利用相关迹函数构造新的APN函数
Constructing new APN functions through relative trace functions
摘要：
在2020年，Budaghyan，Helleseth和Kaleyski [IEEE TIT 66（11）：7081-7087，2020]考虑了形式为$x^3+a(x^{2^s+1})^{2^k}+bx^{3\cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$的$\mathbb{F}_{2^{n}}$上的四项多项式的无限族，其中$n=2m$具有$m$奇数。证明了当$\gcd(3,m)=1$，$ k=0 $，$(s,a,b,c)=(m-2,ω, ω^2,1)$或$((m-2)^{-1}~{\rm mod}~n,ω, ω^2,1)$中有$ω\in\mathbb{F}_4\setminus \mathbb{F}_2$时，这类四项多项式能提供新的几乎完全非线性（APN）函数.通过取$a=ω$和$b=c=ω^2$，我们观察到这类四项式可改写为$a {\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(cx^{2^s+1})$，其中$q=2^m$和$ {\rm Tr}^n_{m}(x)=x+x^{2^m} $为$ n=2m$ .受四项多项式和我们的观察的启发，本文研究了一类具有$f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^q{\rm Tr}^{n}_{m}(G(x))$形式的函数，并确定了这类新函数的APN性，其中$a \in \mathbb{F}_{2^n} $使得$ a+a^q\neq 0$，且$F$和$G$都是$\mathbb{F}_{2^n}$上的二次函数.我们首先得到的$f(x)$的条件，使$f(x) $是一个APN功能的表征。借助于这个刻画，我们得到了$ n=2m $的APN函数的一个无限族，其中$m$是一个奇正整数：$ f(x)=a{\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(b^3x^9) $，其中$ a\in \mathbb{F}_{2^n}$使得$ a+a^q\neq 0 $和$ b $是$ \mathbb{F}_{2^n} $中的一个非立方.
英文摘要：
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 $(s,a,b,c)=(m-2,ω, ω^2,1)$ or $((m-2)^{-1}~{\rm mod}~n,ω, ω^2,1)$ in which $ω\in\mathbb{F}_4\setminus \mathbb{F}_2$. By taking $a=ω$ and $b=c=ω^2$, 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} $.
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