Algorithmic Problems in Number Theory

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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.