Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 185–202 | Cite as

Two notions of differential equivalence on Sboxes

  • Christina BouraEmail author
  • Anne Canteaut
  • Jérémy Jean
  • Valentin Suder
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


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 \(\gamma \)-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 \(\gamma \)-equivalent functions is invariant under EA- and CCZ-equivalence and provide an algorithm for computing the DDT-equivalence and the \(\gamma \)-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.


Boolean function Sbox APN Difference distribution table Equivalence 

Mathematics Subject Classification




The authors wish to thank Jean-Pierre Flori, Itai Dinur and Orr Dunkelman for helpful discussions.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of VersaillesVersaillesFrance
  2. 2.InriaParisFrance
  3. 3.ANSSIParisFrance

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