Abstract
We consider the boomerang uniformity of an infinite class of (locally-APN) power maps and show that their boomerang uniformity over the finite field \(\mathbb {F}_{2^n}\) is 2 and 4, when \(n \equiv 0 \pmod 4\) and \(n \equiv 2 \pmod 4\), respectively. As a consequence, we show that for this class of power maps, the differential uniformity is strictly greater than their boomerang uniformity.
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References
Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991).
Blondeau C., Canteaut A., Charpin P.: Differential properties of \(x \mapsto x^{2^t-1}\). IEEE Trans. Inf. Theory 57(12), 8127–8137 (2011).
Boura C., Canteaut A.: On the boomerang uniformity of cryptographic S-boxes. IACR Trans. Symmetric Cryptol. 3, 290–310 (2018).
Browning K.A., Dillon J.F., McQuistan M.T., Wolfe A.J.: An APN permutation in dimension six. In: McGuire G., et al. (eds.) Proceedings of the 9th International Conference on Finite Fields and Applications, Contemporary Mathematics, vol. 518, pp. 33–42. American Mathematical Society, Providence (2010).
Calderini M.: Differentially low uniform permutations from known \(4\)-uniform functions. Des. Codes Cryptogr. 89, 33–52 (2021).
Calderini M., Villa I.: On the boomerang uniformity of some permutation polynomials. Cryptogr. Commun. 12, 1161–1178 (2020).
Cid C., Huang T., Peyrin T., Sasaki Y., Song L.: Boomerang connectivity table: a new cryptanalysis tool. In: Nielsen J., Rijmen V. (eds.) Advances in Cryptology—EUROCRYPT 2018, LNCS 10821, pp. 683–714. Springer, Cham (2018).
Li K., Qu L., Sun B., Li C.: New results about the boomerang uniformity of permutation polynomials. IEEE Trans. Inf. Theory 65(11), 7542–7553 (2019).
Li K., Li C., Helleseth T., Qu L.: Cryptographically strong permutations from the butterfly structure. Des. Codes Cryptogr. 89, 737–761 (2021).
Li N., Hu Z., Xiong M., Zeng X.: \(4\)-uniform BCT permutations from generalized butterfly structure. Preprint at https://arxiv.org/abs/2001.00464
Mesnager S., Tang C., Xiong M.: On the boomerang uniformity of quadratic permutations. Des. Codes Cryptogr. 88(10), 2233–2246 (2020).
Nyberg K.: Differentially uniform mappings for cryptography. In: Helleseth T. (ed.) Advances in Cryptology—EUROCRYPT’93, LNCS 765, pp. 55–64. Springer, Berlin (1993).
Tu Z., Li N., Zeng X., Zhou J.: A class of quadrinomial permutation with boomerang uniformity four. IEEE Trans. Inf. Theory 66(6), 3753–3765 (2020).
Wagner D.: The boomerang attack. In: Knudsen L.R. (ed.) Proceedings of Fast Software Encryption—FSE 1999, LNCS 1636, pp. 156–170. Springer, Heidelberg (1999).
Zha Z., Hu L.: The Boomerang uniformity of power permutations \(x^{2^k-1}\) over \(\mathbb{F}_{2^n}\). In: 2019 Ninth International Workshop on Signal Design and Its Applications in Communications (IWSDA), pp. 1–4 (2019)
Acknowledgements
We would like to express our sincere appreciation to the editors for handling our paper and to the reviewers for their careful reading, beneficial comments and constructive suggestions.
The research of Sartaj Ul Hasan is partially supported by MATRICS Grant MTR/2019/000744 from the Science and Engineering Research Board, Government of India. The first and second named authors would also like to thank Sumit Kumar Pandey for some useful discussions.
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Hasan, S.U., Pal, M. & Stănică, P. Boomerang uniformity of a class of power maps. Des. Codes Cryptogr. 89, 2627–2636 (2021). https://doi.org/10.1007/s10623-021-00944-x
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DOI: https://doi.org/10.1007/s10623-021-00944-x