SAO 1-Resilient Functions With Lower Absolute Indicator in Even Variables

In 2018, Tang and Maitra presented a class of balanced Boolean functions in
$n$
variables with the absolute indicator
$\Delta _{f}< 2^{n/2}$
and the nonlinearity
$NL(f)> 2^{n-1}-2^{n/2}$
, that is,
$f$
is SAO (strictly almost optimal), for
$n=2k\equiv 2\,\,({\mathrm {mod}\,\,}4)$
and
$n\geq 46$
in [IEEE Ttans. Inf. Theory 64
(1)
: 393-402, 2018]. However, there is no evidence to show that the absolute indicator of any 1-resilient function in
$n$
variables can be strictly less than
$2^{\lfloor ({n+1})/{2}\rfloor }$
, and the previously best known upper bound of which is
$5\cdot 2^{n/2}-2^{n/4+2}+4$
. In this paper, we concentrate on two directions. Firstly, to complete Tang and Maitra’s work for
$k$
being even, we present another class of balanced functions in
$n$
variables with the absolute indicator
$\Delta _{f}< 2^{n/2}$
and the nonlinearity
$NL(f)> 2^{n-1}-2^{n/2}$
for
$n\equiv 0~({\mathrm {mod}~}4)$
and
$n\geq 48$
. Secondly, we obtain two new classes of 1-resilient functions possessing very high nonlinearity and very low absolute indicator, from bent functions and plateaued functions, respectively. Moreover, one class of them achieves the currently known highest nonlinearity
$2^{n-1}-2^{n/2-1}-2^{n/4}$
, and the absolute indicator of which is upper bounded by
$2^{n/2}+2^{n/4+1}$
that is a new upper bound of the minimum of absolute indicator of 1-resilient functions, as it is clearly optimal than the previously best known upper bound
$5\cdot 2^{n/2}-2^{n/4+2}+4$
.