# Constructing new APN functions through relative trace functions

In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over ๐ฝ_2^n of the form x^3+a(x^2^s+1)^2^k+bx^3ยท 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 (3,m)=1, k=0, and (s,a,b,c)=(m-2,ฯ, ฯ^2,1) or ((m-2)^-1ย modย n,ฯ, ฯ^2,1) in which ฯโ๐ฝ_4โ๐ฝ_2. By taking a=ฯ and b=c=ฯ^2, we observe that such kind of quadrinomials can be rewritten as a Tr^n_m(bx^3)+a^q Tr^n_m(cx^2^s+1), where q=2^m and 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 Tr^n_m(F(x))+a^q Tr^n_m(G(x)) and determine the APN-ness of this new kind of functions, where a โ๐ฝ_2^n such that a+a^qโ 0, and both F and G are quadratic functions over ๐ฝ_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 Tr^n_m(bx^3)+a^q Tr^n_m(b^3x^9), where aโ๐ฝ_2^n such that a+a^qโ 0 and b is a non-cube in ๐ฝ_2^n.

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