High-Meets-Low: Construction of Strictly Almost Optimal Resilient Boolean Functions via Fragmentary Walsh Spectra

This paper considers the construction of resilient Boolean functions on an odd number of variables with strictly almost optimal (SAO) nonlinearity. Through introducing the fragmentary Walsh transform, a construction technique called “High-Meets-Low” is proposed. The detailed design procedures of a 39-variable 3-resilient Boolean function with SAO nonlinearity
$2^{38}-2^{19}+2^{16}+2^{14}$
are given. It is shown that the nonlinearity of an
$n$
-variable
$t$
-resilient Boolean function can reach
$2^{n-1}-2^{(n-1)/2}+5\cdot 2^{(n-11)/2}$
or
$2^{n-1}-2^{(n-1)/2}+2^{(n-7)/2}$
, which are the largest known values for the corresponding
$n$
and
$t$
values. Finally, by constructing a 29-variable balanced Boolean function with SAO nonlinearity
$2^{28}-2^{14}+2^{10}+2^{9}$
, we show an alternative method to realize the High-Meets-Low construction technique.