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Boomerang uniformity of normalized permutation polynomials of low degree

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Abstract

Differential uniformity of permutation polynomials has been studied intensively in recent years due to the differential cryptanalysis of S-boxes. The boomerang attack is a variant of differential cryptanalysis which combines two differentials for the upper part and the lower part of the block cipher. The boomerang uniformity measures the resistance of block ciphers to the boomerang attack. In this paper, by using the resultant elimination method, we study the boomerang uniformity of normalized permutation polynomials of the low degree over finite fields. As a result, we determine the boomerang uniformity of all normalized permutation polynomials of degree up to six over the finite field \({\mathbb {F}}_{q}\).

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Acknowledgements

The authors would like to thank the anonymous referees and Professor Zhengbang Zha for their valuable comments and helpful suggestions which improved both the quality and presentation of this paper. This work was supported in part by the China Scholarship Council, the Fundamental Research Funds for the Central Universities and the Innovation Fund of Xidian University, the National Natural Science Foundation of China (Grant 61972303, 61672414, 61602361), the National Cryptography Development Fund (Grant MMJJ20170113) and the NSERC of Canada.

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Authors and Affiliations

  1. State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, 710071, China

    Yan-Ping Wang & Wei-Guo Zhang

  2. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Yan-Ping Wang & Qiang Wang

  3. State Key Laboratory of Information Security, Chinese Academy of Sciences, Beijing, 10009, China

    Wei-Guo Zhang

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  1. Yan-Ping Wang
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Correspondence to Qiang Wang or Wei-Guo Zhang.

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Wang, YP., Wang, Q. & Zhang, WG. Boomerang uniformity of normalized permutation polynomials of low degree. AAECC 31, 307–322 (2020). https://doi.org/10.1007/s00200-020-00431-1

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  • DOI: https://doi.org/10.1007/s00200-020-00431-1

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