Several Classes of Permutation Polynomials over Finite Fields

Several classes of permutation polynomials of the form () () k p t x x + δ + L x − over finite fields are presented in this paper, which is a further investigation on a recent work of Li et al.


Introduction
Let p be a prime and F to itself.Permutation polynomials were first studied by Hermite [1] for the case of finite prime fields and by Dickson [4] for arbitrary finite fields.Permutation polynomials have been studied extensively and have important applications in coding theory, cryptography, combinatorics, and design theory [3][4][5][6].In the recent years, there has been significant progress in finding new permutation polynomials [7][8][9][10][11][12].
The determination of permutation polynomials is not an easy problem.An important class of permutation polynomials is of the form ( ) ( ), where , k t are integers, n p F δ ∈ and ( ) L x is a linearized polynomial.In [13], Helleseth and Zinoviev de- rived new identities on Kloosterman sums by making use of this kind of permutation polynomials.Followed by the research of Helleseth and Zinoviev, many researchers began to study such class of permutation polynomials and numerical results were obtained [14 -18].
In this paper, inspired by permutation polynomials obtained by Zha and Hu in [18]} and This paper is organized as follows.In Section 2, we present several classes of permutation polynomials over 3 n F , which are not covered by [18,19], and over n p F .We conclude the paper in Section 3.
G. H. Sun

Permutation Polynomials over
3 n

F
In this section, we study the permutation polynomials of the form (1) with Let α be a primitive element of ( 1) [19] investigates the permutation polynomials with the form of ) The following theorem shows that it is also a permutation polynomial when where k is a positive integer, and Tr δ = .Then ) Proof For any , it is sufficient to prove the Equation ) has exactly one solution over 3 n F .
If x is a solution of the Equation ( 2), then we consider three cases in the following.
Raising the 3 k th power on the both sides of Equation (3), we have .
. Raising the 3 k th and 2 3 k th power on the both sides of the above Equation, respectively, we have . By the three Equations, we have . Hence only one case will happen in the above three cases.So The following theorems give several new classes of permutation polynomials which are not covered by [18,19].
where k is a positive integer, and Proof For all F for any 0,1 i = or 2. For 0,1 i = or 2, the proof is similar, so we only prove the permutation polynomial for ) If x is a solution of the Equation ( 5), then we consider three cases in the following.
. Then ) Proof Since the proof is similar for any 0,1 i = or 2, we only prove the permutation polynomial when has exactly one solution over x is a solution of the Equation ( 6), then we consider three cases in the following.
1) Case A: 3 0 k x x δ − + =.By (6), we have 2 3 3 By the two Equations, we have If Case A occurs, then we have F .The theorem holds.
In the above section, we consider permutation polynomials over Similarly, the proof is also similar for 1 i = or 3, so we only consider 1 i = .
For   Hence only one case will occur in the above three cases.Therefore, .Then raising the k p th power, 2k p th power, and 3k p th power on the both sides for Equation (7), respectively, we have Adding (11) to (12), we have p th power, and 3k p th power on the both sides of Equation (13), respectively, then we obtain Subtracting (13) from ( 14), then we obtain Adding (15) to ( 16), then we have Subtracting ( 18) from (17), then we obtain

Conclusion
In the paper, we obtain some permutation polynomials of the form ( ) ( ) which are not covered by [18,19].It is possible that they have some applications in coding theory, cryptography, combinatorics, design theory and so on.

FF
denote a finite field with n p elements.A polynomial ( ) [ ] if it induces a one-to-one map from n p Raising the 3 k th power on both sides, we have .Raising the 3 k th power and 2 3 k th power on the both sides of the above Equation, respectively, then we have .Raising the 3 k th power and2  3 k th power on the both sides of the above Equation, respectively, then we have By the three Equations, we have 23 3

3 nFF 4
. In the following, we investigate permutation polynomials over n p for an odd prime p .Theorem Let 4 n k = ,where k is a positive integer and p be an odd prime.

F
Hence only one case will occur in the above three cases.Therefore, .The theorem holds.