On higher order nonlinearities of Boolean functions

When analyzing the security of block ciphers and stream ciphers, the r -th order nonlinearity of a Boolean function is crucial. They also have a prominent place in coding theory because the r - th order nonlinearity of Boolean functions is connected to the covering radius of RM(r, m), i.e., Reed-Muller code. In this study, we determine the lower bound for the higher-order nonlinearity of the two classes of Boolean functions listed below. 1. fa(u) = trm1(aud), where d is the Niho exponent constructed by Dobbertin et al. (J. Comb. Theory Ser. A 113:779-798, 2006). 2. ga(u) = tr(1)(m)(au(d)), where d = 2p - 2. For all u ? F2m, a ? F*(m)(2) and m = 2p.

Automated summary

This study determines the lower bound for the higher-order nonlinearity of two classes of Boolean functions, which are crucial in security analysis and coding theory.