Abstract
Using recent results on solving the equation \(X^{2^k+1}+X+a=0\) over a finite field \({{\mathbb {F}}}_{2^n}\) provided by the second and the third authors, we address an open question raised by the first author in WAIFI 2014 concerning the APN-ness of the Kasami functions \(x\mapsto x^{2^{2k}-2^k+1}\) with \(\gcd (k,n)=1\), \(x\in {{\mathbb {F}}}_{2^n}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beth, T., Ding, C.: On almost perfect nonlinear permutations. Proceedings of Eurocrypt’ 93, Lecture Notes in Computer Science 765, pp. 65–76, (1994).
Carlet, C.: Open Questions on nonlinearity and on APN Functions. Koç, Ç, Mesnager, S., Savas, E. (Eds.): WAIFI 2014, LNCS 9061, 83–107, Springer, New York (2015).
Carlet C.: Vectorial Boolean functions for cryptography, chapter of the monography Boolean models and methods in mathematics. In: Crama Y., Hammer P. (eds.) Computer Science, and Engineering, pp. 398–469. Cambridge University Press, Cambridge (2010).
Carlet C., Ding C.: Highly nonlinear mappings. Special issue “Complexity Issues in Coding and Cryptography”, dedicated to Prof. Harald Niederreiter on the occasion of his 60th birthday. J. Complex. 20, 205–244 (2004).
Cohen S.D., Matthews R.W.: A class of exceptional polynomials. Trans. Am. Math. Soc. 345, 897–909 (1994).
Dillon J.F., Dobbertin H.: New cyclic difference sets with singer parameters. Finite Fields Appl. 10(3), 342–389 (2004).
Dobbertin, H.: Kasami power functions, permutation polynomials and cyclic difference sets. In: Proceedings of the NATO-A.S.I.Workshop Difference sets, sequences and their correlation properties, Bad Windsheim, Kluwer Verlag, pp. 133–158 (1998).
Dobbertin H.: Another proof of Kasami’s Theorem. Des. Codes Cryptogr. 17, 177–180 (1999).
Helleseth T., Kholosha A.: \(x^{2^l+1} + x + a\) and related affine polynomials over \({\mathbb{F}_{{2}^{k}}}\). Crypt. Commun. 2(1), 85–109 (2010).
Janwa, H., Wilson, R.: Hyperplane sections of Fermat varieties in P3 in char. \(2\) and some applications to cyclic codes. Proceedings of AAECC-10 Conference, pp. 139–152 (1998)
Kim K.H., Mesnager S.: Solving \(x^{2^k+1}+x+a=0\) in \({\mathbb{F}_{{2}^{n}}}\) with \(\gcd (n, k)=1\). Finite Fields Their Appl. 63, 12 (2020).
Kim, K.H., Choe, J., Mesnager, S.: Solving \(X^{q+1}+X+a=0\) over finite fields. Cryptologyhttp://arxiv.org/abs/1912.12648.
Nyberg, K., Knudsen, L.R.: Provable security against differential cryptanalysis. Proceedings of CRYPT0’ 92, Lecture Notes in Computer Science 740, pp. 566–574, (1993)
Nyberg, K.: Differentially uniform mappings for cryptography. Proceedings of EUROCRYPT’ 93, Lecture Notes in Computer Science 765, pp. 55–64, (1994).
Nyberg, K.: On the construction of highly nonlinear permutations. Proceedings of EUROCRYPT’ 92, Lecture Notes in Computer Science 658, pp. 92–98, (1993)
Nyberg, K.: Perfect non-linear S-boxes. Proceedings of EUROCRYPT’ 91, Lecture Notes in Computer Science. 547, pp. 378–386 (1992).
Williams K.S.: Note on cubics over \(GF(2^n)\) and \(GF(3^n)\). J. Number Theory 7(4), 361–365 (1975).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Ding.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Carlet, C., Kim, K.H. & Mesnager, S. A direct proof of APN-ness of the Kasami functions. Des. Codes Cryptogr. 89, 441–446 (2021). https://doi.org/10.1007/s10623-020-00830-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-020-00830-y
Keywords
- APN function
- Equation
- Müller–Cohen–Matthews (MCM) polynomial
- Dickson polynomial
- Zeros of a polynomial
- Irreducible polynomial