THE CLASS OF PERFECT TERNARY ARRAYS
https://doi.org/10.21122/2309-4923-2018-2-47-54
Abstract
In recent decades, perfect algebraic constructions are successfully being use to signal systems synthesis, to construct block and stream cryptographic algorithms, to create pseudo-random sequence generators as well as in many other fields of science and technology. Among perfect algebraic constructions a significant place is occupied by bent-sequences and the class of perfect binary arrays associated with them. Bent-sequences are used for development of modern cryptographic primitives, as well as for constructing constant amplitude codes (C-codes) used in code division multiple access technology. In turn, perfect binary arrays are used for constructing correction codes, systems of biphase phase- shifted signals and multi-level cryptographic systems. The development of methods of many-valued logic in modern information and communication systems has attracted the attention of researchers to the improvement of methods for synthesizing many-valued bent-sequences for cryptography and information transmission tasks. The new results obtained in the field of the synthesis of ternary bent-sequences, make actual the problem of researching the class of perfect ternary arrays. In this paper we consider the problem of extending the definition of perfect binary arrays to three-valued logic case, as a result of which the definition of a perfect ternary array was introduced on the basis of the determination of the unbalance of the ternary function. A complete class of perfect ternary arrays of the third order is obtained by a regular method, bypassing the search. Thus, it is established that the class of perfect ternary arrays is a union of four subclasses, in each of which the corresponding methods of reproduction are determined. The paper establishes the relationship between the class of ternary bent-sequences and the class of perfect ternary arrays. The obtained results are the basis for the introduction of perfect ternary arrays into modern cryptographic and telecommunication algorithms.
About the Authors
A. V. Sokolov
Odessa National Polytechnic University
Ukraine
Artem V. Sokolov PhD., Senior Lecturer of the department of Informatics and control of information systems protection
Ukraine
Artem V. Sokolov PhD., Senior Lecturer of the department of Informatics and control of information systems protection
O. N. Zhdanov
Siberian State University of Science and Technology named after Academician
M. F. Reshetnev
Russian Federation
Zhdanov Oleg Nikolaevich is Associate Professor of Informational Technologies subdepartment
Russian Federation
Zhdanov Oleg Nikolaevich is Associate Professor of Informational Technologies subdepartment
References
1. Hnatiuk, S. O. Method for quality evaluation of trit pseudorandom sequence to cryptographic applications / S. O. Hnatiuk, T. O. Zhmurko, V. N. Kinzeriavyi, N. A. Seilova. – Information Technology and Security, 2015. – Vol. 3. – No. 2(5). – С. 108–116.
2. Sokolov, A. V. Pseudo-random key sequence generator based on triple sets of bent-functions / A. V. Sokolov, O. N. Zhdanov, N. A. Barabanov. – Problems of physics, mathematics and technics, 2016. – No. 1(26). – P. 85–91.
3. Zhdanov, O. N. Block symmetric cryptographic algorithm based on principles of variable block length and many-valued logic / O. N. Zhdanov, A. V. Sokolov. – Far East Journal of Electronics and Communications, 2016. – Vol. 16, No. 3. – P. 573–589.
4. Tao, Wu. Stream cipher by reed-solomon code / Wu Tao, Wang Ruomei. – Information and Communication Technology Convergence (ICTC), 2017. – P. 422–427.
5. Tokareva, N. N. Bent functions: results and applications. Survey of works / N. N. Tokareva. – Applied Discrete Mathematics. – Tomsk, 2009. – Ser. № 1(3). – P. 15–37.
6. Mesnager, S. Several New Infinite Families of Bent Functions and Their Duals / S. Mesnager.— IEEE Transactions on Information Theory, 2014. – Vol. 60. – No. 7. – P. 4397–4407.
7. Qingshu, Meng. A novel algorithm enumerating bent functions / Qingshu Meng, Min Yang, Huanguo Zhang, Jingsong Cui. – Discrete Mathematics, 2008. – Vol. 308. – Issue 23. – P. 5576–5584.
8. Kopilovich, L. E. On perfect binary arrays / L. E. Kopilovich. – Electronics Letters, 1988. —Vol. 24. – No. 9. – P. 566–567.
9. Mazurkov, M. I. The regular rules of constructing the complete class of bent-sequences of length 16 / M. I. Mazurkov, A. V. Sokolov. – Proceedings of ONPU, 2013. – No. 2(41). – P.231–237.
10. Wild P. Infinite families of perfect binary arrays / P. Wild. – Electron. Lett, 1988. – Vol. 24. – No. 14. – P. 845–847.
11. Trakhtman, A. M. Fundamentals of the theory of discrete signals on finite intervals / A. M. Trakhtman, V. A. Trachtman. – Moscow: Sov. radio, 1975. – p. 208.
12. Stankovic, R. S. Representation of Multiple-Valued Logic Functions / R. S. Stankovic, J. T. Astola, C. Moraga. – Morgan & Claypool Publishers, Synthesis lectures on digital circuits and systems, 2012. – p. 170.
13. Sokolov, A. V. Synthesis methods of algebraic normal form of many-valued logic functions / A. V. Sokolov, O. N. Zhdanov, A. O. Ayvazyan. – System analysis and applied information science, 2016. – No. 1. – P. 69–76.
14. Mazurkov, M. I. Synthesis method for bent sequences in the Vilenkin-Chrestenson basis / M. I. Mazurkov, A. V. Sokolov, N. A. Barabanov. – Radioelectronics and Communications Systems, 2016. – Vol. 59. – No. 11. – P. 510–517.
15. Sokolov, A. V. Regular synthesis method of a complete class of ternary bent-sequences and their nonlinear properties / A.V. Sokolov, O. N. Zhdanov. – Journal of Telecommunication, Electronic and Computer Engineering. – Vol. 8. – No. 9. – P. 39–43.
Review
For citations:
Sokolov A.V., Zhdanov O.N. THE CLASS OF PERFECT TERNARY ARRAYS. «System analysis and applied information science». 2018;(2):47-54. (In Russ.) https://doi.org/10.21122/2309-4923-2018-2-47-54
Views:
774
Email this article 