An L(1/3) discrete logarithm algorithm for low degree curves

Abstract

We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in X and Y are low with respect to their genera. The finite base fields ℱ q are arbitrary, but their sizes should not grow too fast compared to the genus. For such families, the group structure and discrete logarithms can be computed in subexponential time of L qg (1/3, 0(1)). The runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve. © 2010 International Association for Cryptologic Research.