Computing discrete logarithms with quadratic number rings

Abstract

At present, there are two competing index calculus variants for computing discrete logarithms in (ℤ/pℤ)* in practice. The purpose of this paper is to summarize the recent practical experience with a generalized implementation covering both a variant of the Number Field Sieve and the Gaussian integer method. By this implementation we set a record with p consisting of 85 decimal digits. With regard to computational results, including the running time, we provide a comparison of the two methods for this value of p.