Permutation polynomials EA-equivalent to the inverse function over GF (2 n )

Abstract

Abstract. It is proved that there does not exist a linearized polynomialis a permutation on F2n when n ≥ 5, which is proposed as a conjecture in [15]. As a consequence, a permutation is EA-equivalent to the inverse function over F2n if and only if it is affine equivalent to it when n ≥ 5.

Bent and Permutational Properties of Budaghyan-Carlet Hexanomials

“…Proof. By Proposition 3, we have C(2 , ) = F 2 2 \ ⋃ 2 =0 2 , where | 2 | = 2 for 0 ≤ ≤ 2 by (11) and (12). To prove |C(2 , )| = 2 2 −1 −2 , it suffices to prove | ⋃ 2 =0 2 | = 2 2 −1 + 2 .…”

“…This motivates people to investigate the conditions such that the function of the form ( ) + ( ) is a permutation, where ( ) is a function with a low differential uniformity and ( ) is a linearized polynomial. The case that ( ) is a power function is discussed in [10][11][12]. When ( ) is a quadratic APN function on fields of even extensions, it is known that ( ) + ( ) can never be permutations.…”

On CCZ-equivalence of the inverse function

“…A partial result in the classification of permutations of the form L 1 (x −1 ) + L 2 (x) was found in [18]. We want to remark that the techniques we develop in this paper to tackle the more general question differ considerably from the ones employed in [18].…”

On the EA-classes of known APN functions in small dimensions

“…For the inverse function x 255 we have two simplex codes but only one EA-class. In [16] the authors investigate EA-equivalence of the inverse function to a permutation. They concluded that for n ≥ 5 the inverse function is EA-equivalent to a permutation if and only if it is affine equivalent to it.…”

“…This implies that we have only the EA-class of x −1 since it is an involution. Now, the permutations in the EA-class of x −1 can be obtained only with the affine equivalence [16].…”