Abstract
Very recently, a class of cryptographically strong permutations with boomerang uniformity 4 and the best known nonlinearity is constructed from the closed butterfly structure in Li et al. (Des Codes Cryptogr 89(4):737–761, 2021). In this note, we provide two additional results concerning these permutations. We first represent the conditions of these permutation obtained in Li et al. (Des Codes Cryptogr 89(4):737–761, 2021) in a much simpler form, and then show that they are linear equivalent to Gold functions. We also prove a criterion for solving a new type of equations over finite fields, which is useful and may be of independent interest.
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References
Bluher A.W.: On \(x^{q+1}+ax+b\). Finite Fields Appl. 10(3), 285–305 (2004).
Boura C., Canteaut A.: On the boomerang uniformity of cryptographic Sboxes. IACR Trans. Symmetric Cryptol. 3, 290–310 (2018).
Canteaut A., Duval S., Perrin L.: A generalisation of Dillon’s APN permutation with the best known differential and nonlinear properties for all fields of size \(2^{4k+2}\). IEEE Trans. Inf. Theory 63(11), 7575–7591 (2017).
Cid C., Huang T., Peyrin T., Sasaki Y., Song L.: Boomerang Connectivity Table: A New Cryptanalysis Tool, Advances in Cryptology-EUROCRYPT 2018, Part II, pp. 683–714, Lecture Notes in Comput. Sci., vol. 10821. Springer, Cham (2018).
Fu S., Feng X., Wu B.: Differentially \(4\)-uniform permutations with the best known nonlinearity from butterflies. IACR Trans. Symmetric Cryptol. 2, 228–249 (2017).
Helleseth T., Kholosha A.: On the equation \(x^{2^l+1}+x+a\) over \({\rm GF}(2^k)\). Finite Fields Appl. 14(1), 159–176 (2008).
Li K., Qu L., Li C., Chen H.: On a conjecture about a class of permutation quadrinomials. Finite Fields Appl. 66, 101690 (2020).
Li K., Li C., Helleseth T., Qu L.: Cryptographically strong permutations from the butterfly structure. Des. Codes Cryptogr. 89(4), 737–761 (2021).
Li N., Hu Z., Xiong M., Zeng X.: \(4\)-Uniform BCT permutations from generalized butterfly structure, arXiv:2001.00464.
Li N., Xiong M., Zeng X.: On permutation quadrinomials and \(4\)-uniform BCT. IEEE Trans. Inf. Theory 67(7), 4845–4855 (2021).
Li Y., Tian S., Yu Y., Wang M.: On the generalization of butterfly structure. IACR Trans. Symmetric Cryptol. 2, 160–179 (2018).
Lidl R., Niederreiter H.: Finite Fields, Encyclopedia of Mathematics, vol. 20. Cambridge University Press, Cambridge (1997).
Mesnager S., Kim K., Choe J., Lee D., Go D.: Solving \(x+x^{2^l}+\cdots +x^{2^{ml}}=a\) over \(\mathbb{F}_{2^n}\). Cryptogr. Commun. 12(4), 809–817 (2020).
Perrin L., Udovenko A., Biryukov A.: Cryptanalysis of a Theorem: Decomposing the only known solution to the big APN problem. In: Robshaw M., Katz J. (eds.) LNCS, vol. 9816, pp. 93–122. Springer (2016).
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).
Tu Z., Liu X., Zeng X.: A revisit of a class of permutation quadrinomial. Finite Fields Appl. 59, 57–85 (2019).
Wagner D.: The boomerang attack. In: Knudsen L.R. (ed.) FSE’1999, LNCS, vol. 1636, pp. 156–170. Springer, Heidelberg (1999).
Acknowledgements
The authors would like to thank Dr. Chunming Tang and Haode Yan for helpful discussions. This work was supported by the National Natural Science Foundation of China (Nos. 62072162, 61761166010, 12001176), the Research Grants Council (RGC) of Hong Kong (No. N_HKUST619/17), the National Key Research and Development Project (No. 2018YFA0704702) and the Application Foundation Frontier Project of Wuhan Science and Technology Bureau (No. 2020010601012189).
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Communicated by G. Kyureghyan.
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Li, N., Hu, Z., Xiong, M. et al. A note on “Cryptographically strong permutations from the butterfly structure”. Des. Codes Cryptogr. 90, 265–276 (2022). https://doi.org/10.1007/s10623-021-00974-5
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DOI: https://doi.org/10.1007/s10623-021-00974-5