Semigroup of Matrices over GF{2^s} and Its Relation to AES

Abstract

We introduce to readers a new matrix C for MixColumn operation for AES algorithm for discussion. This matrix has significantly larger multiplicative order, ord(C) = 340, than the used one which is 4 only. This makes so called XSL attack less effective. It is possible to find such a matrix due to our new Euler-Fermat-like theorem and its corollaries for regular circulant matrices over GF(p^s).