The Nonexistence of Permutations EA-Equivalent to Certain AB Functions

Carlet and colleagues conjectured that for any almost bent (AB) function F , there exists a linear function L such that F+L is a permutation. Budaghyan and colleagues found a new class of AB functions which is extended affine (EA)-inequivalent to any power functions and can also serve as a counterexample for the conjecture. They checked with the help of a computer that there are no linear functions L on F₂⁵ such that x²i⁺¹+(x²i+x) Tr (x²i⁺¹+x)+L(x) is a permutation. In this paper, we prove that there are no permutations EA-equivalent to the AB function x²i⁺¹+(x²i+x) Tr (x²i⁺¹+x) on F₂^2m+1 for any m ≥ 2 and there are no permutations EA-equivalent to the APN function x²i⁺¹+(x²i+x+1) Tr (x²i⁺¹) on BBF₂²m for m ≥ 2 either. Furthermore, we present some results about characterizations of permutation polynomials of the type L(x²i⁺¹)+L^´(x) on BBF₂²m, which is essential in the construction of functions Carlet-Charpin-Zinoviev-equivalent to the Gold functions. We obtain all the linear functions L(x) such that x+L(x²i⁺¹) is a permutation on BBF₂²m when |ker(L)| ≥ 2²m-2.