Volume 23, No 1, 2016, P. 65-81
V. A. Vitkup
On symmetric properties of APN-functions
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 ≤ 6. A lower bound for the number of different values of an arbitrary APN function is proven.
Keywords: vectorial Boolean function, APN function, symmetric function.
Valeriya A. Vitkup 1,2
1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
2. Novosibirsk State University,
2 Pirogov St., 630090 Novosibirsk, Russia
Received 11 April 2015
Revised 16 August 2015
 A. A. Gorodilova, A characterization of almost perfect nonlinear functions in terms of subfunctions, Diskretn. Mat., 27, No. 3, 3–16, 2015.
 M. E. Tuzhilin, Almost perfect nonlinear functions, Prikl. Diskretn. Mat., No. 3, 14–20, 2009.
 T. Beth and C. Ding, On almost perfect nonlinear permutations, in T. Helleseth, ed., Advances in Cryptology — EUROCRYPT’93 (Proc. Workshop Theory Appl. Cryptogr. Tech., Lofthus, Norway, May 23–27, 1993), pp. 65–76, Springer-Verl., Heidelberg, 1994 (Lect. Notes in Comput. Sci., Vol. 765).
 E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, J. Cryptol., 4, No. 1, 3–72, 1991.
 M. Brinkman and G. Leander, On the classification of APN functions up to dimension five, Des. Codes Cryptogr., 49, No. 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 Finite Fields: Theory and Applications (Proc. 9th Int. Conf. Finite Fields Appl., Dublin, Ireland, July 13–17, 2009), pp. 33–42, AMS, 2010 (Contemp. Math., Vol. 518).
 L. Budaghyan, Construction and Analysis of Cryptographic Functions, Springer-Verl., Cham, Switzerland, 2014.
 C. Carlet, Vectorial Boolean functions for cryptography, in Y. Crama and P. Hammer, eds., Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 398–472, Cambridge Univ. Press, New York, 2010 (Encycl. Math. Its Appl., Vol. 134).
 C. Carlet, Open questions on nonlinearity and on APN functions, in Arithmetic of Finite Fields (Revised Sel. Pap. 5th Int. Workshop Arith. Finite Fields, Gebze, Turkey, Sept. 27–28, 2014), pp. 83–107, Springer-Verl., Cham, Switzerland, 2015. (Lect. Notes Comput. Sci., Vol. 9061).
 C. Carlet, P. Charpin, and V. A. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15, No. 2, 125–156, 1998.
 J. Daemen and V. Rijmen, The Design of Rijdael: AES — The Advanced Encryption Standard, Springer-Verl., Heidelberg, 2002.
 H. Dobbertin, Another proof of Kasami’s Theorem, Des. Codes Cryptogr., 17, No. 1–3, 177–180, 1999.
 K. Nyberg, Differentially uniform mappings for cryptography, in T. Helleseth, ed., Advances in Cryptology — EUROCRYPT’93 (Proc. Workshop Theory Appl. Cryptogr. Tech., Lofthus, Norway, May 23–27, 1993), pp. 55–64, Springer-Verl., Heidelberg, 1994 (Lect. Notes Comput. Sci., Vol. 765).
 J. Pieprzyk and Ch. X. Qu, Fast hashing and rotation-symmetric functions, J. UCS, 5, No. 1, 20–31, 1999.