Mathematical Sciences: Topics in Number Theory

本研究主要围绕三个方面展开。第一个是关于Artin关于合数的猜想。也就是说，关于模为合成数的最大阶元，人们能说些什么呢？第二个问题是寻找高效的离散对数算法和大数因式分解的高效算法。这位首席研究员和他的同事们建造了一台特殊用途的计算机，用于实现他的大整数因式分解的二次筛选算法。他将研究在这台计算机和其他计算机上实现线性和三次筛算法的实际方面，这些算法是很有前途的实用方法，他还将尝试寻找新的方法，特别是对于离散对数问题。第三个问题涉及有限域上的椭圆曲线及其在素性检验中的应用。人们希望找到一种构造以给定素数为模的预定阶数的椭圆曲线的PolyLog算法。这是从首席调查员给出PolyLog素数检验的结果中得知的。这项研究是在计算数论领域，主要集中在整数因式分解和测试它们是否为素数的算法上。

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.