Quadratic relations for s-boxes: Their minimum representations and bounds

Abstract

We introduce polynomial approximations and consider the particular case of quadratic approximations. We establish an isomorphism between the set of quadratic Boolean functions and graphs. As its consequence, we can reduce problems involving quadratic Boolean functions into problems with graphs and vice-versa. We present the problem of finding a minimum representation of quadratic functions, and prove bounds on the number of terms and variables. With these bounds, we were able to find quadratic relations with the highest probabilities for SERPENT and CRYPTON, former AES candidates. © 2001 Springer-Verlag Berlin Heidelberg.