Algebraic attacks over GF(q)

Abstract

Recent algebraic attacks on LFSR-based stream ciphers and S-boxes have generated much interest as they appear to be extremely powerful. Theoretical work has been developed focusing around the Boolean function case. In this paper, we generalize this theory to arbitrary finite fields and extend the theory of annihilators and ideals introduced at Eurocrypt 2004 by Meier, Pasalic and Carlet. In particular, we prove that for any function / in the multivariate polynomial ring over GF(q), f has a low degree multiple precisely when two low degree functions appear in the same coset of the annihilator of fq-1 -1. In this case, many such low degree multiples exist. © Springer-Verlag 2004.