Bi-braces in connection to private-key cryptography

Abstract

Starting from an n-dimensional vector space (V,+) over F2, we define a class of group operations * whose corresponding group of translations is elementary abelian and regular. With the idea of using the new operation to detect undesired biases in the distribution of differences in a secure block cipher, we select * in such a way the resulting structure (V,+,*) turns out to be a brace with a mutual normalization property of the two translation groups. We show how understanding the group of automorphisms of structures as above is crucial in the application to cryptanalysis, as well as in the classification of braces in terms of binary skew symmetric matrices.