On the Niho Type Locally-APN Power Functions and Their Boomerang Spectrum

This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers
$m$
and
$k$
such that
$\gcd (2^{m}+1,2^{k}+1)=1$
, we investigate the locally-APN-ness property of the Niho type power function
$F(x)=x^{s(2^{m}-1)+1}$
over the finite field
$\mathbb {F}_{2^{2m}}$
for
$s=(2^{k}+1)^{-1}$
, where
$(2^{k}+1)^{-1}$
denotes the multiplicative inverse modulo
$2^{m}+1$
. By employing finer studies of the number of solutions of certain equations over finite fields, we prove that
$F(x)$
is locally-APN and determine its differential spectrum. We emphasize that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for
$2\leq m\leq 10$
. In addition, we also determine the boomerang spectrum of
$F(x)$
by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang and Li.