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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Acknowledgements
The authors are grateful to Dr. Dhananjoy Dey, Associate Professor, Indian Institute of Information Technology, Lucknow, Uttar Pradesh, India for the encouragement and support during the research. The second author would like to acknowledge the financial support by National Board for Higher Mathematics (NBHM), Department of Atomic Energy, India vide sanction no. 0203/6/2019/R&D-II/3659 and 0203/6/2019/R&D-II/10309.
The authors are extremely thankful to the referees for their valuable comments and suggestions. This led to the over all improvement in the quality and presentation of the paper.
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Mishra, P.R., Gupta, P. & Gaur, A. On full differential uniformity of permutations on the ring of integers modulo n. AAECC 34, 301–319 (2023). https://doi.org/10.1007/s00200-021-00503-w
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DOI: https://doi.org/10.1007/s00200-021-00503-w