Algebraic methods in difference sets and bent functions

Year 2021, Volume: 8 Issue: 2, 139 - 148, 20.05.2021

Pradipkumar H. Keskar Priyanka Kumari

https://doi.org/10.13069/jacodesmath.940192

Abstract

We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the $\mathcal{C}$-condition introduced by him doesn't hold.

Keywords

Hilbert functions, $\mathcal{C}$-condition, Flat, Difference set, Bent functions, Hilbert functions, $\mathcal{C}$-condition, Flat, Difference set, Bent functions

References

• [1] T. Becker, V. Weispfennig, Groebner bases a computational approach to commutative algebra, Springer (1993).
• [2] C. Carlet, Two new classes of bent functions, Advances in Cryptology-EUROCRYPT’93, LNCS vol 765 (Ed. T. Hellseth) Springer-Verlag (1994) 77–101.
• [3] D. Cox, J. Little, D. O’Shea, Ideals, varieties and algorithms, Springer Verlag, New York Inc (2007).
• [4] J. F. Dillon, Elementary hadamard difference sets, Ph. D. Thesis, University of Maryland (1974).
• [5] P. H. Keskar, P. Kumari, Polynomial criterion for abelian difference sets, Indian Journal of Pure and Applied Mathematics 51(1) (2020) 233–249.
• [6] N. Kolomeec, The graph of minimal distances of bent functions and its properties, Designs, Codes and Cryptography 85 (2017) 395–410.
• [7] B. Mandal, P. Stanica, S. Gangopadhyay, E. Pasalic, An analysis of the C class of bent functions, Fundamenta Informaticae 146(3) (2016) 271–292.
• [8] E. H. Moore, H. S. Pollatsek, Difference sets, connecting algebra, combinatorics, and geometry, american mathematical society (2013).
• [9] N.Tokareva, Bent functions: Results and applications to cryptography, Elsevier (2015).
• [10] A. van den Essen, Polynomial automorphisms and the jacobian conjecture, Birkhauser (2000).