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$F: \mathbb F_{p^n} \rightarrow \mathbb F_{p^n}$，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$F(x)=x^3+{\rm Tr}(x^9)$的情形，Budaghyan、Carlet和Pott（以及Budaghyan、Carlet和Leander）\cite{BCP06, BCL09FFTA} 分别发现了与$F$ CCZ等价但EA不等价的函数。本文中，当给定函数$F$具有线性结构的分量函数时，我们构造出与$F$ CCZ等价的函数，并在满足适当条件时证明所构造函数与$F$ EA不等价。作为推论，对于$\mathbb F_{2^n}$（$n\geq 4$）上非线性度$>0$且差分均匀度$\leq 2^{n-3}$的任意二次函数$F$，我们显式构造了与$F$ CCZ等价但EA不等价的函数。同时，对于$\mathbb F_{p^n}$（$p>2, n\geq 4$）上满足$|\mathcal W_F|\leq p^{n-1}$且差分均匀度$\leq p^{n-3}$的任意非平面二次函数$F$，我们也显式构造了与$F$ CCZ等价但EA不等价的函数。