The index calculus method using non-smooth polynomials

Abstract:

We study a generalized version of the index calculus method for the discrete logarithm problem in ${\mathbb F}_{\! q}$, when $q=p^n$, $p$ is a small prime and $n\rightarrow \infty$. The database consists of the logarithms of all irreducible polynomials of degree between given bounds; the original version of the algorithm uses lower bound equal to one. We show theoretically that the algorithm has the same asymptotic running time as the original version. The analysis shows that the best upper limit for the interval coincides with the one for the original version. The lower limit for the interval remains a free variable of the process. We provide experimental results that indicate practical values for that bound. We also give heuristic arguments for the running time of the Waterloo variant and of the Coppersmith method with our generalized database.