The Nonexistence of Permutations EA-Equivalent to Certain AB Functions
Yongqiang Li, Ming‐Sheng Wang · IEEE Transactions on Information Theory · 2012
Carlet and colleagues conjectured that for any almost bent (AB) functionF, there exists a linear functionLsuch thatF+Lis 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 functionsLonF25such thatx2i+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 functionx2i+1+(x2i+x) Tr (x2i+1+x) onF22m+1for anym≥ 2 and there are no permutations EA-equivalent to the APN functionx2i+1+(x2i+x+1) Tr (x2i+1) on \BBF22mform≥ 2 either. Furthermore, we present some results about characterizations of permutation polynomials of the typeL(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 functionsL(x) such thatx+L(x2i+1) is a permutation on \BBF22mwhen |ker(L)| ≥ 22m-2.