zbMATH — the first resource for mathematics

Bent and generalized bent Boolean functions. (English) Zbl 1322.94094
Summary: In this paper, we investigate the properties of generalized bent functions defined on \({\mathbb{Z}_2^n}\) with values in \({\mathbb{Z}_q}\) , where \(q \geq 2\) is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana-McFarland type bent functions and Dillon’s functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana-McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on \({\mathbb{Z}_2^n}\) with values in \({\mathbb{Z}_4}\) and \({\mathbb{Z}_8}\). Moreover, we propose several constructions of such generalized bent functions for both \(n\) even and \(n\) odd.

94A60 Cryptography
94D10 Boolean functions
06E30 Boolean functions
Full Text: DOI
[1] Bey, C.; Kyureghyan, G.M., On Boolean functions with the sum of every two of them being bent, Des. Codes Cryptogr., 49, 341-346, (2008) · Zbl 1196.05010
[2] Carlet, C., Generalized partial spreads, IEEE Trans. Inf. Theory, 41, 1482-1487, (1995) · Zbl 0831.94022
[3] Carlet, C.; Guillot, P., A characterization of binary bent functions, J. Comb. Theory (A), 76, 328-335, (1996) · Zbl 0861.94014
[4] Carlet, C.; Guillot, P., An alternate characterization of the bentness of binary functions, with uniqueness, Des. Codes Cryptogr., 14, 133-140, (1998) · Zbl 0956.94008
[5] Carlet C.: Boolean functions for cryptography and error correcting codes. In: Crama Y., Hammer P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257-397. Cambridge University Press, Cambridge (2010). · Zbl 1209.94035
[6] Carlet C.: Vectorial Boolean functions for cryptography. In: Crama Y., Hammer P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 398-469. Cambridge University Press, Cambridge (2010). · Zbl 1209.94036
[7] Cusick T.W., Stănică P.: Cryptographic Boolean Functions and Applications. Elsevier, Amsterdam (2009) · Zbl 1173.94002
[8] Dillon J.F.: Elementary Hadamard difference sets. In: Proceedings of the Sixth S.E. Conference of Combinatorics, Graph Theory, and Computing, Congressus Numerantium No. XIV, Utilitas Math., Winnipeg, pp. 237-249 (1975). · Zbl 0346.05003
[9] Hirschhorn, M.D., A simple proof of jacobi’s four-square theorem, Proc. Am. Math. Soc., 101, 436-438, (1987) · Zbl 0632.10046
[10] Kumar, P.V.; Scholtz, R.A.; Welch, L.R., Generalized bent functions and their properties, J. Comb. Theory (A), 40, 90-107, (1985) · Zbl 0585.94016
[11] Laigle-Chapuy, Y., Permutation polynomials and applications to coding theory, Finite Fields Appl., 13, 58-70, (2007) · Zbl 1107.11048
[12] Lam, T.Y.; Leung, K.H., On vanishing sums of roots of unity, J. Algebra, 224, 91-109, (2000) · Zbl 1099.11510
[13] Rothaus, O.S., On bent functions, J. Comb. Theory Ser. A, 20, 300-305, (1976) · Zbl 0336.12012
[14] Sarkar, P.; Maitra, S., Cross-correlation analysis of cryptographically useful Boolean functions and S-boxes, Theory Comput. Syst., 35, 39-57, (2002) · Zbl 0993.68032
[15] Schmidt K.-U.: Quaternary constant-amplitude codes for multicode CDMA. In: IEEE International Symposium on Information Theory, ISIT’2007, Nice, France, June 24-29, 2007, pp. 2781-2785. Available at http://arxiv.org/abs/cs.IT/0611162. · Zbl 0831.94022
[16] Solé P., Tokareva N.: Connections Between Quaternary and Binary Bent Functions. http://eprint.iacr.org/2009/544.pdf; see also, Prikl. Diskr. Mat. 1, 16-18 (2009). · Zbl 1196.05010
[17] Stănică P., Gangopadhyay S., Chaturvedi A., Kar Gangopadhyay A., Maitra S.: Nega-Hadamard transform, bent and negabent functions. In: Carlet C., Pott A. (eds.) Sequences and Their Applications—SETA 2010, LNCS 6338, 359-372 (2010). · Zbl 1233.94023
[18] Stănică P., Gangopadhyay S., Singh B.K.: Some Results Concerning Generalized Bent Functions. http://eprint.iacr.org/2011/290.pdf.
[19] Stănică P., Martinsen T.: Octal Bent Generalized Boolean Functions. http://eprint.iacr.org/2011/089.pdf. · Zbl 0861.94014
[20] Zhao, Y.; Li, H., On bent functions with some symmetric properties, Discret. Appl. Math., 154, 2537-2543, (2006) · Zbl 1106.94028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.