Index calculus for abelian varieties of small dimension and the elliptic curve discrete logarithm problem

被引：59
|
作者：
Gaudry, Pierrick [1 ,2 ]
机构：
[1] Ecole Polytech, LIX, F-91128 Palaiseau, France
[2] LORIA, F-54506 Vandoeuvre Les Nancy, France
来源：
JOURNAL OF SYMBOLIC COMPUTATION | 2009年 / 44卷 / 12期
关键词：
Discrete logarithm problem; Elliptic curve; Index calculus; Weil descent; WEIL DESCENT; HYPERELLIPTIC CURVES; XEDNI CALCULUS; ALGORITHM; ATTACK;
D O I：
10.1016/j.jsc.2008.08.005
中图分类号：
TP301 [理论、方法];
学科分类号：
081202 ;
摘要：
We propose an index calculus algorithm for the discrete logarithm problem on general abelian varieties of small dimension. The main difference with the previous approaches is that we do not make use of any embedding into the Jacobian of a well-suited curve. We apply this algorithm to the Weil restriction of elliptic curves and hyperelliptic curves over small degree extension fields. In particular, our attack can solve an elliptic curve discrete logarithm problem defined over F-q3 in heuristic asymptotic running time (O) over tilde (q(4/3)); and an elliptic problem over F-q4 or a genus 2 problem over F-q2 in heuristic asymptotic running time (O) over tilde (q(3/2)). (c) 2008 Elsevier Ltd. All rights reserved.
引用
收藏
页码：1690 / 1702
页数：13
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