Mathematical Sciences: Topics in Number Theory

本研究有三个重点。第一个是关于合数的Artin猜想。也就是说，对于一个合数的最大阶模的元素，我们能说些什么呢？第二个问题是寻找有效的离散对数算法和有效的分解大数的算法。首席研究员和他的同事们已经建造了一台特殊用途的计算机，用于实现他的二次筛算法，以分解大整数。他将研究在这台计算机和其他计算机上实现线性和立方筛算法的实际方面，这是很有前途的实用方法，他也将尝试寻找新的方法，特别是对于离散对数问题。第三个问题是关于有限域上的椭圆曲线及其在素数检验中的应用。希望能找到一种构造具有给定素数的定阶模的椭圆曲线的多元对数算法。这是从主要研究者给出的一个多元素数检验的结果中得知的。本研究属于计算数论领域，重点研究分解整数和检验它们是否为素数的算法。

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.