Binary bibraces and applications to cryptography

Skew braces have been found to be relevant in many areas of algebra, most notably in the study of set-theoretic solutions to the Yang-Baxter equation. In the context of differential cryptanalysis, alternative operations for attacking block ciphers, different from the XOR, can be obtained from a special family of braces, which we introduce here as binary bibraces. We show that binary bibraces can also be described in terms of alternating binary algebras of nilpotency class two, and therefore we provide a complete classification of binary bibraces and binary alternating algebras. We also present a characterisation of the automorphisms of these structures which allows their explicit construction by algorithms implemented in the Magma computational algebra system. By doing so, we characterise the XOR-linear permutations for which the differential transitions with respect to the new difference are deterministic, solving an open problem which can now facilitate differential attacks.