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Published in: Applicable Algebra in Engineering, Communication and Computing 2/2023

08-04-2021 | Original Paper

On full differential uniformity of permutations on the ring of integers modulo n

Authors: P. R. Mishra, Prachi Gupta, Atul Gaur

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 2/2023

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Abstract

In this paper, we report some interesting results on permutations on \({\mathbb {Z}}_{n}\), the ring of integers modulo n, having full differential uniformity. By full differential uniformity of a permutation f on \({\mathbb {Z}}_{n}\), we mean that the cardinality of the set \(\{x\in {\mathbb {Z}}_{n}: f(x+a)-f(x)=b\}\) is exactly n for some \(a,b\in {\mathbb {Z}}_{n}\setminus \{0\}\). We give a sufficient condition for an arbitrary map on \({\mathbb {Z}}_{n}\) to have full differential uniformity. A necessary and sufficient condition for a permutation to have full differential uniformity over the ring of integers modulo n is also given. Further, we propose an upper bound and two lower bounds on permutations with full differential uniformity on \({\mathbb {Z}}_{n}\). We prove that these bounds are non-trivial bounds and give the exact number of permutations with full differential uniformity for a certain class of moduli.

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Metadata
Title
On full differential uniformity of permutations on the ring of integers modulo n
Authors
P. R. Mishra
Prachi Gupta
Atul Gaur
Publication date
08-04-2021
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 2/2023
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-021-00503-w

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