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May  2017, 11(2): 293-299. doi: 10.3934/amc.2017021

Explicit constructions of bent functions from pseudo-planar functions

Kanat Abdukhalikov 1,1, and  Sihem Mesnager 2,

1. 

Department of Mathematical Sciences, UAE University, PO Box 15551, Al Ain, UAE
2. 

Department of Mathematics, University of Paris VIII and Paris XIII and Télécom ParisTech, LAGA, UMR 7539, CNRS, Sorbonne Paris Cité

1The first author was supported by UAEU grant 31S107.

Received  February 2016 Revised  March 2016 Published  May 2017

We investigate explicit constructions of bent functions which are linear on elements of spreads. Our constructions are obtained from symplectic presemifields which are associated to pseudo-planar functions. The following diagram gives an indication of the main interconnections arising in this paper: $pseudo-planar\ functions \longleftrightarrow\ commutaive\ presemifields \longrightarrow bent\ functions$

Keywords: Bent functions, spreads, presemifields, pseudo-planar functions.
Mathematics Subject Classification: 51E23, 12K10, 94A60.
Citation: Kanat Abdukhalikov, Sihem Mesnager. Explicit constructions of bent functions from pseudo-planar functions. Advances in Mathematics of Communications, 2017, 11 (2) : 293-299. doi: 10.3934/amc.2017021
References:
[1]

K. Abdukhalikov, Symplectic spreads, planar functions and mutually unbiased bases, J. Algebraic Combin., 41 (2015), 1055-1077.  doi: 10.1007/s10801-014-0565-y.  Google Scholar

[2]

K. Abdukhalikov and S. Mesnager, Bent functions linear on elements of some classical spreads and semifields spreads, Crypt. Commun., 9 (2017), 3-21.  doi: 10.1007/s12095-016-0195-4.  Google Scholar

[3]

C. Carlet, More $\mathcal PS$ and $\mathcal H$-like bent functions Crypt. ePrint Arch. Report 2015/168. Google Scholar

[4]

C. Carlet and S. Mesnager, On Dillon's class H of bent functions, Niho bent functions and o-polynomials, J. Combin. Theory Ser. A, 118 (2011), 2392-2410.  doi: 10.1016/j.jcta.2011.06.005.  Google Scholar

[5]

C. Carlet and S. Mesnager, Four decades of research on bent functions, J. Des. Codes Crypt., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8.  Google Scholar

[6]

A. Çeşmelioğlu, W. Meidl and A. Pott, Bent functions, spreads, and o-polynomials, SIAM J. Discrete Math., 29 (2015), 854-867.  doi: 10.1137/140963273.  Google Scholar

[7]

J. Dillon, Elementary Hadamard Difference Sets Ph. D thesis, Univ. Maryland, 1974.  Google Scholar

[8]

S. Hu, S. Li, T. Zhang, T. Feng and G. Ge, New pseudo-planar binomials in characteristic two and related schemes, J. Des. Codes Crypt., 76 (2015), 345-360.  doi: 10.1007/s10623-014-9958-0.  Google Scholar

[9]

N. Knarr, Quasifields of symplectic translation planes, J. Combin. Theory Ser. A, 116 (2009), 1080-1086.  doi: 10.1016/j.jcta.2008.11.012.  Google Scholar

[10]

S. Mesnager, Bent functions from spreads, J. Amer. Math. Soc. Contemp. Math., 632 (2015), 295-316.  doi: 10.1090/conm/632/12634.  Google Scholar

[11]

S. Mesnager, On $p$-ary bent functions from (maximal) partial spreads in Int. Conf. Finite Field Appl. Fq12 New York, 2015. doi: 10.1090/conm/632/12634.  Google Scholar

[12]

S. Mesnager, Binary Bent Functions: Fundamentals and Results Springer-Verlag, 2016. doi: 10.1007/978-3-319-32595-8.  Google Scholar

[13]

O. S. Rothaus, On ''bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.   Google Scholar

[14]

Z. Scherr and M. E. Zieve, Some planar monomials in characteristic 2, Ann. Comb., 18 (2014), 723-729.  doi: 10.1007/s00026-014-0248-3.  Google Scholar

[15]

K.-U. Schmidt and Y. Zhou, Planar functions over fields of characteristic two, J. Algebraic Combin., 40 (2014), 503-526.  doi: 10.1007/s10801-013-0496-z.  Google Scholar

show all references

References:
[1]

K. Abdukhalikov, Symplectic spreads, planar functions and mutually unbiased bases, J. Algebraic Combin., 41 (2015), 1055-1077.  doi: 10.1007/s10801-014-0565-y.  Google Scholar

[2]

K. Abdukhalikov and S. Mesnager, Bent functions linear on elements of some classical spreads and semifields spreads, Crypt. Commun., 9 (2017), 3-21.  doi: 10.1007/s12095-016-0195-4.  Google Scholar

[3]

C. Carlet, More $\mathcal PS$ and $\mathcal H$-like bent functions Crypt. ePrint Arch. Report 2015/168. Google Scholar

[4]

C. Carlet and S. Mesnager, On Dillon's class H of bent functions, Niho bent functions and o-polynomials, J. Combin. Theory Ser. A, 118 (2011), 2392-2410.  doi: 10.1016/j.jcta.2011.06.005.  Google Scholar

[5]

C. Carlet and S. Mesnager, Four decades of research on bent functions, J. Des. Codes Crypt., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8.  Google Scholar

[6]

A. Çeşmelioğlu, W. Meidl and A. Pott, Bent functions, spreads, and o-polynomials, SIAM J. Discrete Math., 29 (2015), 854-867.  doi: 10.1137/140963273.  Google Scholar

[7]

J. Dillon, Elementary Hadamard Difference Sets Ph. D thesis, Univ. Maryland, 1974.  Google Scholar

[8]

S. Hu, S. Li, T. Zhang, T. Feng and G. Ge, New pseudo-planar binomials in characteristic two and related schemes, J. Des. Codes Crypt., 76 (2015), 345-360.  doi: 10.1007/s10623-014-9958-0.  Google Scholar

[9]

N. Knarr, Quasifields of symplectic translation planes, J. Combin. Theory Ser. A, 116 (2009), 1080-1086.  doi: 10.1016/j.jcta.2008.11.012.  Google Scholar

[10]

S. Mesnager, Bent functions from spreads, J. Amer. Math. Soc. Contemp. Math., 632 (2015), 295-316.  doi: 10.1090/conm/632/12634.  Google Scholar

[11]

S. Mesnager, On $p$-ary bent functions from (maximal) partial spreads in Int. Conf. Finite Field Appl. Fq12 New York, 2015. doi: 10.1090/conm/632/12634.  Google Scholar

[12]

S. Mesnager, Binary Bent Functions: Fundamentals and Results Springer-Verlag, 2016. doi: 10.1007/978-3-319-32595-8.  Google Scholar

[13]

O. S. Rothaus, On ''bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.   Google Scholar

[14]

Z. Scherr and M. E. Zieve, Some planar monomials in characteristic 2, Ann. Comb., 18 (2014), 723-729.  doi: 10.1007/s00026-014-0248-3.  Google Scholar

[15]

K.-U. Schmidt and Y. Zhou, Planar functions over fields of characteristic two, J. Algebraic Combin., 40 (2014), 503-526.  doi: 10.1007/s10801-013-0496-z.  Google Scholar

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