Fast arithmetic on hyperelliptic curves via continued fraction expansions

Abstract

In this paper, we present a new algorithm for computing the reduced sum of two divisors of an arbitrary hyperelliptic curve. Our formulas and algorithms are generalizations of Shanks’s NUCOMP algorithm, which was suggested earlier for composing and reducing positive definite binary quadratic forms. Our formulation of NUCOMP is derived by approximating the irrational continued fraction expansion used to reduce a divisor by a rational continued fraction expansion, resulting in a relatively simple and efficient presentation of the algorithm as compared to previous versions. We describe a novel, unified framework for divisor reduction on an arbitrary hyperelliptic curve using the theory of continued fractions, and derive our formulation of NUCOMP based on these results. We present numerical data demonstrating that our version of NUCOMP is more efficient than Cantor’s algorithm for most hyperelliptic curves, except those of very small genus defined over small finite fields.