Weakness of F_(3~(6·1429)) and F_(2~(4·3041)) for discrete logarithm cryptography

摘要

In the past two years, there have been several dramatic improvements in algorithms for computing discrete logarithms in small-characteristic finite fields. In this paper, we examine the effectiveness of these new algorithms for computing discrete logarithms in F_(3~(6·1429)) and F_(2~(4·3041)). The intractability of the discrete logarithm problem in these fields is necessary for the security of bilinear pairings derived from supersingular curves with embedding degree 6 and 4 defined, respectively, over F_(3~(1429)) and F_(2~(3041)); these curves were believed to enjoy a security level of 192 bits against attacks by Coppersmith's algorithm. Our analysis shows that these pairings offer security levels of at most 96 and 129 bits, respectively, leading us to conclude that they are dead for pairing-based cryptography.