Section: New Results
Improving the Complexity of Index Calculus Algorithms in Elliptic Curves over Binary Fields
In  , we 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. 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. We emphasize that the complexity obtained in the paper is very conservative in comparison to experimental results. We hope the new ideas provided here may lead to efficient index calculus based methods for solving ECDLP in theory and practice.