Geodesic
On symmetric properties of APN functions
Diskretnyj analiz i issledovanie operacij, Tome 23 (2016) no. 1, pp. 65-81.

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We study symmetric properties of APN functions and the structure of their images. It is proven that there is no permutation of variables which keeps an APN function values. Upper bounds for the number of symmetric coordinate Boolean functions in APN function are obtained. Also, there are proven upper bounds for the number of coordinate Boolean functions of an APN function which are invariant under circular translation of indices. Upper bounds for the maximal number of coincidental values are obtained for $n\le6$. A lower bound for the number of different values of an arbitrary APN function is proven. Bibliogr. 14.
Keywords: vectorial Boolean function, APN function, symmetric function. 
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V. A. Vitkup. On symmetric properties of APN functions. Diskretnyj analiz i issledovanie operacij, Tome 23 (2016) no. 1, pp. 65-81. http://geodesic.mathdoc.fr/item/DA_2016_23_1_a4/

[1] A. A. Gorodilova, “A characterization of almost perfect nonlinear functions in terms of subfunctions”, Diskretn. Mat., 27:3 (2015), 3–16 | DOI

[2] M. E. Tuzhilin, “Almost perfect nonlinear functions”, Prikl. Diskretn. Mat., 2009, no. 3, 14–20

[3] Beth T., Ding C., “On almost perfect nonlinear permutations”, Advances in cryptology, Proc. Workshop Theory Appl. Cryptogr. Tech. (Lofthus, Norway, May 23–27, 1993), Lect. Notes Comput. Sci., 765, Springer-Verl., Heidelberg, 1994, 65–76 | DOI | MR | Zbl

[4] Biham E., Shamir A., “Differential cryptanalysis of DES-like cryptosystems”, J. Cryptology, 4:1 (1991), 3–72 | DOI | MR | Zbl

[5] Brinkman M., Leander G., “On the classification of APN functions up to dimension five”, Des. Codes Cryptogr., 49:1–3 (2008), 273–288 | DOI | MR | Zbl

[6] Browning K. A., Dillon J. F., McQuistan M. T., Wolfe A. J., “An APN permutation in dimension six”, Proc. 9th Int. Conf. Finite Fields Appl. (Dublin, Ireland, July 13–17, 2009), Contemp. Math., 518, AMS, 2010, 33–42 | DOI | MR | Zbl

[7] Budaghyan L., Construction and analysis of cryptographic functions, Springer-Verl., Cham, Switzerland, 2014, 168 pp. | MR | Zbl

[8] Carlet C., “Vectorial Boolean functions for cryptography”, Boolean models and methods in mathematics, computer science, and engineering, Encycl. Math. Its Appl., 134, Cambridge Univ. Press, New York, 2010, 398–472

[9] Carlet C., “Open questions on nonlinearity and on APN functions”, Arithmetic of finite fields, Proc. 5th Int. Workshop on the Arithmetic of Finite Fields, WAIFI 2014 (Gebze, Turkey, Sept. 27–28, 2014), Lect. Notes Comput. Sci., 9061, Springer-Verl., Cham, Switzerland, 2015, 83–107 | DOI | MR | Zbl

[10] Carlet C., Charpin P., Zinoviev V., “Codes, bent functions and permutations suitable for DES-like cryptosystems”, Des. Codes Cryptogr., 15:2 (1998), 125–156 | DOI | MR | Zbl

[11] Daemen J., Rijmen V., The Design of Rijdael: AES – the Advanced Encryption Standard, Springer-Verl., Heidelberg, 2002, 256 pp. | MR

[12] Dobbertin H., “Another proof of Kasami's theorem”, Des. Codes Cryptogr., 17:1–3 (1999), 177–180 | DOI | MR | Zbl

[13] Nyberg K., “Differentially uniform mappings for cryptography”, Advances in cryptology, Proc. Workshop Theory Appl. Cryptogr. Tech. (Lofthus, Norway, May 23–27, 1993), Lect. Notes Comput. Sci., 765, Springer-Verl., Heidelberg, 1994, 55–64 | DOI | MR | Zbl

[14] Pieprzyk J., Qu C. X., “Fast hashing and rotation-symmetric functions”, J. UCS, 5:1 (1999), 20–31 | MR