Abstract
Differentially \(4\)-uniform permutations on \({\mathbb {F}}_{2^{2k}}\) with high nonlinearity are chosen as Substitution boxes in many block ciphers and some stream ciphers. Recently, Qu et al. (IEEE Trans Inf Theory, 59(7), 4675–4686, 2013) introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially \(4\)-uniform permutations over \({\mathbb {F}}_{2^{2k}}\). Hence they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZ-inequivalent differentially \(4\)-uniform permutations over \({\mathbb {F}}_{2^{2k}}\) grows exponentially when \(k\) increases, which gives a positive answer to an open problem proposed in Qu et al.(IEEE Trans Inf Theory, 59(7), 4675–4686, 2013).
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Acknowledgments
We would like to thank the anonymous reviewer for the valuable comments which significantly improve the quality and presentation of this paper. Part of this work was done when the first author visited Hong Kong University of Science and Technology. He would like to thank Prof. Cunsheng Ding for the kind hospitality and discussions during this period. The research of Chao Li is supported by the National Basic Research Program of China 2013CB338002 and the open research fund of Science and Technology on Information Assurance Laboratory (Grant No. KJ-12-02). The research of Longjiang Qu is supported by the National Natural Science Foundation of China (No. 61272484), the Research Project of National University Defense Technology under Grant CJ 13-02-01 and the Program for New Century Excellent Talents in University (NCET).
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Qu, L., Tan, Y., Li, C. et al. More constructions of differentially 4-uniform permutations on \({\mathbb {F}}_{2^{2k}}\) . Des. Codes Cryptogr. 78, 391–408 (2016). https://doi.org/10.1007/s10623-014-0006-x
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DOI: https://doi.org/10.1007/s10623-014-0006-x
Keywords
- Differentially \(4\)-uniform permutation
- Substitution box
- Preferred function
- Preferred Boolean function