超椭圆曲线离散对数的Weil Descent代数攻击的分析

An analysis of Weil Descent algebraic attack on the hyperelliptic curves discrete logarithm

本文首先介绍了Galbraith的Weil Descent代数攻击方法,然后对定义在GF(qn)上的形如y2+xy=f(x)的HCDLP能否用Weil Descent代数方法攻击作了详细讨论,作为例子,研究了GF(4)和GF(8)上的这类曲线。得到结论:(1)Weil Descent代数攻击法只能适用于极少部分这类超椭圆曲线;(2)当亏格或基域增大时,Weil Descent方法攻击成功的概率趋向于0。所以说Weil Descent代数攻击法对建立在GF(2n)上的这类超椭圆曲线密码体制并没有太大的威胁。

In this paper, we analyze the Weil Descent algebraic method generalized by Galbraith, and discuss whether or not the Weil Descent?method can be used to attack the DLP in the Jacobian of hyperelliptic curves with the form of y2+xy=f(x) defined over GF(qn) in detail. As examples we study the hyperelliptic curves with 2 0f this form over GF(4) and GF(8). We get the following conclusions: (1) Weil Descent' can be applied to a small proportion of the set of such curves only; (2) the probability of successful attack by Weil Descent' tends to 0 with the increase of the genus or the size of base field. This implies that the Weil Descent' algebraic attack method is not a threat to hyperelliptic curves with the form of y2+xy=f(x) defined over GF(2r) when r is large.