Constructing new APN functions through relative trace functions

Abstract: In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $\mathbb{F}_{2^{n}}$ of the form $$x^3+a(x^{2^s+1})^{2^k}+bx^{3\cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$$, where $n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $\gcd(3,m)=1$, $k=0$, and $(s,a,b,c)=(m-2,\omega, \omega^2,1)$ or $((m-2)^{-1}~{\rm mod}~n,\omega, \omega^2,1)$ in which $\omega\in\mathbb{F}_4\setminus \mathbb{F}_2$. By taking $a=\omega$ and $b=c=\omega^2$, we observe that such kind of quadrinomials can be rewritten as $$a {\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(cx^{2^s+1})$$, where $q=2^m$ and ${\rm Tr}^n_{m}(x)=x+x^{2^m}$ for $n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $$f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^q{\rm Tr}^{n}_{m}(G(x))$$ and determine the APN-ness of this new kind of functions, where $a \in \mathbb{F}_{2^n}$ such that $a+a^q\neq 0$, and both $F$ and $G$ are quadratic functions over $\mathbb{F}_{2^n}$. We first obtain a characterization of the conditions for $f(x)$ such that $f(x)$ is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for $n=2m$ with $m$ being an odd positive integer: $$f(x)=a{\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(b^3x^9)$$, where $a\in \mathbb{F}_{2^n}$ such that $a+a^q\neq 0$ and $b$ is a non-cube in $\mathbb{F}_{2^n}$.