On the Δ-equivalence of Boolean functions
-
Oleg A. Logachev
,
Sergey N. Fedorov
Abstract
A new equivalence relation on the set of Boolean functions is introduced: functions are declared to be Δ-equivalent if their autocorrelation functions are equal. It turns out that this classification agrees well with the cryptographic properties of Boolean functions: for functions belonging to the same Δ-equivalence class a number of their cryptographic characteristics do coincide. For example, all bent-functions (of a fixed number of variables) make up one class.
Note: Originally published in Diskretnaya Matematika (2018) 30, №4, 29–40 (in Russian).
Funding: The study was supported by the RFBR grant No. 16-01-00226 A.
References
[1] Alekseev E. K., “On some measures of Boolean functions nonlinearity”, Prikl. diskr. matem., 2011,№2 (12), 5–16 (in Russian).10.17223/20710410/12/1Search in Google Scholar
[2] Alekseev E. K., Karelina E. K., “Classification of correlation-immune and minimal correlation-immune Boolean functions of 4 and 5 variables”, Discrete Math. Appl., 25:4 (2015), 193–202.10.1515/dma-2015-0019Search in Google Scholar
[3] de la Cruz Jimenez R. A., Kamlovskiy O. V., “The sum of modules of Walsh coefficients of Boolean functions”, Discrete Math. Appl., 26:5 (2016), 259–272.10.1515/dma-2016-0023Search in Google Scholar
[4] Kamlovskiy O. V., “The sums of modulus of the Walsh-Hadamard coefficients of some balanced Boolean functions”, Matematicheskie voprosy kriptografii, 8:4 (2017), 75–98.Search in Google Scholar
[5] Logachev O. A., Salnikov A. A., Yashchenko V. V., Boolean Functions in Coding Theory and Cryptology, Providence, Rhode Island: Amer. Math. Soc., 2011, 334 pp.Search in Google Scholar
[6] Logachev O. A., Fedorov S. N., Yashchenko V. V., “Boolean functions as points on the hypersphere in the Euclidean space”, Discrete Math. Appl., 29:2 (2019), 89–101.10.1515/dma-2019-0009Search in Google Scholar
[7] MacWilliams E. J., Sloane N. J. A., The Theory of Error-Correcting Codes. Parts I, II., North-Holland, Amsterdam, 1977.Search in Google Scholar
[8] Cheremushkin A. V., “Linear and affine classification of discrete functions (a review of publications)”, Matematicheskie voprosy kibernetiki, 2005,№14, 261–280.10.1515/1569392042572159Search in Google Scholar
[9] O’Donnell R., Analysis of Boolean Functions, Cambridge Univ. Press, 2014, 417 pp.10.1017/CBO9781139814782Search in Google Scholar
[10] Rothaus O. S., “On ‘Bent’ Functions”, J. Comb. Theory (A), 20:3 (1976), 300–305.10.1016/0097-3165(76)90024-8Search in Google Scholar
[11] Terras A., Fourier Analysis on Finite Groups and Applications, Cambridge Univ. Press, 1999, 442 pp.10.1017/CBO9780511626265Search in Google Scholar
[12] Zhou Y., Xie M., Xiao G. Z., “On the global avalanche characteristics between two Boolean functions and the higher order nonlinearity”, Inf. Sci., 180:2 (2010), 256–265.10.1016/j.ins.2009.09.012Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- On a method of synthesis of correlation-immune Boolean functions
- On the Δ-equivalence of Boolean functions
- On diagnostic tests of contact break for contact circuits
- On stabilization of an automaton model of migration processes
- Bounds on the discrepancy of linear recurring sequences over Galois rings
- Asymptotically best method for synthesis of Boolean recursive circuits
Articles in the same Issue
- On a method of synthesis of correlation-immune Boolean functions
- On the Δ-equivalence of Boolean functions
- On diagnostic tests of contact break for contact circuits
- On stabilization of an automaton model of migration processes
- Bounds on the discrepancy of linear recurring sequences over Galois rings
- Asymptotically best method for synthesis of Boolean recursive circuits
