A SUPER-SET OF PATTERSON-WIEDEMANN FUNCTIONS: UPPER BOUNDS AND POSSIBLE NONLINEARITIES

Date

2018-01-01

Author

Kavut, Selcuk
MAİTRA, Subhamoy
Özbudak, Ferruh

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Construction of Boolean functions on an odd number of variables with nonlinearity exceeding the bent concatenation bound is one of the most difficult combinatorial problems within the domain of Boolean functions. This problem also has deep implications in coding theory and cryptology. Patterson and Wiedemann demonstrated instances of such functions back in 1983. For more than three decades efforts have been channeled into obtaining such instances. For the first time, in this paper we explore nontrivial upper bounds on nonlinearity for such classes of functions that are invariant not only under several group actions but also for larger sets of functions than what have been considered so far. Further, we present tight upper bounds on the nonlinearity in several cases. To support our claims, we present computational results for functions on n variables, where n is an odd composite integer in the interval [9, 39]. In particular, our results for n = 15 and 21 are of immediate interest given recent research results in this domain. In addition to the upper bounds, we also discover the nonlinearities that can actually be achieved above the bent concatenation bound for such a class of functions. Finally, we obtain all possible values in the absolute Walsh spectra of the functions considered.

Subject Keywords

Covering Radius, First Order Reed-Muller Code, Nonlinearity Bound, Patterson-Wiedemann Type Functions

URI

https://hdl.handle.net/11511/36742

Journal

SIAM JOURNAL ON DISCRETE MATHEMATICS

DOI

https://doi.org/10.1137/17m1144581

Collections

Department of Mathematics, Article

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Citation Formats

S. Kavut, S. MAİTRA, and F. Özbudak, “A SUPER-SET OF PATTERSON-WIEDEMANN FUNCTIONS: UPPER BOUNDS AND POSSIBLE NONLINEARITIES,” SIAM JOURNAL ON DISCRETE MATHEMATICS, pp. 106–122, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/36742.