Advanced Search
Article Contents
Article Contents

Generalized bent functions -sufficient conditions and related constructions

Abstract Full Text(HTML) Related Papers Cited by
  • The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \to \mathbb{Z}_q$, where $q ≥q 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of $f$, which then allows us to derive sufficient conditions that $f$ is generalized bent for any even $q$. The case when $q$ is not a power of two, which has not been addressed previously, is treated separately and a suitable representation in terms of the component functions is employed. Consequently, the derived results lead to generic construction methods of this class of functions. The main remaining task, which is not answered in this article, is whether the sufficient conditions are also necessary. There are some indications that this might be true which is also formally confirmed for generalized bent functions that belong to the class of generalized Maiorana-McFarland functions (GMMF), but still we were unable to completely specify (in terms of necessity) gbent conditions.

    Mathematics Subject Classification: Primary: 06E30, 94A60; Secondary: 94A55.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87. 
    [2] S. Hodžić and E. Pasalic, Generalized bent functions -Some general construction methods and related necessary and sufficient conditions, Crypt. Commun., 7 (2015), 469-483.  doi: 10.1007/s12095-015-0126-9.
    [3] S. Hodžić and E. Pasalic, Construction methods for generalized bent functions preprint, arXiv: 1604.02730
    [4] P. V. KumarR. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.
    [5] H. LiuK. Feng and R. Feng, Nonexistence of generalized bent functions from $\mathbb Z^n_2$ to $\mathbb Z_m$, Des. Codes Crypt., 82 (2017), 647-662.  doi: 10.1007/s10623-016-0192-9.
    [6] P. Sarkar and S. Maitra, Cross-correlation analysis of cryptographically useful Boolean functions and S-boxes, Theory Comp. Syst., 35 (2002), 39-57.  doi: 10.1007/s00224-001-1019-1.
    [7] K. U. Schmidt, Complementary sets, generalized Reed-Muller Codes, and power control for OFDM, IEEE Trans. Inf. Theory, 52 (2007), 808-814.  doi: 10.1109/TIT.2006.889723.
    [8] K. U. Schmidt, Quaternary constant-amplitude codes for multicode CDMA, in IEEE Int. Symp. Inf. Theory – ISIT'2007, Nice, France, 2007. doi: 10.1109/TIT.2009.2013041.
    [9] J. Seberry and X. -M. Zhang, Highly nonlinear 0-1 balanced Boolean functions satisfying strict avalanche criterion, in Advances in Cryptography -Auscrypt'92, Springer, Berlin, 1993,145–755. doi: 10.1007/3-540-57220-1.
    [10] B. K. Singh, Secondary constructions on generalized bent functions IACR Crypt. ePrint Arch. 2012, p. 17.
    [11] B. K. Singh, On cross-correlation spectrum of generalized bent functions in generalized Maiorana-McFarland class, Inf. Sci. Lett., 2 (2013), 139-145. 
    [12] P. Solé and N. Tokareva, Connections between quaternary and binary bent functions Crypt. ePrint Arch. 2009, available at https://eprint.iacr.org/2009/544.pdf
    [13] V. I. Solodovnikov, Bent functions from a finite Abelian group into a finite Abelian group, Discr. Math. Appl., 12 (2002), 111-126.  doi: 10.1515/dma-2002-0203.
    [14] P. Stanica and T. Martinsen, Octal bent generalized Boolean functions preprint, arXiv: 1102.4812
    [15] P. StanicaT. MartinsenS. Gangopadhyay and B. K. Singh, Bent and generalized bent Boolean functions, Des. Codes Crypt., 69 (2013), 77-94.  doi: 10.1007/s10623-012-9622-5.
    [16] N. N. Tokareva, Generalizations of bent functions –a survey, J. Appl. Industr. Math., 5 (2011), 110-129.  doi: 10.1134/S1990478911010133.
  • 加载中

Article Metrics

HTML views(1338) PDF downloads(417) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint