2021/01/27 by Zheng, Lijing, Kan, Haibin, Li, Yanjun +2
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2101.11535
In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over \mathbbF2n of the form x3+a(x2s+1)2k+bx3⋅ 2m+c(x^2s+m+2m)2k, 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,ω, ω2,1) or ((m-2)-1~\rm mod~n,ω, ω2,1) in which ω∈\mathbbF4∖ \mathbbF2. By taking a=ω and b=c=ω2, we observe that such kind of quadrinomials can be rewritten as a \rm Trnm(bx3)+aq\rm Trnm(cx2s+1), where q=2m and \rm Trnm(x)=x+x2m 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 Trnm(F(x))+aq\rm Trnm(G(x)) and determine the APN-ness of this new kind of functions, where a ∈ \mathbbF2n such that a+aq≠ 0, and both F and G are quadratic functions over \mathbbF2n. 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 Trnm(bx3)+aq\rm Trnm(b3x9) , where a∈ \mathbbF2n such that a+aq≠ 0 and b is a non-cube in \mathbbF2n .