Two Notions of Differential Equivalence on Sboxes

Christina Boura (1, 2) , Anne Canteaut (1) , Jérémy Jean (3) , Valentin Suder (2)

1 SECRET - Security, Cryptology and Transmissions
2 LMV - Laboratoire de Mathématiques de Versailles
3 LCR - Laboratoire de cryptographie de l'ANSSI

In this work, we discuss two notions of differential equivalence on Sboxes. First, we introduce the notion of DDT-equivalence which applies to vectorial Boolean functions that share the same difference distribution table (DDT). Next, we compare this notion to what we call the $γ$-equivalence, applying to vectorial Boolean functions whose DDTs have the same support. We discuss the relation between these two equivalence notions, demonstrate that the number of DDT- or $γ$--equivalent functions is invariant under EA- and CCZ-equivalence and provide an algorithm for computing the DDT-equivalence and the $γ$-equivalence classes of a given function. We study the sizes of these classes for some families of Sboxes. Finally, we prove a result that shows that the rows of the DDT of an APN permutation are pairwise distinct.

Mots clés: Sbox; Equivalence; Boolean function; Difference distribution table; APN

Domaines: Cryptographie et sécurité [cs.CR]

Christina Boura, Anne Canteaut, Jérémy Jean, Valentin Suder. Two Notions of Differential Equivalence on Sboxes. Designs, Codes and Cryptography, In press, 87 (2-3), pp.185-202. ⟨10.1007/s10623-018-0496-z⟩. ⟨hal-01944565⟩