On the Functions Which are CCZ-equivalent but not EA-equivalent to Quadratic Functions over 𝔽_p^n

06/23/2023

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by   Jaeseong Jeong, et al.

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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 <cit.> 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+ Tr(x^9) over 𝔽_2^n, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) <cit.> 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≥ 4) with nonlinearity >0 and differential uniformity ≤ 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≥ 4) with |𝒲_F|≤ p^n-1 and differential uniformity ≤ p^n-3, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to F.