Balanced NUCOMP

Abstract : Arithmetic in the divisor class group of a hyperelliptic curve is a fundamental component of algebraic geometry packages implemented in computer algebra systems such as Magma and Sage. In this paper, we present an adaptation of Shanks’ NUCOMP algorithm for split model hyperelliptic curves of arbitrary genus that uses balanced divisors and includes a number of enhancements to optimize its efficiency in that setting. Our version of NUCOMP offers better performance than Cantor’s algorithm in the balanced divisor setting. Compared with Magma’s built-in arithmetic, our Magma implementation shows significant speed-ups for curves of all but the smallest genera, with the improvement increasing as the genus grows.