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} $.
Categories: cs.IT math.IT