A note on inverses of cyclotomic mapping permutation polynomials over finite fields

Abstract

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.

Permutations from an arithmetic setting

“…There are some cryptographic attacks that explore the number of fixed points of a permutation and according to [4], for secure implementations, involutions should have few fixed points. In the particular case m = 2 of Proposition 4.8, the θ q -lift F θq,π recover a family of involutions that were previously obtained in [14]. This is presented in the following corollary, which is just a straightforward application of the previous proposition.…”

On Inverses of Permutation Polynomials of Small Degree Over Finite Fields

“…Hence Theorems 7, 8 and 9 are all true for n = 1. If γ = θ = 0, then Lemmas 5, 6 and 7 are the special cases of [18,Corollary 2.3], and their inverses are given in [19,29].…”

“…However, the problem of determining the inverse of a PP seems to be an even more complicated problem. In fact, there are few known classes of PPs whose inverses have been obtained explicitly; see for example [8,11,17,29] for PPs of the form x r h(x (q−1)/d ), [20,22] for linearized PPs, [19,29] for generalized cyclotomic mapping PPs, [30] for general piecewise PPs, [3] for involutions over F 2 n , and [14,15] for more general classes of PPs. The general results in [14,15] also contain some concrete classes, such as bilinear PPs in [21] and linearized PPs of the form L(x) + K(x) in [12].…”

On inverse of permutation polynomials of small degree over finite fields, II

“…In fact, finding the inverse of a PP of a large finite field is a hard problem except for the well-known classes such as the inverses of linear polynomials, monomials, and some Dickson polynomials [4]. There are only several papers on the inverses of some special classes of PPs, see [4,6,10,18] for PPs of the form x r h(x (q−1)/d ), [7,[12][13][14]16] for linearized PPs, [2,15] for two classes of bilinear PPs, [11,18] for generalized cyclotomic mapping PPs, [1] for involutions over F 2 n , [18,19] for more general piecewise PPs, [8,9] for more general classes of PPs. The general results in [8,9] also contain some concrete classes mentioned earlier such as bilinear PPs [15], linearized PPs of the form L(x) + K(x) [7], and PPs of the form x + γf (x) with b-linear translator γ [3].…”