Spectral and Nonlinear Properties of the Sum of Boolean Functions

Authors

  • O.N. Zhdanov Siberian State University of Science and Technology named after Academician M.F.Reshetnev, Russian
  • A.V. Sokolov Odessa National Polytechnic University, Ukraine

Keywords:

Boolean function, Walsh-Hadamard Transform, Distance of Nonlinearity, Cryptography,

Abstract

Boolean functions are the mathematical basis of modern cryptographic algorithms. However, in practice, a set of interrelated Boolean functions is often used to construct a cryptographic algorithm. This circumstance makes the task of research of cryptographic quality, in particular, the distance of the nonlinearity of the sum of few Boolean functions important. The nonlinearity distance of a Boolean function is determined by the maximum value of its Walsh-Hadamard transform coefficients. In this paper, we proposed a formula that is the equivalent of the summation of Boolean functions in the Walsh-Hadamard transform domain. The application of this formula, as well as the Walsh-Hadamard spectral classification made it possible to determine the structure of WalshHadamard transform coefficients, and the distance of the nonlinearity when summing the Boolean functions lengths N  8 and N 16 , indicating valuable practical application for information protection.

References

B. Schneier, Applied Cryptography. Second edition, New York, John Wiley & Sons(1996).

I.V. Agafonova, Cryptographic properties of non-linear Boolean functions, Seminar on discrete harmonic analysis and geometeric modeling, St. Petersburg DHA & CAGD (2007), 1–24.

O.N. Zhdanov, The method of selection of key information for the block cipher algorithm, Moscow, INFRA-M(2013), 90.

Rodier, F. (2003). On the nonlinearity of Boolean functions. In Proceedings of WCC2003, Workshop on coding and cryptography (pp. 397-405).

A.V. Sokolov, New methods for synthesizing nonlinear transform of modern ciphers, Lap Lambert Academic Publishing (2015), 100.

Canteaut, A., Carlet, C., Charpin, P., & Fontaine, C. (2000, May). Propagation characteristics and correlation-immunity of highly nonlinear Boolean functions. In International Conference on the Theory and Applications of Cryptographic Techniques (pp. 507-522). Springer, Berlin, Heidelberg.

M.I. Mazurkov, Broadband radio communication systems, Odessa, Science and Technology(2010), 340.

M.I. Mazurkov, A.V. Sokolov, The regular rules of constructing the complete class of bent-sequences of length 16, Proceedings of ONPU (2013), No.2(41), 231–237.

O.S. Rothaus, On "bent" functions J. Comb. Theory Ser. A, USA: Academic Press Inc (1976), No.20(3), 300–305.

A.V. Sokolov, Constructive method for the synthesis of nonlinear Sboxes satisfying the strict avalanche criterion, Radioelectronics and Communications Systems(2013), 56, 8, 415–423.

M.I. Mazurkov, N.A. Barabanov, A.V. Sokolov, The key sequences generator based on bent functions dual couples, Proceedings of ONPU(2013), No.3(42), 150–156.

K. G. Paterson, Sequences For OFDM and Multi-code CDMA: two problems in algebraic coding theory, Sequences and their applications. Seta(2001,2002). Second Int. Conference. Proc. Berlin: Springer, 46–71.

Mazurkov M.I. Synthesis method for families of constant amplitude correcting codes based on an arbitrary bent-square / M.I. Mazurkov, A.V. Sokolov, I.V. Tsevukh. — Journal of Telecommunication, Electronic and Computer Engineering (JTEC). — Vol. 2. — No.9. — P. 99-103.

M.I. Mazurkov, A.V. Sokolov, Fast orthogonal transforms based on bent-sequences, Informatics and mathematical methods in simulation(2014), No. 1, 5–13.

A.V. Sokolov, N.A. Barabanov, Algorithm for removing the spectral equivalence of component Boolean functions of Nyberg-design Sboxes. Radioelectronics and Communications

Systems(2015), Vol. 58, Vo. 5, 220-227.

Downloads

Published

2019-05-24

How to Cite

Zhdanov, O., & Sokolov, A. (2019). Spectral and Nonlinear Properties of the Sum of Boolean Functions. Journal of Telecommunication, Electronic and Computer Engineering (JTEC), 11(2), 31–35. Retrieved from https://jtec.utem.edu.my/jtec/article/view/4139

Issue

Section

Articles