Abstract
In this paper, according to the known results of some normalized permutation polynomials with degree 5 over , we determine sufficient and necessary conditions on the coefficients such that permutes . Meanwhile, we obtain a class of complete permutation binomials over .
1. Introduction
Denote as a finite field with elements, where is a prime power; then, is its multiplicative group. The bijectivity of the associated polynomial mapping(s) from into itself (and ) makes a polynomial as a (complete) permutation polynomial [1]. Research interests in (complete) permutation polynomials over finite fields have been aroused due to their widespread applications in cryptography [2, 3], combinatorial designs [4], design theory [1, 5], coding theory [6], and other areas of mathematics and engineering [1, 7]. Readers could refer to [1, 8] for a comprehensive survey about (complete) permutation polynomials. So, it is important to realize that there is significance in discovering new methods to construct permutation polynomials. Readers can find recent progress in [9, 10].
Few-term permutation polynomials, especially binomials and trinomials, have not only simple algebraic form but excellent properties. The following form over from Niho exponents has drawn much attention:where , , . Note that can be viewed as modulo . It is a hard problem to find necessary and sufficient conditions on for (1) being a permutation polynomial over with given . In most known cases, the coefficients are assumed to be trivial, such as ; some pairs in (1) were made in [10–15]. For , Hou [16] completely characterized all non-trivial over through the Hermite criterion. After that, Tu et al. [17] studied the case over where by solving low-degree equations with variable in the unit circle. The latter only obtained the sufficient conditions but conjectured their necessity based on numerical experiments, which were then proved by Bartoli [18] and Hou [19], respectively, using algebraic curves over finite fields and the Hasse–Weil bound. In 2018, Tu and Zeng [20] determined all with and gave sufficient conditions for over , the necessity of which was later proved by Hou [21], using a similar method described in [19]. Recently, Zheng et al. [22] claimed sufficient conditions for both and over . Furthermore, they conjectured the necessity of the former based on numerical experiments. Very recently, by making use of the similar approach in [18], Bartoli and Timpanella [23] provided necessary conditions for over finite fields with characteristic 2.
To our knowledge, only above seven different pairs had been characterized completely. However, the exponents of in (1) are also Niho exponents. This motivates us to explore new permutation polynomials with general coefficients from non-Niho exponents with even characteristics. By transforming the problem into studying some normalized permutation polynomials with degree 5 over even characteristics, we determine the coefficients for being a permutation over .
The rest of this paper is arranged as follows. Some notations and useful results are presented in Section 2. In Section 3, the sufficient and necessary conditions are shown to determine the coefficients of a class of permutation polynomials over . Finally, Section 4 provides some concluding remarks.
2. Preliminaries
Let and be two positive integers with and be a finite field with elements; the trace function from to is denoted by , where
If the usual complex conjugation of any is defined as , the following relations hold.(i)For all , .(ii)For all , .
denotes the unit circle of as
Lemma 1 (see [24]). with is a permutation polynomial over iff from into itself is a bijection.
Hereinafter, we claim that is a permutation polynomial over upon the bijectivity of from to , which is the only case we consider.
Lemma 2 (see [1]). The irreducibility of with degree remains over iff .
Definition 1 (see [25]). A permutation polynomial of degree is said to be normalized if the following properties hold:(i) is monic and the value of at 0 is equal to 0.(ii)The coefficient of equals 0 if , where is the characteristic of .
Remark 1. Let and ; for any permutation polynomial , there exists a unique normalized form provided by .
Theorem 1 (Hermite’s criterion) (see [1]). Let be the characteristic of . Thus, is a permutation polynomial over iff(i) has exactly one solution over .(ii), where , , the degree of the reduction of is no greater than .
From Hermite’s criterion, the characterization of all normalized permutation polynomials with degree no greater than 5 in is due to ([1], Section 7.2), and they are listed in Table 1.
3. A Class of Permutation Polynomials over
In this section, we consider the coefficients such that the polynomial permutes over .
The main results in this paper are given in the following theorems.
Theorem 2. For two positive integers and with , let . Define . Then, the polynomialpermutes iff one of the following two cases holds:(1)If , permutes .(2)If is odd, permutes .
Proof. Based on Lemma 1, in order to prove that is a permutation polynomial in , we only consider that for any , the equationhas exactly one root in .
Let , where and . Then, equation (5) becomesRaising both sides of equation (6) to the power leads toSince , we can let , where and . Then, equation (7) can be rewritten asRaising both sides of equation (8) to the power yieldsMultiplying equations (8) and (9), we can obtainThe substitution of with in equation (10) leads toLet , and we can getwhere and .
From the above discussion, we can deduce permutes iff permutes .
Based on Table 1, we know that if is a normalized permutation polynomial of degree 5 over , then it must have the following forms:(1), .(2) ( arbitrary), .When , we have , which includes that and is odd. When is odd, that is, , we mainly study the permutation polynomial of ( arbitrary). When , we consider the permutation polynomial of . Hence, permutes iff one of the following two cases holds:(1)If , permutes .(2)If is odd, permutes .In conclusion, we deduce that the polynomialpermutes iff the following two cases are satisfied:(1)If , permutes .(2)If is odd, permutes .
Remark 2. When and are both 0, if , then , so we achieve that the monomial permutes . If are not both 0, then is binomial or trinomial in . We will investigate the permutation behavior of those polynomials in the sequel.
Theorem 3. For two positive integers and with and being even, let which are not both 0. Then, the polynomialpermutes iff , , and is a root of , where and is a primitive third root of unity.
Proof. Based on Theorem 2, we deduce that permutes iff permutes . When , we get that permutes iff permutes .
“” Comparing the coefficients of and , we haveFrom equation (16), we can directly obtain that , i.e., . Hence, there exists some element such that .
Plugging into equation (15), we haveBy substituting with in equation (17), we obtainCombining equations (18) and (19), we know that . This means that and , where is a primitive third root of unity.
Therefore, we conclude that and are exactly two roots of the equationHence, the above analysis shows that for , if permutes , then and is a root of , where and is a primitive third root of unity.
In what follows, we will proceed to prove the necessity.
“” For some , let , and then we know , and this is equivalent to , which implies that .
Suppose that and are two different roots of equation (20) in ; then, we have and . Recall that , and we havewhich implies thatwhere is a primitive third root of unity.
Therefore, equations (15)–(17) are satisfied. Furthermore, we know that if , and is a root of , where and is a primitive third root of unity, and thus is a permutation polynomial over .
To summarize, we conclude that the polynomialpermutes iff , , and is a root of , where and is a primitive third root of unity.
Theorem 4. For two positive integers and with and being odd, let which are not both 0. Then, the polynomialpermutes iff one of the following two cases holds:(1), or , where and is a primitive third root of unity.(2) or , or , where , , and is a primitive third root of unity.
Proof. Based on Theorem 2, if is odd, then permutes iff permutes , where .
Since , is a irreducible polynomial in , and we deduce that also remains irreducible over by Lemma 2. Let be one of the roots of in ; then, and is a primitive third root of unity. Furthermore, we can view 1 and as a basis of vector space upon .
“” Comparing the coefficients of and , we haveFrom equation (26), we know thatThen, we can discuss the solutions of as follows.
Case 1. . Then, equation (27) turns toSince 1 and are basis of vector space upon , for any element , we can set with .
By plugging into equation (28), we obtainSince is odd, we know that . Then, equation (29) can be written aswhich implies that or . Consequently, we conclude that or .
The above analysis indicates that if permutes and is odd, then , or , where and is a primitive third root of unity.
Case 2. . Thus, equation (25) becomesand this is equivalent toFrom equation (32), we can directly obtain that . Hence, there exists some element such that , i.e., .
Plugging into equation (27) givesComparing the coefficients on both sides of equation (33), we haveDiscussion similar to that in Case 1 shows that or . Meanwhile, we obtain or , which follows from .
Therefore, the above analysis demonstrates that if permutes and is odd, then or , or , where , , and is a primitive third root of unity.
All in all, when is odd, if the polynomial permutes , then one of the following two cases is satisfied:(1), or , where and is a primitive third root of unity.(2) or , or , where , , and is a primitive third root of unity.Next, we will continue to prove the necessity.
“” If , or , then , which means that equation (25) is satisfied. Since is a primitive third root of unity and , we can calculate that and , and thus the left-hand side of equation (27) turns to , which implies that equation (27) is satisfied. Therefore, when m is odd, we conclude that if , , or , ( and is a primitive third root of unity), the polynomial is a permutation polynomial over .
If , recall that or , or , and we havewhich implies thatwhere the third equation holds only if . Consequently, we conclude that equation (25) is satisfied.
Next we considerwhere the second equation holds only if . Hence, we can get that equation (27) is satisfied.
Therefore, the above analysis shows that for is odd, if or , or , where , , and is a primitive third root of unity, then the polynomial permutes .
All in all, for is odd, we know that permutes if one of the following two cases is satisfied:(1), or , where and is a primitive third root of unity.(2) or , or , where , , and is a primitive third root of unity.In conclusion, we deduce that the polynomialpermutes iff one of the following two cases is met:(1), or , where and is a primitive third root of unity.(2) or , or , where , , and is a primitive third root of unity.
Applying Theorem 4 to and , we have the following.
Corollary 1. For two positive integers and satisfying and is odd, the binomial is a complete permutation polynomial over .
Remark 3. Observe that a complete permutation binomial proposed in Corollary 1 is obtained by Zieve [26]. However, the approach we used to prove the permutation property is different from that in [26].
4. Concluding Remarks
In this paper, by transforming the problem into studying some normalized permutation polynomials of degree five with even characteristics, we investigate the coefficient pairs making to be a permutation polynomial over . The sufficient and necessary conditions are shown in Theorems 3 and 4. Furthermore, a class of complete permutation binomials with the form over is obtained.
Data Availability
No data were used to support the findings of this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported in part by the Educational Research Project of Young and Middle-Aged Teachers of Fujian Province under grant no. JAT200033, the Talent Fund Project of Fuzhou University under grant no. GXRC-20002, and the National Natural Science Foundation of China under grant nos. 61902073, 62072109, and U1804263.