# American Institute of Mathematical Sciences

February  2014, 8(1): 21-33. doi: 10.3934/amc.2014.8.21

## Special bent and near-bent functions

 1 IMATH(IAA), Université du Sud Toulon-Var, 83957 La Garde Cedex, France

Received  December 2011 Revised  November 2013 Published  January 2014

Starting from special near-bent functions in dimension $2t-1$ we construct bent functions in dimension $2t$ having a specific derivative. We deduce new families of bent functions.
Citation: Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21
##### References:
 [1] A. Canteault, C. Carlet, P. Charpin and C. Fontaine, On cryptographic properties of the cosets of R(1,m), IEEE Trans. Inform. Theory, 47 (2001), 1494-1513. doi: 10.1109/18.923730. [2] A. Canteault and P. Charpin, Decomposing bent functions, IEEE Trans. Inform. Theory, 49 (2003), 2004-2019. doi: 10.1109/TIT.2003.814476. [3] J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D thesis, University of Maryland, 1974. [4] J. F. Dillon, Multiplicative difference sets via additive characters, Des. Codes Cryptogr., 17 (1999), 225-235. doi: 10.1023/A:1026435428030. [5] R. Gold, Maximal recursive squences with 3-valued recursive cross-correlation functions, IEEE Trans. Inform. Theory, 14 (1968), 154-156. doi: 10.1109/TIT.1968.1054106. [6] X. D. Hou, Cubic bent functions, Discrete Math., 189 (1998), 149-161. doi: 10.1016/S0012-365X(98)00008-9. [7] G. Leander and G. McGuire, Construction of bent functions from near-bent functions, J. Combin. Theory Ser. A, 116 (2009), 960-970. doi: 10.1016/j.jcta.2008.12.004. [8] O. S. Rothaus, On bent functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8. [9] J. Wolfmann, Bent functions and coding theory, in Difference Sets, Sequences and their Correlation Properties (eds. A. Pott, P. V. Kumar, T. Helleseth and D. Jungnickel), Kluwer Academic Publishers, 1999, 393-418. [10] J. Wolfmann, Cyclic code aspects of bent functions, in Finite Fields Theory and Applications, Amer. Math. Soc., 2010, 363-384. doi: 10.1090/conm/518/10218.

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##### References:
 [1] A. Canteault, C. Carlet, P. Charpin and C. Fontaine, On cryptographic properties of the cosets of R(1,m), IEEE Trans. Inform. Theory, 47 (2001), 1494-1513. doi: 10.1109/18.923730. [2] A. Canteault and P. Charpin, Decomposing bent functions, IEEE Trans. Inform. Theory, 49 (2003), 2004-2019. doi: 10.1109/TIT.2003.814476. [3] J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D thesis, University of Maryland, 1974. [4] J. F. Dillon, Multiplicative difference sets via additive characters, Des. Codes Cryptogr., 17 (1999), 225-235. doi: 10.1023/A:1026435428030. [5] R. Gold, Maximal recursive squences with 3-valued recursive cross-correlation functions, IEEE Trans. Inform. Theory, 14 (1968), 154-156. doi: 10.1109/TIT.1968.1054106. [6] X. D. Hou, Cubic bent functions, Discrete Math., 189 (1998), 149-161. doi: 10.1016/S0012-365X(98)00008-9. [7] G. Leander and G. McGuire, Construction of bent functions from near-bent functions, J. Combin. Theory Ser. A, 116 (2009), 960-970. doi: 10.1016/j.jcta.2008.12.004. [8] O. S. Rothaus, On bent functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8. [9] J. Wolfmann, Bent functions and coding theory, in Difference Sets, Sequences and their Correlation Properties (eds. A. Pott, P. V. Kumar, T. Helleseth and D. Jungnickel), Kluwer Academic Publishers, 1999, 393-418. [10] J. Wolfmann, Cyclic code aspects of bent functions, in Finite Fields Theory and Applications, Amer. Math. Soc., 2010, 363-384. doi: 10.1090/conm/518/10218.
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