# Regular Synthesis Method of a Complete Class of Ternary Bent-Sequences and Their Nonlinear Properties

## Keywords:

Bent-Sequence, Many-Valued Logic, VilenkinChristenson Transform, Nonlinearity,## Abstract

A regular synthesis method of the full class of ternary bent-sequences based on their representation in the form of 3-bent-squares is developed. The formula for the nonlinearity estimation of ternary sequences, based on their VilenkinChristenson spectral coefficients is introduced. The distribution of nonlinearity of the full code of length*N=9*is researched.

## References

M.I. Mazurkov, V.Ya. Chechel'nitskii, “The classes of equivalent and generative perfect binary arrays for CDMA-technologies,” Radioelectron. Commun. Syst., vol. 46, no. 5, pp. 48-57, 2003.

M.I. Mazurkov, A.V. Sokolov, “Synthesis methods of pseudo-random binary sequences with the property of the k-gram distribution for encryption tasks,” Works of the Odessa polytechnic univer., no. 1(38), pp.188-198, 2012.

M. I. Mazurkov, A.V. Sokolov, “Constructive method for synthesis of complete classes of multilevel de Bruijn sequences,” Radioelectron. Commun. Syst., vol. 56, no. 1, pp. 36-41, 2013.

M. I. Mazurkov, A. V. Sokolov, “Nonlinear transformations based on complete classes of isomorphic and automorphic representations of field GF(256),” Radioelectron. Commun. Syst., vol. 56, no. 11, pp. 513-521, 2013.

O.S. Rothaus, “On “bent” functions,” J. Comb. Theory Ser. A. — USA: Academic Press Inc, no. 20(3), pp. 300-305, 1976.

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

A.V. Sokolov, “Constructive method for the synthesis of nonlinear Sboxes satisfying the strict avalanche criterion”, Radioelectron. Commun. Syst., vol. 56, no. 8, pp. 415-423, 2013.

M.I. Mazurkov, N.A. Barabanov, A.V. Sokolov, “The key sequences generator based on bent functions dual couples,” Works of the Odessa Polytechnic Univer., vol. 3(42), pp. 150-156, 2013.

A.V.Sokolov, “The cellular automata key sequences generator,” Works of the Odessa Polytechnic University, vol. 1 (43), pp. 180-186, 2014.

L. Bolc, P. Borowik, “Many-valued Logics 2: Automated reasoning and practical applications,” Springer Science & Business Media, 2003, p. 303.

M.I. Mazurkov, A.V. Sokolov, N.A. Barabanov, “On the effect of the type of orthogonal transform on PAPR of signal spectrum in CDMA systems,” Informatics and Math. Methods in Sim., vol. 5, no. 1, pp. 28-37, 2015.

O.N. Zhdanov, A.V. Sokolov, “Algorithm of construction of optimal according to criterion of zero correlation non-binary S-boxes,” PFMT, no. 3(24), pp. 94-97, 2015.

A.V. Sokolov, O.N. Zhdanov, N.A. Barabanov, “Pseudo-random key sequence generator based on triple sets of bent-functions,” PFMT, no. 1 (26), pp. 85-91, 2016.

S.O. Gnatyuk, T.S. Zhmurko, V.M. Kіnzeryavy, N.A. Seylova, “Method of trit pseudorandom sequences generating for quantum cryptography systems,” Inf. Sec., vol. 22, no. 2, pp. 140-147, 2015.

A.M. Trakhtman, V.A. Trakhtman, “Elements of the Theory of

Discrete Signals on Finite Intervals,” Moscow, Sov. Radio, 1975 p. 208.

A.V. Sokolov, O.N. Zhdanov, O.A. Ayvazian “Synthesis methods of algebraic normal form of many-valued logic functions,” System analysis and applied inform. sc., no.1, pp. 69-76, 2016.

S.V. Agievich “On the representation of bent functions by bent rectangles,” Probabilistic Methods in Discrete Mathematics: roceedings of the Fifth International Petrozavodsk Conference. Utrecht, Boston: VSP, pp. 121-135, 2002.

N. Tokareva “Bent Functions: Results and Applications to Cryptography,” Academic Press, 2015, p. 220.

W. Maier, O. Staffelbach “Nonlinearity criteria for cryptographic functions,” In Advances in Cryptology, EUROCRYPT’89, vol.434, Lecture

Notes in Computer Science, Springer-Verlag, pp.549-562, 1990.

G.B. Arfken, H. J. Weber, “Mathematical Methods for Physicists,” Harcourt: San Diego, 2005, p. 1182.

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*Journal of Telecommunication, Electronic and Computer Engineering (JTEC)*,

*8*(9), 39–43. Retrieved from https://jtec.utem.edu.my/jtec/article/view/835

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