On the functions which are CCZ-equivalent but not EA-equivalent to quadratic functions over Fpn

Abstract

For a given function F from $F_{p^n}$ 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, Kölsch [33] 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 $F\left(x\right)=x^{2^i+1}$ and $F\left(x\right)=x^3+\mathrm{Tr}\left(x^9\right)$ over $F_{2^n}$, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) [12], [14] 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 $F_{2^n}$ ($n\ge 4$) with nonlinearity greater than 0 and differential uniformity not exceeding $2^{n-3}$, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to F. Also for every non-planar quadratic function on $F_{p^n}$ $(p>2,n\ge 4)$ with $\left|W_F\right|\le p^{n-1}$ and differential uniformity not exceeding $p^{n-3}$, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to F. As an application, for a proper divisor m of n, we present many examples of $(n,m)$-functions F on $F_{p^n}$ such that the CCZ-equivalence class of F is strictly larger than the EA-equivalence class of F.