Constructions Of Permutation Polynomials Over Finite Fields With Even Characteristic

Permutation polynomials play an important role in algebra.It has a wide range of applications in the fields of number theory,combinatorics,coding,and cryptography.In recent years,a great progress has been made in the study of permutation polynomi-als.Scholars have presented many methods to construct permutation polynomials over finite fields,such as the AGW criterion,piecewise construction,switching construc-tion and so on.Permutation polynomials with fewer terms over finites fields always attract researchers’ interests due to their simple algebric forms.In 2014,Ding etc.presented nine classes of permutation trinomials by studying the number of roots to special equations.In what follows,Li and Qu used fractional polynomils to constuct six classes of permutaion trinomials over finite fields with characteristic 2 and three classes of permutaion trinomials over finite fields with characteristic 3.Later,Li and Helleseth further investigated permutation trinomials of the form xrh(x(q-1)/d)from Niho exponents,and they presented two classes of permutation trinomials of the form x+xs(2m-1)+ xt(2m-1)+1 over F22m.Recently,Gupta and Sharma investigated the per-mutation trinomials of the form xrh(x(q-1)/d)over Fq,where q = 22m,d = 2m+1 and h(x)∈F2[x].They constructed four classes of permutation trinomials from cer-tain bijections on the unit circle.Moreover,two conjectures were proposed as follows.Conjectures 1.f(x):= x5 + x3·2m+2 + x4·2m+1∈F22m[x]permutes F22m if and only if m≡2(mod 4).Conjectures 2.g(x):x5 + x2m+4 + x5·2m∈F22m[x]permutes F22m if and only if m≡2(mod 4).Very recently,Zha etc.further investigated permuta-tion trinomials and presented six classes permutation trinomials.Meanwihle using the property of the unit circle,the two conjectures proposed by Gupta and Sharma were fi-nally resolved.In this paper,we construct several classes of permutation pentanomials with the form xrh(x(q-1)/d).Further,we propose a general method for constructing permutation polynomials with odd terms.