Mathematical Sciences: Topics in Number Theory

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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.