For a given function $F$ from $\mathbb F_{p^n}$ to itself, determining whether there exists a function which is CCZ-equivalent but EA-inequivalent to $F$ is a very important and interesting problem. For example, K\"olsch \cite{KOL21} showed that there is no function which is CCZ-equivalent but EA-inequivalent to the inverse function. On the other hand, for the cases of Gold function $F(x)=x^{2^i+1}$ and $F(x)=x^3+{\rm Tr}(x^9)$ over $\mathbb F_{2^n}$, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) \cite{BCP06, BCL09FFTA} found functions which are CCZ-equivalent but EA-inequivalent to $F$. In this paper, when a given function $F$ has a component function which has a linear structure, we present functions which are CCZ-equivalent to $F$, and if suitable conditions are satisfied, the constructed functions are shown to be EA-inequivalent to $F$. As a consequence, for every quadratic function $F$ on $\mathbb F_{2^n}$ ($n\geq 4$) with nonlinearity $>0$ and differential uniformity $\leq 2^{n-3}$, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to $F$. Also for every non-planar quadratic function on $\mathbb F_{p^n}$ $(p>2, n\geq 4)$ with $|\mathcal W_F|\leq p^{n-1}$ and differential uniformity $\leq p^{n-3}$, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to $F$.

翻译：对于给定函数 $F: \mathbb F_{p^n} \to \mathbb F_{p^n}$，判断是否存在与 $F$ 满足 CCZ 等价但 EA 不等价的函数，是一个非常重要且有趣的问题。例如，Kölsch \cite{KOL21} 证明了不存在与逆函数 CCZ 等价但 EA 不等价的函数。另一方面，对于 $\mathbb F_{2^n}$ 上的 Gold 函数 $F(x)=x^{2^i+1}$ 和 $F(x)=x^3+{\rm Tr}(x^9)$，Budaghyan、Carlet 和 Pott（以及 Budaghyan、Carlet 和 Leander）\cite{BCP06, BCL09FFTA} 分别找到了与 $F$ CCZ 等价但 EA 不等价的函数。在本文中，当给定函数 $F$ 具有线性结构的组件函数时，我们给出了与 $F$ CCZ 等价的函数构造；若满足适当条件，则证明所构造的函数与 $F$ 是 EA 不等价的。由此，对于 $\mathbb F_{2^n}$（$n\geq 4$）上每个非线性度 $>0$ 且差分均匀度 $\leq 2^{n-3}$ 的二次函数 $F$，我们显式构造了与 $F$ CCZ 等价但 EA 不等价的函数。此外，对于 $\mathbb F_{p^n}$（$p>2, n\geq 4$）上每个满足 $|\mathcal W_F|\leq p^{n-1}$ 且差分均匀度 $\leq p^{n-3}$ 的非平面二次函数 $F$，我们也显式构造了与 $F$ CCZ 等价但 EA 不等价的函数。