新的 方法，特别是离散对数问题。 的 第三个问题涉及有限域上的椭圆曲线， 它们在素性测试中的应用 希望A 构造椭圆曲线的polylog算法 可以找到给定素数模的规定阶数。这是 根据主要研究者的结果， 多对数素性检验 这项研究是在计算数论领域 专注于整数分解的算法 测试它们是否是素数