Trace Monomial Boolean Functions with Large High-Order Nonlinearities. (arXiv:2309.11229v1 [cs.CR])

Exhibiting an explicit Boolean function with a large high-order nonlinearity
is an important problem in cryptography, coding theory, and computational
complexity. We prove lower bounds on the second-order, third-order, and
higher-order nonlinearities of some trace monomial Boolean functions.

We prove lower bounds on the second-order nonlinearities of functions
$\mathrm{tr}_n(x^7)$ and $\mathrm{tr}_n(x^{2^r+3})$ where $n=2r$. Among all
trace monomials, our bounds match the best second-order nonlinearity lower
bounds by \cite{Car08} and \cite{YT20} for odd and even $n$ respectively. We
prove a lower …

coding complexity computational cryptography explicit function functions high higher important large order problem prove second-order theory third trace