Faster Index Calculus for the Medium Prime Case Application to 1175-bit and 1425-bit Finite Fields

Many index calculus algorithms generate multiplicative relations between smoothness basis elements by using a process called

Sieving

. This process allows us to quickly filter potential candidate relations, without spending too much time to consider bad candidates. However, from an asymptotic point of view, there is not much difference between sieving and straightforward testing of candidates. The reason is that even when sieving, some small amount of time is spent for each bad candidate. Thus, asymptotically, the total number of candidates contributes to the complexity.

In this paper, we introduce a new technique:

Pinpointing

, which allows us to construct multiplicative relations much faster, thus reducing the asymptotic complexity of relations’ construction. Unfortunately, we only know how to implement this technique for finite fields which contain a medium-sized subfield. When applicable, this method improves the asymptotic complexity of the index calculus algorithm in the cases where the sieving phase dominates. In practice, it gives a very interesting boost to the performance of state-of-the-art algorithms. We illustrate the feasability of the method with discrete logarithm records in two medium prime finite fields, the first of size 1175 bits and the second of size 1425 bits.