Improving the Complexity of Index Calculus Algorithms in Elliptic Curves over Binary Fields

Abstract. The goal of this paper is to further study the index calculus method that was first introduced by Semaev for solving the ECDLP and later developed by Gaudry and Diem. In particular, we focus on the step which consists in decomposing points of the curve with respect to an appropriately chosen factor basis. This part can be nicely reformulated as a purely algebraic problem consisting in finding solutions to a multivariate polynomial f(x1, . . . , xm) = 0 such that x1, . . . , xm all belong to some vector subspace of F2n /F2. Our main contribution is the identification of particular structures inherent to such polynomial systems and a dedicated method for tackling this problem. We solve it by means of Gröbner basis techniques and analyze its complexity using the multi-homogeneous structure of the equations. A direct consequence of our results is an index calculus algorithm solving ECDLP over any binary field F2n in time O(2ω t ), with t ≈ n/2 (provided that a certain heu...

• Type: Journal
• Year: 2012
• Where: EUROCRYPT
• Authors: Jean-Charles Faugère, Ludovic Perret, Christophe Petit, Guénaël Renault
• Keywords: Cryptology, EUROCRYPT 2012, Multivariate Polynomial, Polynomial Systems, Vector Subspace