×

zbMATH — the first resource for mathematics

Generalized bent functions – some general construction methods and related necessary and sufficient conditions. (English) Zbl 1343.94064
Summary: In this article we present a broader theoretical framework useful in studying the properties of so-called generalized bent functions. We give the sufficient conditions (and in many cases also necessary) for generalized bent functions when these functions are represented as a linear combination of: generalized bent; Boolean bent; and a mixture of generalized bent and Boolean bent functions. These conditions are relatively easy to satisfy and by varying the variables that specify these linear combinations many different classes of generalized bent functions can be derived. In particular, based on these results, we provide some generic construction methods of these functions and demonstrate that some previous methods are just special cases of the results given in this article.

MSC:
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Carlet, C.: Two new classes of bent functions. Eurocrypt ’93, LNCS, 765, pp. 77-101 (1994) · Zbl 0951.94542
[2] Dillon, J. F.: Elementary Hadamard difference sets. PhD Thesis, University of Maryland (1974) · Zbl 0346.05003
[3] Dobbertin, H.: Construction of bent functions and balanced Boolean functions with high nonlinearity. Fast Software Encryption, Leuven 1994, LNCS 1008, Springer-Verlag, 61-74 (1995) · Zbl 0939.94563
[4] Golay, MJE, Complementary series, IRE Trans. Inf. Theory, 7, 82-87, (1961)
[5] Kumar, PV; Scholtz, RA; Welch, LR, Generalized bent functions and their properties, J. Comb. Theory Series A, 40, 90-107, (1985) · Zbl 0585.94016
[6] McFarland, RL, A family of noncyclic difference sets, J. Comb. Theory Series A, 15, 1-10, (1973) · Zbl 0268.05011
[7] Schmidt, K.U.: Quaternary constant-amplitude codes for multicode CDMA, IEEE International Symposium on Information Theory, ISIT’2007, Nice, France. Available at arXiv:0611162 (2007) · Zbl 1367.94344
[8] Schmidt, KU, Complementary sets, generalized Reed-muller codes, and power control for OFDM, IEEE Trans. Inf. Theory, 52, 808-814, (2007) · Zbl 1310.94203
[9] Singh, B.K.: Secondary constructions on generalized bent functions. IACR. Cryptol. ePrint. Arch., 17-17 (2012)
[10] Singh, BK, On cross-correlation spectrum of generalized bent functions in generalized maiorana-mcfarland class, Inf. Sci. Lett., 2, 139-145, (2013)
[11] Solé, P., Tokareva, N.: Connections between quaternary and binary bent functions. Available at https://eprint.iacr.org/2009/544.pdf (2009)
[12] Solodovnikov, VI, Bent functions from a finite abelian group into a finite abelian group, Discret. Math. Appl., 12, 111-126, (2002) · Zbl 1047.94011
[13] Stanica, P., Gangopadhyay, S., Singh, B.K.: Some results concerning generalized bent functions. Available at https://eprint.iacr.org/2011/290.pdf (2011)
[14] Stanica, P; Martinsen, T; Gangopadhyay, S; Singh, BK; Bent and generalized bent Boolean functions, No article title, Des. Codes Crypt., 69, 77-94, (2013) · Zbl 1322.94094
[15] Stanica, P., Martinsen, T.: Octal bent generalized Boolean Functions. IACR. Cryptol. ePrint. Arch., 89-89 (2011) · Zbl 0585.94016
[16] Tokareva, NN, Generalizations of bent functions - a survey, J. Appl. Ind. Math., 5, 110-129, (2011)
[17] 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.