Compositional Inverses of Permutation Polynomials of the Form xrh(xs) over Finite Fields
he study of computing compositional inverses of permutation polynomials over finite fields efficiently is motivated by an open problem proposed by G.L.Mullen 20, as well as the potential applications of these permutation polynomials 6, 8, 12, 13, 24, 25.It is well known that every permutation polynomial over a finite field IFq can be reduced to a permutation polynomial of the form xrh(xs) with s | q-1 and h(x) ∈ IFq 1, 30.he study of computing compositional inverses of permutation polynomials over finite fields efficiently is motivated by an open problem proposed by G.L.Mullen 20, as well as the potential applications of these permutation polynomials 6, 8, 12, 13, 24, 25.It is well known that every permutation polynomial over a finite field IFq can be reduced to a permutation polynomial of the form xrh(xs) with s | q-1 and h(x) ∈ IFq 1, 30.T Recently, several explicit classes of permutation polynomials of the form xrh (xs) over IFq have been constructed.However, all the known methods to compute the compositional inverses of permutation polynomials of this form seem to be inadequately explicit, which could be a hurdle to potential applications.In this paper, for any prime power q, we introduce a new approach to explicitly compute the compositional inverse of a permutation polynomial of the form xrh (xs) over IFq, where s | q-1 and gcd(r, q-1) =1.The main idea relies on transforming the problem of computing the compositional inverses of permutation polynomials over IFq into computing the compositional inverses of two restricted permutation mappings, where one of them is a monomial over IFq and the other is the polynomial xrh(x)s over a particular subgroup of IFq with order (q-1)/s.This is a multiplicative analogue of 26, 33.We demonstrate that the inverses of these two restricted permutations can be explicitly obtained in many cases.As consequences, many explicit compositional inverses of permutation polynomials given in 40-42 are obtained using this method.
Finite Fields Permutation Polynomials Compositional Inverses
Kangquan Li Longjiang Qu Qiang Wang
College of Science, National University of Defense Technology, Changsha, 410073,China School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K
国际会议
广州
英文
53-73
2018-05-01（万方平台首次上网日期，不代表论文的发表时间）