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Explicit constructions of bent functions from pseudo-planar functions
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 |
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: $pseudo-planar\ functions \longleftrightarrow\ commutaive\ presemifields \longrightarrow bent\ functions$
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. |
[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. |
[3] |
C. Carlet,
More $\mathcal PS$ and $\mathcal H$-like bent functions Crypt. ePrint Arch. Report 2015/168. |
[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. |
[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. |
[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. |
[7] |
J. Dillon,
Elementary Hadamard Difference Sets Ph. D thesis, Univ. Maryland, 1974. |
[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. |
[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. |
[10] |
S. Mesnager,
Bent functions from spreads, J. Amer. Math. Soc. Contemp. Math., 632 (2015), 295-316.
doi: 10.1090/conm/632/12634. |
[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. |
[12] |
S. Mesnager,
Binary Bent Functions: Fundamentals and Results Springer-Verlag, 2016.
doi: 10.1007/978-3-319-32595-8. |
[13] |
O. S. Rothaus,
On ''bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.
|
[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. |
[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. |
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. |
[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. |
[3] |
C. Carlet,
More $\mathcal PS$ and $\mathcal H$-like bent functions Crypt. ePrint Arch. Report 2015/168. |
[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. |
[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. |
[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. |
[7] |
J. Dillon,
Elementary Hadamard Difference Sets Ph. D thesis, Univ. Maryland, 1974. |
[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. |
[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. |
[10] |
S. Mesnager,
Bent functions from spreads, J. Amer. Math. Soc. Contemp. Math., 632 (2015), 295-316.
doi: 10.1090/conm/632/12634. |
[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. |
[12] |
S. Mesnager,
Binary Bent Functions: Fundamentals and Results Springer-Verlag, 2016.
doi: 10.1007/978-3-319-32595-8. |
[13] |
O. S. Rothaus,
On ''bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.
|
[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. |
[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. |
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