A simple method for obtaining relations among factor basis elements for special hyperelliptic curves

Abstract

Nagao proposed a decomposition method for divisors of hyperelliptic curves defined over a field ${\mathbb {F}}_{q^n}$ with $n\ge 2$. Joux and Vitse later proposed a variant which provided relations among the factor basis elements. Both Nagao’s and the Joux–Vitse methods require solving a multi-variate system of polynomial equations. In this work, we revisit Nagao’s approach with the idea of avoiding the requirement of solving a multi-variate system. While this cannot be done in general, we are able to identify special cases for which this is indeed possible. Our main result is for curves $C:y^2=f(x)$ of genus g defined over ${\mathbb {F}}_{q^2}$ having characteristic >2. If there is no restriction on f(x), we show that it is possible to obtain a relation in $(4g+4)!$ trials. The number of trials, though high, quantifies the computation effort needed to obtain a relation. This is in contrast to the methods of Nagao and Joux–Vitse which are based on solving systems of polynomial equations, for which the computation effort is hard to precisely quantify. The new method combines well with a sieving technique proposed by Joux and Vitse. If f(x) has a special form, then the number of trials can be significantly lower. For example, if f(x) has at most g consecutive coefficients which are in ${\mathbb {F}}_{q^2}$ while the rest are in ${\mathbb {F}}_q$, then we show that it is possible to obtain a single relation in about $(2g+3)!$ trials. Our implementation of the resulting algorithm provides examples of factor basis relations for $g=5$ and $g=6$. To the best of our knowledge, none of the previous methods known in the literature can provide such relations faster than our method. Other than obtaining such decompositions, we also explore the applicability of our approach for $n>2$ and for binary characteristic fields.