Curves and Their Applications to Factoring Polynomials

Abstract

We present new methods for computing square roots and factorization of polynomials over finite fields. We also describe a method for computing in the Jacobian of a singular hyperelliptic curve. There is a compact representation of an element in the Jacobian of a smooth hyperelliptic curve over any field. This compact representation leads an efficient method for computing in Jacobians which is called Cantor's Algorithm. In one part of the dissertation, we show that an extension of this compact representation and Cantor's Algorithm is possible for singular hyperelliptic curves. This extension lead to the use of singular hyperelliptic curves for factorization of polynomials and computing square roots in finite fields. Our study shows that computing the square root of a number mod p is equivalent to finding any of the particular group elements in the Jacobian of a certain singular hyperelliptic curve. This is also true in the case of polynomial factorizations. Therefore the efficiency of our algorithms depends on only the efficiency of the algorithms for computing in the Jacobian of a singular hyperelliptic curve. The algorithms for computing in Jacobians of hyperelliptic curves are very fast especially for small genus and this makes our algorithms especially computing square roots algorithms competitive with the other well-known algorithms. In this work we also investigate superelliptic curves for factorization of polynomials.