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

show all references

##### 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. [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. [3] A. E. Brouwer, Web database of strongly regular graphs, http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html (online). [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. [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. [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. [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. [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. [9] E. Z. Chen, Web database of two-weight codes, http://moodle.tec.hkr.se/~chen/research/2-weight-codes/search.php (online). [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. [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. [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. [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. [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. [15] S. L. Ma, A survey of partial difference sets, Des., Codes, Cryptogr., 4 (1994), 221-261. doi: 10.1007/BF01388454. [16] S. Mesnager, Bent Functions. Fundamentals and Results, Springer, 2016. doi: 10.1007/978-3-319-32595-8. [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. [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. [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.
 [1] J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413 [2] 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 [3] Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721 [4] Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1399-1406. doi: 10.3934/mbe.2017072 [5] 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 [6] Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939 [7] Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669 [8] Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42. [9] Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741 [10] Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37 [11] Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2018, 8 (0) : 1-42. doi: 10.3934/mcrf.2019012 [12] Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51 [13] Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475 [14] Luc Robbiano. Counting function for interior transmission eigenvalues. Mathematical Control & Related Fields, 2016, 6 (1) : 167-183. doi: 10.3934/mcrf.2016.6.167 [15] Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857 [16] Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569 [17] Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795 [18] 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 [19] Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101 [20] Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391

2017 Impact Factor: 0.564