-
Previous Article
Explicit formulas for monomial involutions over finite fields
- AMC Home
- This Issue
-
Next Article
Certain sextics with many rational points
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. |
| [1] |
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 |
| [2] |
Sihem Mesnager, Fengrong Zhang, Yong Zhou. On construction of bent functions involving symmetric functions and their duals. Advances in Mathematics of Communications, 2017, 11 (2) : 347-352. doi: 10.3934/amc.2017027 |
| [3] |
Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069 |
| [4] |
Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305 |
| [5] |
Sihem Mesnager, Fengrong Zhang. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions. Advances in Mathematics of Communications, 2017, 11 (2) : 339-345. doi: 10.3934/amc.2017026 |
| [6] |
Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041 |
| [7] |
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 |
| [8] |
Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249 |
| [9] |
Junchao Zhou, Nian Li, Xiangyong Zeng, Yunge Xu. A generic construction of rotation symmetric bent functions. Advances in Mathematics of Communications, 2021, 15 (4) : 721-736. doi: 10.3934/amc.2020092 |
| [10] |
Kanat Abdukhalikov, Duy Ho. Vandermonde sets, hyperovals and Niho bent functions. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021048 |
| [11] |
Jyrki Lahtonen, Gary McGuire, Harold N. Ward. Gold and Kasami-Welch functions, quadratic forms, and bent functions. Advances in Mathematics of Communications, 2007, 1 (2) : 243-250. doi: 10.3934/amc.2007.1.243 |
| [12] |
Joan-Josep Climent, Francisco J. García, Verónica Requena. On the construction of bent functions of $n+2$ variables from bent functions of $n$ variables. Advances in Mathematics of Communications, 2008, 2 (4) : 421-431. doi: 10.3934/amc.2008.2.421 |
| [13] |
Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425 |
| [14] |
Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037 |
| [15] |
Sihong Su. A new construction of rotation symmetric bent functions with maximal algebraic degree. Advances in Mathematics of Communications, 2019, 13 (2) : 253-265. doi: 10.3934/amc.2019017 |
| [16] |
Wenying Zhang, Zhaohui Xing, Keqin Feng. A construction of bent functions with optimal algebraic degree and large symmetric group. Advances in Mathematics of Communications, 2020, 14 (1) : 23-33. doi: 10.3934/amc.2020003 |
| [17] |
Natalia Tokareva. On the number of bent functions from iterative constructions: lower bounds and hypotheses. Advances in Mathematics of Communications, 2011, 5 (4) : 609-621. doi: 10.3934/amc.2011.5.609 |
| [18] |
Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019 |
| [19] |
Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou. A new class of $ p $-ary regular bent functions. Advances in Mathematics of Communications, 2021, 15 (1) : 55-64. doi: 10.3934/amc.2020042 |
| [20] |
Rumi Melih Pelen. Three weight ternary linear codes from non-weakly regular bent functions. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022020 |
2021 Impact Factor: 1.015
Tools
Metrics
Other articles
by authors
[Back to Top]