Solving a discrete logarithm problem with auxiliary input on a 160-bit elliptic curve

Abstract

A discrete logarithm problem with auxiliary input (DLPwAI) is a problem to find α from G, αG, α dG in an additive cyclic group generated by an element G of prime order r, and a positive integer d satisfying d|(r - 1). The infeasibility of this problem assures the security of some cryptographic schemes. In 2006, Cheon proposed a novel algorithm for solving DLPwAI (Cheon's algorithm). This paper reports our experimental results of Cheon's algorithm by implementing it with some speeding-up techniques. In fact, we have succeeded to solve DLPwAI on a pairing-friendly elliptic curve of 160-bit order in 1314 core days. Implications of our experiments on cryptographic schemes are also discussed. © 2012 International Association for Cryptologic Research.