Abstract
The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two authors showed that the elements which a vectorial dual-bent function with certain additional properties maps to 0, form a partial difference set, which generalizes the connection between Boolean bent functions and Hadamard difference sets, and some later established connections between p-ary bent functions and partial difference sets to vectorial bent functions. We discuss the effects of coordinate transformations. As all currently known vectorial dual-bent functions \(F:{\mathbb {F}}_{p^n}\rightarrow {\mathbb {F}}_{p^s}\) are linear equivalent to l-forms, i.e., to functions satisfying \(F(\beta x) = \beta ^lF(x)\) for all \(\beta \in {\mathbb {F}}_{p^s}\), we investigate properties of partial difference sets obtained from l-forms. We show that they are unions of cosets of \({\mathbb {F}}_{p^s}^*\), which also can be seen as certain cyclotomic classes. We draw connections to known results on partial difference sets from cyclotomy. Motivated by experimental results, for a class of vectorial dual-bent functions from \({\mathbb {F}}_{p^n}\) to \({\mathbb {F}}_{p^s}\), we show that the preimage set of the squares of \({\mathbb {F}}_{p^s}\) forms a partial difference set. This extends earlier results on p-ary bent functions.
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References
Bernasconi A., Codenotti B.: Spectral analysis of Boolean functions as a graph eigenvalue problem. IEEE Trans. Comput. 48, 345–351 (1999).
Bernasconi A., Codenotti B., VanderKam J.M.: A characterization of bent functions in terms of strongly regular graphs. IEEE Trans. Comput. 50, 984–985 (2001).
Çeşmelioğlu A., McGuire G., Meidl W.: A construction of weakly and non-weakly regular bent functions. J. Combin. Theory Ser. A 119, 420–429 (2012).
Çeşmelioğlu A., Meidl W., Pott A.: On the dual of (non)-weakly regular bent functions and self-dual bent functions. Adv. Math. Commun. 7, 425–440 (2013).
Çeşmelioğlu A., Meidl W., Pott A.: There are infinitely many bent functions for which the dual is not bent. IEEE Trans. Inform. Theory 62, 5204–5208 (2016).
Çeşmelioğlu A., Meidl W., Pott A.: Vectorial bent functions and their duals. Linear Algebra Appl. 548, 305–320 (2018).
Çeşmelioğlu A., Meidl W., Pott A.: Vectorial bent functions in odd characteristic and their components. Cryptogr. Commun. 12, 899–912 (2020).
Çeşmelioğlu A., Meidl W.: Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Adv. Math. Commun. 12, 691–705 (2018).
Chee Y.M., Tan Y., Zhang Y.D.: Strongly regular graphs constructed from \(p\)-ary bent functions. J. Algebraic Combin. 34, 251–266 (2011).
Feng T., Wen B., Xiang Q., Yin J.: Partial difference sets from quadratic forms and p-ary weakly regular bent functions. Number theory and related areas, Adv. Lect. Math. (ALM), 27, pp. 25–40. Int. Press, Somerville, MA (2013).
Helleseth T., Kholosha A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inform. Theory 52(5), 2018–2032 (2006).
Hyun J.Y., Lee Y.: Characterization of \(p\)-ary bent functions in terms of strongly regular graphs. IEEE Trans. Inform. Theory 65, 676–684 (2019).
Joyner D., Melles C.: Perspectives on \(p\)-ary bent functions. Elementary theory of groups and group rings, and related topics (New York, Nov. 1–2, 2018), pp. 103–126. De Gruyter, Berlin (2020).
Kumar P.V., Scholtz R.A., Welch L.R.: Generalized bent functions and their properties. J. Combin. Theory Ser. A 40, 90–107 (1985).
Ma S.L.: A survey of partial difference sets. Des. Codes Cryptogr. 4(4), 221–261 (1994).
Ma S.L.: Partial difference sets. Discret. Math. 52, 75–89 (1984).
Nyberg K.: Perfect nonlinear S-boxes. Advances in cryptology–EUROCRYPT ’91 (Brighton, 1991). Lecture Notes in Comput. Sci., 547, pp. 378–386. Springer, Berlin (1991).
Pott A., Tan Y., Feng T., Ling S.: Association schemes arising from bent functions. Des. Codes Cryptogr. 59, 319–331 (2011).
Tan Y., Pott A., Feng T.: Strongly regular graphs associated with ternary bent functions. J. Combin. Theory Ser. A 117, 668–682 (2010).
Xu Y., Carlet C., Mesnager S., Wu C.: Classification of bent monomials, constructions of bent multinomials and upper bounds on the nonlinearities of vectorial functions. IEEE Trans. Inform. Theory 64, 367–383 (2018).
Acknowledgements
W.M. is supported by the FWF Project P 30966; I.P. is supported by the FWF Project F5508-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. We like to thank the Associate Editor, and the reviewers for their comments.
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Çeşmelioğlu, A., Meidl, W. & Pirsic, I. Vectorial bent functions and partial difference sets. Des. Codes Cryptogr. 89, 2313–2330 (2021). https://doi.org/10.1007/s10623-021-00919-y
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DOI: https://doi.org/10.1007/s10623-021-00919-y