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Differential and boomerang spectrums of some power permutations

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Abstract

The differential (resp. boomerang) spectrum is an important parameter to estimate the resistance of cryptographic functions against some variants of differential (resp. boomerang) cryptanalysis. This paper aims to determine the differential and boomerang spectrums of some power permutations. In 1997, Helleseth and Sandberg proved that the differential uniformity of \(x^{\frac {p^{n}-1}{2}+2}\) over \(\mathbb {F}_{p^{n}}\), where p is an odd prime, is less than or equal to 4. In this paper, we first determine the differential spectrum of \(x^{\frac {3^{n}-1}{2}+2}\) over \(\mathbb {F}_{3^{n}}\) with n odd and then compute its boomerang spectrum based on the differential spectrum. In addition, in 2018, Boura and Canteaut determined the boomerang spectrum of the inverse function over \(\mathbb {F}_{2^{n}}\) with n even. Following their work, we characterize the boomerang spectrum of the inverse function \(x^{p^{n}-2}\) over \(\mathbb {F}_{p^{n}}\) for any odd prime p.

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Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referees for their helpful comments and suggestions, which have highly improved the paper’s technical and editorial qualities.

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Authors and Affiliations

  1. College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

    Sha Jiang, Kangquan Li, Yubo Li & Longjiang Qu

  2. Hunan Engineering Research Center of Commercial Cryptography Theory and Technology Innovation, Changsha, 410073, China

    Sha Jiang, Kangquan Li, Yubo Li & Longjiang Qu

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  1. Sha Jiang
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  2. Kangquan Li
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  4. Longjiang Qu
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Correspondence to Longjiang Qu.

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The research of Longjiang Qu is supported in part by the Nature Science Foundation of China (NSFC) under Grant 62032009.

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Jiang, S., Li, K., Li, Y. et al. Differential and boomerang spectrums of some power permutations. Cryptogr. Commun. 14, 371–393 (2022). https://doi.org/10.1007/s12095-021-00530-x

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