A note on the $ c $-differential spectrum of an AP$ c $N function

Tingting Pang; Nian Li; Xiangyong Zeng; Haiying Zhu

doi:10.3934/amc.2022075

Article Contents

2022, Volume 16, Issue 4: 935-945. Doi: 10.3934/amc.2022075

This issue Previous Article Two new classes of Hermitian self-orthogonal non-GRS MDS codes and their applications Next Article Some new quantum MDS codes derived from generalized Reed-Solomon codes

A note on the $ c $-differential spectrum of an AP$ c $N function

1.
School of Mathematics, Shandong University, Jinan 250100, China
2.
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China

^*Corresponding author: Nian Li
^*Corresponding author: Nian Li

Received: March 2022

Revised: August 2022

Early access: September 2022

Published: November 2022

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Abstract

Motivated by a recent work of Zhang and Yan on the $ c $-differential spectrum of some power functions over finite fields, we further study an AP$ c $N function and express its $ c $-differential spectrum in terms of $ (i, j, k)_2 $, i.e., the cardinality of the intersection $ (\mathcal{C}^{(2)}_i+1)\cap\mathcal{C}^{(2)}_j\cap(\mathcal{C}^{(2)}_k-1) $ for $ i, j, k\in\{0, 1\} $, where $ \mathcal{C}^{(2)}_0, \mathcal{C}^{(2)}_1 $ are the cyclotomic classes of order two over the finite field $ \mathbb{F}_{p^n} $, $ p $ is an odd prime and $ n $ is a positive integer. By virtue of the cyclotomic numbers of orders two and four, we determine the values of $ (i, j, k)_2 $ for $ i, j, k\in\{0, 1\} $, which may be of independent interest. As an application, we give another proof of the $ c $-differential spectrum of an AP$ c $N function over finite fields with characteristic $ 5 $. Further, we refine the result of Zhang and Yan in the sense that we completely characterize the conditions when the $ c $-differential equation of the AP$ c $N function has one solution and two solutions, respectively.

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Mathematics Subject Classification: Primary: 94A60; Secondary: 11T71.

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Table 1. Simplification of Equation (2)

$ x $	Equation	Discriminant	Solutions in $ \mathbb{F}_{5^n} $	$ x_ix'_i $
$ \mathcal{C}^{(2)}_{0, 0} $	$ x^2+(1-2b^{-1})x-b^{-1}=0 $	$ \Delta_1=1-b^{-2} $	$ x_{1}, x'_1=b^{-1}+2\pm2\sqrt{1-b^{-2}} $	$ 4b^{-1} $
$ \mathcal{C}^{(2)}_{0, 1} $	$ x^2+x-b^{-1}=0 $	$ \Delta_2=1-b^{-1} $	$ x_2, x'_2=2\pm2\sqrt{1-b^{-1}} $	$ 4b^{-1} $
$ \mathcal{C}^{(2)}_{1, 0} $	$ x^2+x+b^{-1}=0 $	$ \Delta_3=1+b^{-1} $	$ x_3, x'_3=2\pm2\sqrt{1+b^{-1}} $	$ b^{-1} $
$ \mathcal{C}^{(2)}_{1, 1} $	$ x^2+(1+2b^{-1})x+b^{-1}=0 $	$ \Delta_4=1-b^{-2} $	$ x_4, x'_4=-b^{-1}+2\pm2\sqrt{1-b^{-2}} $	$ b^{-1} $

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References

[1]	D. Bartoli and M. Calderini, On construction and (non)existence of $c$-(almost) perfect nonlinear functions, Finite Fields Appl., 72 (2021), 101835, 16 pp. doi: 10.1016/j.ffa.2021.101835.
[2]	D. Bartoli, M. Calderini, C. Riera and P. Stănică, Low $c$-differential uniformity for functions modified on subfields, Cryptogr. Commun., (2022). doi: 10.1007/s12095-022-00554-x.
[3]	C. Blondeau, A. Canteaut and P. Charpin, Differential properties of power functions, Int. J. Inf. Coding Theory, 1 (2010), 149-170. doi: 10.1504/IJICOT.2010.032132.
[4]	N. Borisov, M. Chew, R. Johnson and D. Wagner, Multiplicative differentials, International Workshop on Fast Software Encryption, 2365 (2002), 17-33. doi: 10.1007/3-540-45661-9_2.
[5]	C. Boura and A. Canteaut, On the boomerang uniformity of cryptographic Sboxes, IACR Trans. Symmetric Cryptol., 3 (2018), 290-310. doi: 10.46586/tosc.v2018.i3.290-310.
[6]	L. Carlitz, Pairs of quadratic equations in a finite fields, Amer. J. Math., 76 (1954), 137-154. doi: 10.2307/2372405.
[7]	P. Ellingsen, P. Felke, C. Riera, P. Stănică and A. Tkachenko, $C$-differentials, multiplicative uniformity, and (almost) perfect $c$-nonlinearity, IEEE Trans. Inf. Theory, 66 (2020), 5781-5789. doi: 10.1109/TIT.2020.2971988.
[8]	S. U. Hasan, M. Pal, C. Riera and P. Stănică, On the $c$-differential uniformity of certain maps over finite fields, Des. Codes Cryptogr., 89 (2021), 221-239. doi: 10.1007/s10623-020-00812-0.
[9]	S. U. Hasan, M. Pal and P. Stănică, Boomerang uniformity of a class of power maps, Des. Codes Cryptogr., 89 (2021), 2627-2636. doi: 10.1007/s10623-021-00944-x.
[10]	S. U. Hasan, M. Pal and P. Stănică, The binary Gold function and its $c$-boomerang connectivity table, Cryptogr. Commun., (2022). doi: 10.1007/s12095-022-00573-8.
[11]	S. U. Hasan, M. Pal and P. Stănică, The $c$-differential uniformity and boomerang uniformity of two classes of permutation polynomials, IEEE Trans. Inf. Theory, 68 (2022), 679-691. doi: 10.1109/TIT.2021.3123104.
[12]	S. Mesnager, B. Mandal and M. Msahli, Survey on recent trends towards generalized differential and boomerang uniformities, Cryptogr. Commun., 14 (2021), 691-735. doi: 10.1007/s12095-021-00551-6.
[13]	S. Mesnager, C. Riera, P. Stănică, H. Yan and Z. Zhou, Investigations on $c$-(almost) perfect nonlinear functions, IEEE Trans. Inf. Theory, 67 (2021), 6916-6925. doi: 10.1109/TIT.2021.3081348.
[14]	K. Nyberg, Differentially uniform mappings for cryptography, Advances in Cryptology-EUROCRYPT '93, Lecture Notes in Comput. Sci., Springer, Berlin, 765 (1994), 55-64. doi: 10.1007/3-540-48285-7_6.
[15]	J. H. Silverman, The Arithmetic of Elliptic Curves, Second edition, Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-09494-6.
[16]	P. Stănică, Investigations on $c$-boomerang uniformity and perfect nonlinearity, Discrete Appl. Math., 304 (2021), 297-314. doi: 10.1016/j.dam.2021.08.002.
[17]	P. Stănică, Low $c$-differential and $c$-boomerang uniformity of the swapped inverse function, Discrete Math., 344 (2021), 112543.
[18]	P. Stănică and A. Geary, The $c$-differential behavior of the inverse function under the EA-equivalence, Cryptogr. Commun., 13 (2021), 295-306. doi: 10.1007/s12095-020-00466-8.
[19]	P. Stănică, C. Riera and A. Tkachenko, Characters Weil sums and $c$-differential uniformity with an application to the perturbed Gold function, Cryptogr. Commun., 13 (2021), 891-907. doi: 10.1007/s12095-021-00485-z.
[20]	T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, III. 1967
[21]	Z. Tu, X. Zeng, Y. Jiang and X. Tang, A class of AP$c$N power functions over finite fields of even characteristic, arXiv: 2107.06464.
[22]	X. Wang and D. Zheng, Several classes of P$c$N power functions over finite fields, arXiv: 2104.12942.
[23]	Y. Wu, N. Li and X. Zeng, New P$c$N and AP$c$N functions over finite fields, Des. Codes Cryptogr., 89 (2021), 2637-2651. doi: 10.1007/s10623-021-00946-9.
[24]	H. Yan, On $(-1)$-differential uniformity of ternary APN power functions, Cryptogr. Commun., 14 (2022), 357-369. doi: 10.1007/s12095-021-00526-7.
[25]	H. Yan and C. Li, Differential spectra of a class of power permutations with characteristic 5, Des. Codes Cryptogr., 89 (2021), 1181-1191. doi: 10.1007/s10623-021-00865-9.
[26]	Z. Zha and L. Hu, Some classes of power functions with low $c$-differential uniformity over finite fields, Des. Codes cryptogr., 89 (2021), 1193-1210. doi: 10.1007/s10623-021-00866-8.
[27]	K. Zhang and H. Yan, On the $c$-differential spectrum of power functions over finite fields, Des. Codes Cryptogr., (2022). doi: 10.1007/s10623-022-01086-4.