Notes on Bent Functions in Polynomial Forms

Year 2012, Volume: 1 Issue: 2, 43 - 48, 02.07.2012

Onur Kocak Onur Kurt Neşe Öztop Zülfükar Saygı

Abstract

The existence and construction of bent functions are two of the most widely studied problems in Boolean functions. For monomial functions f(x) = T rn 1 (axs), these problems were examined extensively and it was shown that the bentness of the monomial functions is complete for n ≤ 20. However, in the binomial function case, i.e. f(x) = T rn 1 (axs1 ) + T rk 1 (bxs2 ), this characterization is not complete and there are still open problems. In this paper, we give a summary of the literature on the bentness of binomial functions and show that there exist no bent functions of the form T rn 1 (axr(2m−1)) + T rm 1 (bxs(2m+1)) where n = 2m, gcd(r, 2m + 1) = 1, gcd(s, 2 m − 1) = 1. Also, we give a bent function example of the form fa,b(x) = T rn 1 (ax2m−1 ) + T r2 1(bx 2n−1 3 ) for n = 4, although, it is stated in [9] that there is no such bent function of this form for any value of a and b.

Keywords

Boolean function, Bent functions, Walsh-Hadamard Transform, Dillon exponents, Kloosterman Sums

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