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

Abstract

The main approach to solve hyperelliptic curve DLP is index calculus algorithm. One of the most important steps is to obtain relations among the factor basis elements. Although Nagao and Joux proposed decomposition methods respectively, their algorithms required solving a multi-variate system of non-linear equations. We propose a new method to find the relations among factor basis elements for a class of special hyperelliptic curves y2+y = x2g+1 + A0 over finite field K with the characteristic 2. Let G(x,y) = R(x)+yT(x) and H(x) be the norm of G(x,y). We prove that supp(div(G(x,y))) ⊆ supp(div(H(x)). Then we show how to obtain relations among factor basis, by solving several linear equations and univariate equations with relatively low degree. Let g be the genus of hyperelliptic curve. After (6g+3)! trials we may obtain a single relation. We also give some examples to illustrate the effectiveness of our algorithm when g = 2,3.