Constructing New APN Functions Through Relative Trace Functions

Let
$n=2m$
. In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered a family of quadrinomials over
$\mathbb {F}_{2^{n}}$
of the form
$x^{3}+a(x^{2^{s}+1})^{2^{k}}+bx^{3\cdot 2^{m}}+c(x^{2^{s+m}+2^{m}})^{2^{k}}$
. They showed that two infinite classes of almost perfect nonlinear (APN) functions belong to this family when
$\gcd (6,m)=1$
. We observe that these two infinite classes of APN quadrinomials and the infinite class of APN polynomials from the Budaghyan-Carlet family belong to a more general family of polynomials over
$\mathbb {F}_{2^{n}} $
with the form
$f(x)=a{\mathrm{ Tr}}^{n}_{m}(F(x))+a^{2^{m}}{\mathrm{ Tr}}^{n}_{m}(G(x))$
, where
$a \in \mathbb {F}_{2^{n}}\backslash \mathbb {F}_{2^{m}} $
, and both
$F$
and
$G$
are quadratic functions over
$\mathbb {F}_{2^{n}}$
. We characterize when
$f(x) $
is APN. With the help of our characterization, letting
$F(x)=bx^{2^{i}+1} $
and
$G(x)=cx^{2^{s}+1}$
with
$b, c\in \mathbb {F}_{2^{n}} $
, we obtain an infinite family of APN functions of the form
$f(x) $
when
${\mathrm{ gcd}}(2,m)=1 $
and verify that for
$n=10 $
two APN instances from this infinite family are CCZ-inequivalent to each other, and to any APN function over
$\mathbb {F}_{2^{10}} $
from the previously known infinite families.