Efficient computation of Cantor's division polynomials of hyperelliptic curves over finite fields

Abstract

Let p be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via p-adic differential equations with a sharp analysis of the loss of precision. Consequently, after having possibly lifted the problem in the p-adics, we derive fast algorithms for computing explicitly Cantor's division polynomials of hyperelliptic curves defined over finite fields.