Foundations of Differential Cryptanalysis in Abelian Groups

Abstract

Differential cryptanalysis is based on differentials having relatively high probability. Hawkes and O’Connor studied maximal differential probabilities for three most common operations used as components in ciphers: XOR, modular addition and modular multiplication. They showed that the upper bound for these probabilities for two latter operations is actually a lower bound for maximal probabilities of differentials based on XOR. In this paper we analyse such bounds in abelian groups of odd order and generalize Hawkes and O’Connor’s result. We show that for all operations in these groups the upper bound is the same as for addition mod 2n. This means that no odd order abelian group leads to such high probabilities of differentials as in an abelian group with XOR operation.