The Nonexistence of Permutations EA-Equivalent to Certain AB Functions

Yongqiang Li Yongqiang Li,Mingsheng Wang Mingsheng Wang

doi:10.1109/tit.2012.2213064

Abstract

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 F 2 5 such that x 2 i +1 +( x 2 i + x ) Tr ( x 2 i +1 + x )+ L ( x ) is a permutation. In this paper, we prove that there are no permutations EA-equivalent to the AB function x 2 i +1 +( x 2 i + x ) Tr ( x 2 i +1 + x ) on F 2 2m+1 for any m ≥ 2 and there are no permutations EA-equivalent to the APN function x 2 i +1 +( x 2 i + x +1) Tr ( x 2 i +1 ) on \BBF 2 2 m for m ≥ 2 either. Furthermore, we present some results about characterizations of permutation polynomials of the type L ( x 2 i +1 )+ L ' ( x ) on \BBF 2 2 m , 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 ( x 2 i +1 ) is a permutation on \BBF 2 2 m when |ker( L )| ≥ 2 2 m -2.

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