Algorithmic Problems in Number Theory

数域筛选是将整数分解为素数的最新算法。调查员将继续研究数场筛的速度。该计划是第一，研究在数域筛中构造的某些向量的精确分布;第二，开发工具来分析从非均匀分布中选择的向量将是独立的概率;第三，联合收割机将这些结果结合起来，以锐化已知的数域筛所花费的时间。 调查人员还将考虑从几何的观点在多精度算术的各种问题。 本文的研究属于数论中的一般数学领域福尔斯。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在各种 在诸如数据传输和处理以及通信系统的领域中的应用。

The number field sieve is the latest algorithm for factoring integers into primes. The investigator will continue his research into the speed of the number field sieve. The plan is first, to study the precise distribution of certain vectors constructed in the number field sieve; second, to develop tools to analyze the probability that vectors chosen from a non-uniform distributimn will be independent; third, combine these results to sharpen the known heuristics for the time taken by the number field sieve. The investigator will also consider various problems in multiprecision arithmetic from a geometric viewpoint. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.