# Two notions of differential equivalence on Sboxes

Two notions of differential equivalence on Sboxes 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 $$\gamma$$ 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 $$\gamma$$ γ -equivalent functions is invariant under EA- and CCZ-equivalence and provide an algorithm for computing the DDT-equivalence and the $$\gamma$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Designs, Codes and Cryptography Springer Journals

# Two notions of differential equivalence on Sboxes

Designs, Codes and Cryptography, Volume 87 (3) – Jun 1, 2018
18 pages

/lp/springer_journal/two-notions-of-differential-equivalence-on-sboxes-bQ783bFAY5
Publisher
Springer Journals
Subject
Computer Science; Cryptology; Coding and Information Theory; Data Structures and Information Theory; Cryptology; 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

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

### Journal

Designs, Codes and CryptographySpringer Journals

Published: Jun 1, 2018

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