# American Institute of Mathematical Sciences

November  2018, 12(4): 691-705. doi: 10.3934/amc.2018041

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

 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  September 2018

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.

Citation: 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
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##### References:
 [1] 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. Google Scholar [2] 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. Google Scholar [3] A. E. Brouwer, Web database of strongly regular graphs, http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html (online).Google Scholar [4] 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. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] 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. Google Scholar [8] 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. Google Scholar [9] E. Z. Chen, Web database of two-weight codes, http://moodle.tec.hkr.se/~chen/research/2-weight-codes/search.php (online).Google Scholar [10] 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. Google Scholar [11] 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. Google Scholar [12] 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. Google Scholar [13] 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. Google Scholar [14] 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. Google Scholar [15] S. L. Ma, A survey of partial difference sets, Des., Codes, Cryptogr., 4 (1994), 221-261. doi: 10.1007/BF01388454. Google Scholar [16] S. Mesnager, Bent Functions. Fundamentals and Results, Springer, 2016. doi: 10.1007/978-3-319-32595-8. Google Scholar [17] 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. Google Scholar [18] 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. Google Scholar [19] 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. Google Scholar
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