Abstract
Functions with low c-differential uniformity were proposed in 2020 and attracted lots of attention, especially the PcN and APcN functions, due to their applications in cryptography. The objective of this paper is to study PcN and APcN functions. As a consequence, we propose two classes of PcN functions and three classes of APcN functions by using the cyclotomic technique and the switching method. In addition, four classes of PcN or APcN functions are presented by virtue of the (generalized) AGW criterion.
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References
Akbary A., Ghioca B., Wang Q.: On constructing permutations of finite fields. Finite Fields Appl. 17(1), 51–67 (2011).
Bartoli D., Timpanella M.: On a generalization of planar functions. J. Algebr. Comb. 52, 187–213 (2020).
Bartoli D., Calderini M.: On construction and (non)existence of \(c\)-(almost) perfect nonlinear functions. Finite Fields Appl. (2021). https://doi.org/10.1016/j.ffa.2021.101835.
Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991).
Borisov N., Chew M., Johnson R., Wagner B.: Multiplicative Differentials. In: Daemen J., Rijmen V. (eds.) Fast Software Encryption. LNCS, vol. 2365. Springer, Berlin, Heidelberg (2002).
Budaghyan L., Helleseth T.: New perfect nonlinear multinomials over \({\mathbb{F}}_{p^{2k}}\) for any odd prime \(p\). In: Golomb S.W., Parker M.G., Pott A., Winterhof A. (eds.) Sequences and Their Applications. LNCS, vol. 5203. Springer, Berlin, Heidelberg (2008).
Coulter R., Matthews R.: Planar functions and planes of Lenz–Barlotti class II. Designs. Codes Cryptogr. 10(2), 167–184 (1997).
Coulter R., Henderson M., Hu L., Kosick P., Xiang Q., Zeng X.: Planar polynomials and commutative semifields two dimensional over their middle nucleus and four dimensional over their nucleus [Online]. Available: http://www.math.udel.edu/~coulter/papers/d24.pdf.
Dembowski P., Ostrom T.: Planes of order \(n\) with collineation groups of order \(n^{2}\). Math. Z. 103(3), 239–258 (1968).
Ding C., Yuan J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113(7), 1526–1535 (2006).
Dobbertin H.: Almost perfect nonlinear power functions on GF(\(2^n\)): A new case for \(n\) divisible by 5. In: Proceedings of the Finite Fields and Applications, pp. 113–121. Augsburg, Germany (1999).
Dobbertin H.: Almost perfect nonlinear power functions on GF(\(2^n\)): the Welch case. IEEE Trans. Inf. Theory 45(4), 1271–1275 (1999).
Dobbertin H.: Almost perfect nonlinear power functions on GF(\(2^n\)): the Niho case. Inf. Comput. 151(1–2), 57–72 (1999).
Ellingsen P., Felke P., Riera C., Stănică P., Tkachenko A.: \(C\)-differentials, multiplicative uniformity and (almost) perfect \(c\)-nonlinearity. IEEE Trans. Inf. Theory 66(9), 5781–5789 (2020).
Hasan S., Pal M., Riera C., Stănică P.: On the \(c\)-differential uniformity of certain maps over finite fields. Des. Codes Cryptogr. 89(2), 221–239 (2021).
Li N., Helleseth T., Tang X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Appl. 22, 16–23 (2013).
Mesnager S., Qu L.: On two-to-one mappings over finite fields. IEEE Trans. Inf. Theory 65(12), 7884–7895 (2019).
Mesnager S., Riera C., Stănică P., Yan H., Zhou Z.: Investigations on \(c\)-(almost) perfect nonlinear functions. Early access in IEEE Trans. Inf. Theory. https://doi.org/10.1109/TIT.2021.3081348.
Nyberg K.: Differnetially uniform mappings for cryptography. In: Helleseth T. (ed.) Proceedings of the EUROCRYPT 1993, vol. 765, pp. 55–64. LNCS. Springer, Heidelberg (1994).
Stănică P.: Low \(c\)-differential and \(c\)-boomerang uniformity of the swapped inverse function. Discret. Math. 344(10), (2021).
Stănică P., Geary A.: The \(c\)-differential behavior of the inverse function under the EA-equivalence. Cryptogr. Commun. 13(2), 295–306 (2021).
Stănică P., Riera C., Tkachenko A.: Characters, Weil sums and \(c\)-differential uniformity with an application to the perturbed Gold function. Cryptogr. Commun. https://doi.org/10.1007/s12095-021-00485-z.
Tu Z., Zeng X., Jiang Y., Tang X.: A class of \({\rm {AP}}c{\rm {N}}\) power functions over finite fields of even characteristi. arXiv:2107.06464v1.
Wang X., Zheng D.: Several classes of \({\rm {P}}c{\rm {N}}\) power functions over finite fields. arXiv:2104.12942v1.
Yan H., Mesnager S., Zhou Z.: Power functions over finite fields with low \(c\)-differential uniformity. arXiv:2003.13019v3.
Zha Z., Hu L.: Some classes of power functions with low \(c\)-differential uniformity over finite fields. Des. Codes Cryptogr. 89, 1193–1210 (2021).
Zha Z., Kyureghyan G.M., Wang X.: Perfect nonlinear binomials and their semifields. Finite Fields Appl. 15(2), 125–133 (2009).
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Funding was provided by Application Foundation Frontier Project of Wuhan Science and Technology Bureau (Grant No. 2020010601012189) and National Natural Science Foundation of China (Grant Nos. 62072162 and 61761166010).
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Communicated by G. Kyureghyan.
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Wu, Y., Li, N. & Zeng, X. New PcN and APcN functions over finite fields. Des. Codes Cryptogr. 89, 2637–2651 (2021). https://doi.org/10.1007/s10623-021-00946-9
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DOI: https://doi.org/10.1007/s10623-021-00946-9