# American Institute of Mathematical Sciences

August  2017, 11(3): 549-566. doi: 10.3934/amc.2017043

## Generalized bent functions -sufficient conditions and related constructions

 University of Primorska, FAMNIT, Koper, Slovenia

Received  November 2015 Published  August 2017

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.

Citation: Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043
##### References:
 [1] M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.   Google Scholar [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.  Google Scholar [3] S. Hodžić and E. Pasalic, Construction methods for generalized bent functions preprint, arXiv: 1604.02730 Google Scholar [4] P. V. Kumar, R. 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.  Google Scholar [5] H. Liu, K. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] B. K. Singh, Secondary constructions on generalized bent functions IACR Crypt. ePrint Arch. 2012, p. 17. Google Scholar [11] B. K. Singh, On cross-correlation spectrum of generalized bent functions in generalized Maiorana-McFarland class, Inf. Sci. Lett., 2 (2013), 139-145.   Google Scholar [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 Google Scholar [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.  Google Scholar [14] P. Stanica and T. Martinsen, Octal bent generalized Boolean functions preprint, arXiv: 1102.4812 Google Scholar [15] P. Stanica, T. Martinsen, S. 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.  Google Scholar [16] N. N. Tokareva, Generalizations of bent functions –a survey, J. Appl. Industr. Math., 5 (2011), 110-129.  doi: 10.1134/S1990478911010133.  Google Scholar

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##### References:
 [1] M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.   Google Scholar [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.  Google Scholar [3] S. Hodžić and E. Pasalic, Construction methods for generalized bent functions preprint, arXiv: 1604.02730 Google Scholar [4] P. V. Kumar, R. 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.  Google Scholar [5] H. Liu, K. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] B. K. Singh, Secondary constructions on generalized bent functions IACR Crypt. ePrint Arch. 2012, p. 17. Google Scholar [11] B. K. Singh, On cross-correlation spectrum of generalized bent functions in generalized Maiorana-McFarland class, Inf. Sci. Lett., 2 (2013), 139-145.   Google Scholar [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 Google Scholar [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.  Google Scholar [14] P. Stanica and T. Martinsen, Octal bent generalized Boolean functions preprint, arXiv: 1102.4812 Google Scholar [15] P. Stanica, T. Martinsen, S. 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.  Google Scholar [16] N. N. Tokareva, Generalizations of bent functions –a survey, J. Appl. Industr. Math., 5 (2011), 110-129.  doi: 10.1134/S1990478911010133.  Google Scholar
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