# Two notions of differential equivalence on Sboxes

Two notions of differential equivalence on Sboxes Des. Codes Cryptogr. https://doi.org/10.1007/s10623-018-0496-z 1,2 2 Christina Boura · Anne Canteaut · 3 1 Jérémy Jean · Valentin Suder Received: 21 December 2017 / Accepted: 17 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 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 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. Keywords Boolean function · Sbox · APN · Difference distribution table · Equivalence Mathematics Subject Classiﬁcation 94A60 This is one of several papers published in Designs, Codes and Cryptography comprising http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Designs, Codes and Cryptography Springer Journals

# Two notions of differential equivalence on Sboxes

, Volume OnlineFirst – Jun 1, 2018
18 pages

/lp/springer_journal/two-notions-of-differential-equivalence-on-sboxes-bQ783bFAY5
Publisher
Springer US
Subject
Mathematics; Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits
ISSN
0925-1022
eISSN
1573-7586
D.O.I.
10.1007/s10623-018-0496-z
Publisher site
See Article on Publisher Site

### Abstract

Des. Codes Cryptogr. https://doi.org/10.1007/s10623-018-0496-z 1,2 2 Christina Boura · Anne Canteaut · 3 1 Jérémy Jean · Valentin Suder Received: 21 December 2017 / Accepted: 17 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 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 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. Keywords Boolean function · Sbox · APN · Difference distribution table · Equivalence Mathematics Subject Classiﬁcation 94A60 This is one of several papers published in Designs, Codes and Cryptography comprising

### Journal

Designs, Codes and CryptographySpringer Journals

Published: Jun 1, 2018

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