  In this article, we focus on the concept of locally-APN-ness (``APN" is the
abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced
by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow
larger regarding their differential uniformity and, therefore, possibly, more
suitable candidates against the differential attack (or their variants).
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 an
infinite family of Niho type power functions in the form $F(x)=x^{s(2^m-1)+1}$
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 (with even characteristic) as well as some subtle
manipulations of solving some equations, we prove that $F(x)$ is locally APN
and determine its differential spectrum. It is worth noting that computer
experiments show that this class of locally-APN power functions covers all Niho
type locally-APN power functions for $2\leq m\leq10$. 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.
