Bent and vectorial bent functions, partial difference sets, and strongly regular graphs

Ayça Çeşmelioğlu; Wilfried Meidl

doi:10.3934/amc.2018041

Article Contents

2018, Volume 12, Issue 4: 691-705. Doi: 10.3934/amc.2018041

This issue Previous Article The results on optimal values of some combinatorial batch codes Next Article Self-duality of generalized twisted Gabidulin codes

Bent and vectorial bent functions, partial difference sets, and strongly regular graphs

Ayça Çeşmelioğlu^1, and
Wilfried Meidl^2,

1.
Altınbaş University, School of Engineering and Natural Sciences, Bağcılar, 34217 İstanbul, Turkey
2.
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040-Linz, Austria

Received: May 2017

Published: November 2018

Abstract Full Text(HTML) Related Papers Cited by

Abstract

Bent and vectorial bent functions have applications in cryptography and coding and are closely related to objects in combinatorics and finite geometry, like difference sets, relative difference sets, designs and divisible designs. Bent functions with certain additional properties yield partial difference sets of which the Cayley graphs are always strongly regular. In this article we continue research on connections between bent functions and partial difference sets respectively strongly regular graphs. For the first time we investigate relations between vectorial bent functions and partial difference sets. Remarkably, properties of the set of the duals of the components play here an important role. Seeing conventional bent functions as 1-dimensional vectorial bent functions, some earlier results on strongly regular graphs from bent functions follow from our more general results. Finally we describe a recursive construction of infinitely many partial difference sets with a secondary construction of p-ary bent functions.

Keywords:

Mathematics Subject Classification: 06E30, 05B10, 11T06, 94C10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	A. Bernasconi and B. Codenotti , Spectral analysis of Boolean functions as a graph eigenvalue problem, IEEE Trans. Comput., 48 (1999) , 345-351. doi: 10.1109/12.755000.
	A. Bernasconi , B. Codenotti and J. M. VanderKam , A characterization of bent functions in terms of strongly regular graphs, IEEE Trans. Comput., 50 (2001) , 984-985. doi: 10.1109/12.954512.
	A. E. Brouwer, Web database of strongly regular graphs, http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html (online).
	A. Çeşmelioğlu and W. Meidl , A Construction of bent functions from plateaued functions, Des. Codes Cryptogr., 66 (2013) , 231-242. doi: 10.1007/s10623-012-9686-2.
	A. Çeşmelioğlu , W. Meidl and A. Pott , On the dual of (non)-weakly regular bent functions and self-dual bent functions, Advances in Mathematics of Communications, 7 (2013) , 425-440. doi: 10.3934/amc.2013.7.425.
	A. Çeşmelioğlu , W. Meidl and A. Pott , There are infinitely many bent functions for which the dual is not bent, IEEE Trans. Inform. Theory, 62 (2016) , 5204-5208. doi: 10.1109/TIT.2016.2586081.
	A. Çeşmelioğlu , W. Meidl and A. Pott , Vectorial bent functions and their duals, Linear Algebra and its Applications, 548 (2018) , 305-320. doi: 10.1016/j.laa.2018.03.016.
	Y. M. Chee , Y. Tan and Y. D. Zhang , Strongly regular graphs constructed from $p$-ary bent functions, J. Algebraic Combin., 34 (2011) , 251-266. doi: 10.1007/s10801-010-0270-4.
	E. Z. Chen, Web database of two-weight codes, http://moodle.tec.hkr.se/~chen/research/2-weight-codes/search.php (online).
	E. R. van Dam and M. Muzychuk , Some implications on amorphic association schemes, J. Combin. Theory Ser. A, 117 (2010) , 111-127. doi: 10.1016/j.jcta.2009.03.018.
	T. Feng, B. Wen, Q. Xiang and J. Yin, Partial difference sets from quadratic forms and p-ary weakly regular bent functions, Number Theory and Related Areas, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 27 (2013), 25-40.
	T. Helleseth and A. Kholosha , Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inform. Theory, 52 (2006) , 2018-2032. doi: 10.1109/TIT.2006.872854.
	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.
	J. H. van Lint and A. Schrijver , Constructions of strongly regular graphs, two-weight codes and partial geometries by finite fields, Combinatorica, 1 (1981) , 63-73. doi: 10.1007/BF02579178.
	S. L. Ma , A survey of partial difference sets, Des., Codes, Cryptogr., 4 (1994) , 221-261. doi: 10.1007/BF01388454.
	S. Mesnager, Bent Functions. Fundamentals and Results, Springer, 2016. doi: 10.1007/978-3-319-32595-8.
	K. Nyberg, Perfect nonlinear S-boxes. Advances in cryptology-EUROCRYPT '91 (Brighton, 1991), 378-386, Lecture Notes in Comput. Sci., 547, Springer, Berlin, 1991. doi: 10.1007/3-540-46416-6_32.
	A. Pott , Y. Tan , T. Feng and S. Ling , Association schemes arising from bent functions, Des., Codes, Cryptogr., 59 (2011) , 319-331. doi: 10.1007/s10623-010-9463-z.
	Y. Tan , A. Pott and T. Feng , Strongly regular graphs associated with ternary bent functions, J. Combin. Theory Ser. A, 117 (2010) , 668-682. doi: 10.1016/j.jcta.2009.05.003.