Abstract
We study the symmetric properties of APN functions as well as the structure and properties of the range of an arbitrary APN function. We prove that there is no permutation of variables that preserves the values of an APN function. Upper bounds for the number of symmetric coordinate Boolean functions in an APN function and its coordinate functions invariant under a cyclic shift are obtained. For n ≤ 6, some upper bounds for the maximal number of identical values of an APN function are given and a lower bound is found for different values of an arbitrary APN function of n variables.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Gorodilova, “Characterization of Almost Perfect Nonlinear Functions in Terms of Subfunctions,” Diskret. Mat. 27 (3), 3–16 (2015).
M. E. Tuzhilin, “Almost Perfect Nonlinear Functions,” Prikl. Diskret. Mat. No. 3, 14–20 (2009).
T. Beth and C. Ding, “On Almost Perfect Nonlinear Permutations,” in Advances in Cryptology—EUROCRYPT’93. Workshop on the Theory and Application of Cryptographic Techniques (Lofthus, Norway, May 23–27, 1993) (Springer, Heidelberg, 1994), pp. 65–76.
E. Bihamand A. Shamir. “Differential Cryptanalysis of DES-Like Cryptosystems,” J. Cryptology 4 (1), 3–72 (1991).
M. Brinkman and G. Leander, “On the Classification of APN Functions up to Dimension Five,” Des. Codes Cryptogr. 49 (1–3), 273–288 (2008).
K. A. Browning, J. F. Dillon, M. T. McQuistan, and A. J.Wolfe, “An APN Permutation in Dimension Six,” in Proceedings of the 9th International Conference on Finite Fields Applications (Dublin, Ireland, July 13–17, 2009) (AMS, 2010), pp. 33–42.
L. Budaghyan, Construction and Analysis of Cryptographic Functions (Springer, Cham, Switzerland, 2014).
C. Carlet, “Vectorial Boolean Functions for Cryptography,” in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Ed. by Y. Crama and P. Hammer (Cambridge Univ. Press, New York, 2010), pp. 398–472.
C. Carlet, “Open Questions on Nonlinearity and on APN Functions,” in Arithmetic of Finite Fields. Proceedings of the 5th International Workshop on the Arithmetic of Finite Fields (WAIFI 2014) (Gebze, Turkey, September 27–28, 2014) (Springer, Cham, Switzerland, 2015), pp. 83–107.
C. Carlet, P. Charpin, and V. Zinoviev, “Codes, Bent Functions, and Permutations Suitable for DES-Like Cryptosystems,” Des. Codes Cryptogr. 15 (2), 125–156 (1998).
J. Daemen and V. Rijmen, The Design of Rijdael: AES—the Advanced Encryption Standard (Springer, Heidelberg, 2002).
H. Dobbertin, “Another Proof of Kasami’s Theorem,” Des. Codes Cryptogr. 17 (1–3), 177–180 (1999).
K. Nyberg, “Differentially Uniform Mappings for Cryptography,” in Advances in Cryptology. Proceedings of Workshop on Theory Application Cryptography Techniques (Lofthus, Norway, May 23–27, 1993), Ed. by T. Helleseth (Springer, Heidelberg, 1994), pp. 55–64.
J. Pieprzyk and Ch. X. Qu, “Fast Hashing and Rotation-Symmetric Functions,” J. UCS. 5 (1), 20–31 (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.A. Vitkup, 2016, published in Diskretnyi Analiz i Issledovanie Operatsii, 2016, Vol. 23, No. 1, pp. 65–79.
Rights and permissions
About this article
Cite this article
Vitkup, V.A. On the symmetric properties of APN functions. J. Appl. Ind. Math. 10, 126–135 (2016). https://doi.org/10.1134/S1990478916010142
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478916010142