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

Abstract

For a given function F from Fp 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(x)=x2 and F(x)=x3+Tr(x9) over F2, 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 F2 (n≥4) with nonlinearity greater than 0 and differential uniformity not exceeding 2n−3, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to F. Also for every non-planar quadratic function on Fp (p>2,n≥4) with |WF|≤pn−1 and differential uniformity not exceeding pn−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 Fp such that the CCZ-equivalence class of F is strictly larger than the EA-equivalence class of F. © 2025 Elsevier Inc.