EN|RU
English version:
Journal of Applied and Industrial Mathematics, 2018, 12:1, 112-125

Volume 25, No 1, 2018, P. 98-119

UDC 519.7
A. V. Kutsenko
The Hamming distance spectrum between self-dual Maiorana–McFarland bent functions

Abstract:
A bent function is self-dual if it is equal to its dual function. We study the metric properties of the self-dual bent functions constructed on using available constructions. We find the full Hamming distance spectrum between self-dual Maiorana–McFarland bent functions. Basing on this, we find the minimal Hamming distance between the functions under study.
Bibliogr. 22.

Keywords: Hamming distance, self-dual bent function, Maiorana–McFarland bent function.

DOI: 10.17377/daio.2018.25.557

Alexander V. Kutsenko 1
1. Novosibirsk State University,
2 Pirogov St., 630090 Novosibirsk, Russia
e-mail: AlexandrKutsenko@bk.ru

Received 17 October 2016
Revised 27 July 2017

References

[1] N. A. Kolomeec and A. V. Pavlov, Properties of bent functions with minimal distance, Prikl. Diskretn. Mat., No. 4, 5–20, 2009.

[2] O. A. Logachev, A. A. Sal’nikov, S. V. Smyshlyaev, and V. V. Yashchenko, Bulevy funktsii v teorii kodirovaniya i kriptologii (Boolean functions in coding theory and cryptology), MTsNMO, Moscow, 2012.

[3] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977 (North-Holland Math. Libr., Vol. 16). Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Svyaz’, Moscow, 1979.

[4] A. K. Oblaukhov, Metric complements to subspaces in the Boolean cube, Diskretn. Anal. Issled. Oper., 23, No. 3, 93–106, 2016. Translated in J. Appl. Ind. Math., 10, No. 3, 397–403, 2016.

[5] V. N. Potapov, Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes, Probl. Peredachi Inf., 48, No. 1, 54–63, 2012. Translated in Probl. Inf. Transm., 48, No. 1, 47–55, 2012.

[6] N. N. Tokareva, On decomposition of a dual bent function into sum of two bent functions, Prikl. Discretn. Mat., No. 4, 59–61, 2014.

[7] L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha, and S. Mesnager, Further results on Niho bent functions, IEEE Trans. Inf. Theory, 58, No. 11, 6979–6985, 2012.

[8] C. Carlet, Boolean functions for cryptography and error-correcting codes, Boolean Models and Methods in Mathematics, Computer Science, and Engi?neering, pp. 257–397, Cambridge Univ. Press, New York, 2010 (Encycl. Math. Its Appl., Vol. 134).

[9] C. Carlet, L. E. Danielson, M. G. Parker, and P. Solé, Self-dual bent functions, Int. J. Inf. Coding Theory. 1, No. 4, 384–399, 2010.

[10] T. W. Cusick and P. Stanica, Cryptographic Boolean Functions and Applications, Acad. Press, London, 2017.

[11] T. Feulner, L. Sok, P. Solé, and A. Wassermann, Towards the classification of self-dual bent functions in eight variables, Des. Codes Cryptogr., 68, No. 1, 395–406, 2013.

[12] X.-D. Hou, On the coefficients of binary bent functions, Proc. Amer. Math. Soc., 128, No. 4, 987–996, 2000.

[13] X.-D. Hou, Classification of self dual quadratic bent functions, Des. Codes Cryptogr., 63, No. 2, 183–198, 2012.

[14] N. A. Kolomeec, The graph of minimal distances of bent functions and its properties, Des. Codes Cryptogr., 1–16, 2017. DOI 10.1007/s10623-016-0306-4

[15] N. A. Kolomeec and A. V. Pavlov, Bent functions on the minimal distance, Proc. IEEE Region 8 Int. Conf. Comput. Technol. Electr. Electron. Eng., Irkutsk, Russia, July 11–15, 2010, pp. 145–149, IEEE, Piscataway, 2010.

[16] P. Langevin and G. Leander, Monomial bent functions and Stickelberger’s theorem, Finite Fields Appl., 14, No. 3, 727–742, 2008.

[17] R. L. McFarland, A family of difference sets in non-cyclic groups, J. Comb. Theory, Ser. A, 15, No. 1, 1–10, 1973.

[18] S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory, 60, No. 7, 4397–4407, 2014.

[19] O. Rothaus, On “bent” functions, J. Comb. Theory. Ser. A, 20, No. 3, 300–305, 1976.

[20] N. N. Tokareva, Duality between bent functions and affine functions, Discrete Math., 312, No. 3, 666–670, 2012.

[21] N. N. Tokareva, Bent Functions: Results and Applications to Cryptography, Acad. Press, London, 2015.

[22] B. Xu, Dual bent functions on finite groups and C-algebras, J. Pure Appl. Algebra, 220, No. 3, 1055–1073, 2016.
 © Sobolev Institute of Mathematics, 2015