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Two notions of differential equivalence on Sboxes

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Abstract

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.

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Acknowledgements

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

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Authors and Affiliations

  1. University of Versailles, Versailles, France

    Christina Boura & Valentin Suder

  2. Inria, Paris, France

    Christina Boura & Anne Canteaut

  3. ANSSI, Paris, France

    Jérémy Jean

Authors
  1. Christina Boura
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  2. Anne Canteaut
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  3. Jérémy Jean
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  4. Valentin Suder
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Correspondence to Christina Boura.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Boura, C., Canteaut, A., Jean, J. et al. Two notions of differential equivalence on Sboxes. Des. Codes Cryptogr. 87, 185–202 (2019). https://doi.org/10.1007/s10623-018-0496-z

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