Notes on Bent Functions in Polynomial Forms

Authors : Onur Kocak, Onur Kurt, Neşe Öztop, Zülfükar Saygı

Pages : 43-48

View : 43 | Download : 11

Publication Date : 2012-07-02

Article Type : Research Paper

Abstract :The existence and construction of bent functions are two of the most widely studied problems in Boolean functions. For monomial functions finsert ignore into journalissuearticles values(x); = T rn 1 insert ignore into journalissuearticles values(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. finsert ignore into journalissuearticles values(x); = T rn 1 insert ignore into journalissuearticles values(axs1 ); + T rk 1 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(axrinsert ignore into journalissuearticles values(2m−1);); + T rm 1 insert ignore into journalissuearticles values(bxsinsert ignore into journalissuearticles values(2m+1);); where n = 2m, gcdinsert ignore into journalissuearticles values(r, 2m + 1); = 1, gcdinsert ignore into journalissuearticles values(s, 2 m − 1); = 1. Also, we give a bent function example of the form fa,binsert ignore into journalissuearticles values(x); = T rn 1 insert ignore into journalissuearticles values(ax2m−1 ); + T r2 1insert ignore into journalissuearticles values(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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