Smoothness Test for Polynomials Defined Over Small Characteristic Finite Fields

The problem of determining whether a polynomial defined over a finite field ring is smooth or not with respect to a given degree, is the most intensive arithmetic operation of the so-called descent phase of index-calculus algorithms. In this paper, we present an analysis and efficient implementation of Coppersmith’s smoothness test for polynomials defined over finite fields with characteristic three. As a case study, we review the best strategies for obtaining a fast field and polynomial arithmetic for polynomials defined over the ring F_q[X], with q=3^6, and report the timings achieved by our library when computing the smoothness test applied to polynomials of several degrees defined in that ring. This software library was recently used in Adj et al. (Cryptology 2016 . http://eprint.iacr.org/2016/914 ), as a building block for achieving a record computation of discrete logarithms over the 4841-bit field 𝔽_3^6· 509 .