Improvement of Faugère et al.'s Method to Solve ECDLP

Solving the elliptic curve discrete logarithm problem (ECDLP) by using Gr¨obner basis has recently appeared as a new threat to the security of elliptic curve cryptography and pairing-based cryptosystems. At Eurocrypt 2012, Faug`ere, Perret, Petit and Renault proposed a new method using a multivariable polynomial system to solve ECDLP over finite fields of characteristic 2. At Asiacrypt 2012, Petit and Quisquater showed that this method may beat generic algorithms for extension degrees larger than about 2000. In this paper, we propose a variant of Faug`ere et al.’s attack that practically reduces the computation time and memory required. Our variant is based on the idea of symmetrization. This idea already provided practical improvements in several previous works for composite-degree extension fields, but its application to prime-degree extension fields has been more challenging. To exploit symmetries in an efficient way in that case, we specialize the definition of factor basis used in Faug`ere et al.’s attack to replace the original polynomial system by a new and simpler one. We provide theoretical and experimental evidence that our method is faster and requires less memory than Faug`ere et al.’s method when the extension degree is large enough.