Let be the finite field with elements, where is a prime power. When the map is a bijection on , is called a permutation polynomial (PP) and denotes the compositional inverse of . If there exists an positive integer such that is the identity map, we call an n-cycle permutation, where the -th functional power of is defined inductively by and with our notation. For a small , -cycle permutations are called low-cycle permutations in this paper. When or , is called an involution, a triple-cycle permutation or a quadruple-cycle permutation respectively, and they are low-cycle permutations.
The applications of permutation polynomials in cryptography, coding theory, and combinational designs have been extensively studied, readers are refereed to [25, 15, 22, 33] and the references therein. It is a challenging task to find new classes of permutation polynomials. However in 2011, Akbary et al.  provided a powerful method for constructing PPs over finite fields, which was called the AGW Criterion. Its significance lies in both providing a unified explanation of earlier constructions of PPs and serving a method to construct many new classes of PPs. After then, several authors, such as [12, 16, 14, 21, 39, 8, 17, 36, 38, 30, 20], constructed permutation polynomials of the form over . In many situations, both the permutation polynomial and its compositional inverse are necessary. For instance, in block ciphers, a permutation is used as an S-box to build the confusion layer during the encryption process. The compositional inverse of the S-box is required while decrypting the cipher. Recently, the compositional inverse was also applied to the study of constructing permutations with boomerang uniformity 4, please see. Due to the importance of the compositional inverse, it attracts a lot of attentions and there are also many researches on it, such as [28, 34, 35, 31, 19, 27, 41]. Therefore, if both the permutation and its compositional inverse are efficient in terms of implementation, it is advantageous to the designer.
This motivates the use of low-cycle permutations in the S-box of block ciphers. One immediate practical advantage of a low-cycle permutation is that the implementation of the inverse does not require much resources, which is particularly useful in devices with limited resources as a part of a block cipher. For instance, involutions have been used frequently in block cipher designs, in AES , Khazad , Anubis  and PRINCE . Furthermore, low-cycle permutations (such as involutions) have been also used to construct Bent functions over finite fields [24, 10, 13] and to design codes. Recently, in , behaviours of permutations of an affine equivalent class have been analyzed with respect to some cryptanalytic attacks, and it is shown that low-cycle permutations (such as involutions) are the best candidates against these attacks. In addition, the study of
-cycle permutations will be very helpful in classifying permutations in the view of cycle, since each permutation over finite sets must be an-cycle permutation. Due to the importance of -cycle permutations, there are systematic studies about them in recent years. The explicit study of involutions was started with the paper  for finite fields with even characteristic, where basic tools and constructions of involutions were given. Since then, lots of attentions had been drawn in this direction. Recently, Zheng et al.  gave a more concise criterion for involutory permutations of the form over , where . By using this criterion, they proposed a general method, from a cyclotomic perspective, to construct involutions of such form from given involutions over some subgroups of by solving congruent and linear equations over finite fields. Independently, Niu et al.  started from the AGW Criterion, and proposed an involutory version of the AGW Criterion. Then they demonstrated their results by constructing explicit involutions of the forms and . In 2019,  studied triple-cycle permutations over binary fields of the types of Monomial, Dickson polynomial and Linearized polynomial, respectively. Very recently, Wu et al.  generalized the work of  and obtained some characterizations of triple-cycle permutations of the form . However, there are no studies on general -cycle permutations in the literature as far as we know. This motivates us to propose unified results and to provide new explicit constructions.
The main purpose of this paper is to study the constructions of -cycle permutations of the form over finite sets. First, motivated by the AGW Criterion, we generalize the previously known results about involutions in , and obtain that -cycle permutations on a finite set can be constructed from -cycle permutations on its proper small set under suitable conditions. Next, we obtain a unified criterion for -cycle permutations of the form , a method for constructing -cycle permutations from ones over the subfield recursively, and a cyclotomic perspective to characterize the properties of -cycle permutations. Finally, we consider a piecewise method in the construction of -cycle permutations of cyclotomic form. We will explain our approaches by giving five classes of explicit constructions of PPs.
The rest of this paper is organized as follows. In Section 2, we introduce some basic knowledge about general -cycle permutations. Main tools for constructing -cycle permutations of the form are proposed in Section 3. Explicit -cycle permutations are respectively constructed from two different perspectives in Sections 4 and 5. A conclusion of this paper is given in Section 6.
Before we handling permutations of the form , we prepare and discuss general -cycle permutations in this section.
[9, Definition 4] Let be a permutation of a finite set . Let be a positive integer. A cycle of is a subset of pairwise distinct elements of such that for and . The cardinality of a cycle is called its length.
The result below is a generalization of [9, Proposition 6] for -cycle permutations.
Let be a permutation of a finite set . Then, is an -cycle permutation if and only if the length of each cycle of is no more than and .
Let be elements defined by , and define for any integer . Then since is an -cycle permutation. One can obtain . Thus, the length of each cycle of an -cycle permutation is no more than , and .
Conversely, if the length of each cycle of is no more than and , then for in each cycle, we have , i.e., is an -cycle permutation. ∎
Apparently if is an -cycle permutation, then is also an -cycle permutation, where is a positive integer. Although we focus on the -cycle permutation in this paper, we propose the following results for clarity.
For the least positive integer such that , we call a fundamentally n-cycle permutation.
The least common multiple of lengths of all cycles of is if and only if is a fundamentally -cycle permutation.
According to Proposition 2.2, is an -cycle permutation if and only if the length of each cycle of satisfying . Thus, the least common multiple of lengths of all cycles of is exactly the least positive integer such that . ∎
Assume is a prime, then is a fundamentally -cycle permutation if and only if is an -cycle permutation and .
Thus, for -cycle permutations (involutions) and -cycle permutations (triple-cycle permutations), they are respectively fundamentally -cycle permutations and fundamentally -cycle permutations clearly.
Now we recall a famous result named AGW Criterion, which was proposed by Akbary et al. for constructing PPs in . It will play an important role in our following results.
(, AGW Criterion) Let , and be finite sets with , and let , and be maps such that . If both and are surjective, then the following statements are equivalent:
is a bijection and
is a bijection from to and is injective on for each .
The AGW Criterion can be illustrated as follows:
It not only provided a unified explanation for a lot of earlier constructions, but also motivated more new findings of PPs. Inspired by the AGW Criterion, we obtain the following result for all -cycle permutations on a finite set, which is a generalization of [27, Proposition 2.2].
Let and be finite sets, and let , , be maps such that is surjective and . Assume that is an -cycle permutation on , then is an -cycle permutation on .
We have , where denotes composing itself for times. Since is surjective, we obtain that is an -cycle permutation on from . ∎
Once we obtained an -cycle permutation, it is natural to obtain more -cycle permutations by composing itself by the following lemma.
Assume that is an -cycle permutation on , then is also an -cycle permutation, where .
Furthermore, for different mappings with -cycle permutations, we have the following results.
Assume that are -cycle permutations on , then is also an -cycle permutation if .
It is easy to verify by the definition of -cycle permutation. ∎
Assume that permute . Furthermore, is an -cycle permutation. Then is also an -cycle permutation.
Since , is also an -cycle permutation. ∎
The results above can be used to quickly generate large general -cycle permutations from known ones over the same finite sets. However, in this paper we do not give examples for them, since they are simple to implement.
Now we introduce PPs of the form . Throughout the rest of paper, let be a power of a prime, and be divisors of such that All nonzero elements of a finite set is denoted by . In this paper, we always assume that for any . Otherwise, it is easy to obtain that can not permute . It is well-known that every polynomial over such that has the form for some positive integers in the case .
Based on , the concept of the index of a polynomial was first introduced in . Any nonconstant polynomial of degree can be written uniquely as such that the degree of is less than the index which is defined in  in the following. Namely, write
where . The case is trivial and we have . Thus we shall assume that . Write the vanishing order of at 0 (i.e., the lowest degree of in is ). Then where and . Hence in this case . The integer is called the index of . One can see that the greatest common divisor condition in the definition of makes the index minimal among those possible choices. Note that the index of a polynomial is closely related to the concept of the least index of a cyclotomic mapping polynomial . Let denote the set of -th roots of unity in i.e.
which is also the unique cyclic subgroup of of order . This subgroup can also be represented as . It is easy to verify that
where . For more information about the index approach and piecewise method, readers are refereed to [33, 7] et al. The following well-known lemma, the multiplicative form of the AGW Criterion, gives a necessary and sufficient condition for being a permutation over .
In the rest of the paper, we focus on the constructions of -cycle permutations of the form , which will be helpful for considering general -cycle permutations of other forms.
3 General Results for -Cycle Permutations over Finite Fields
In this section, we give a general approach to -cycle permutations of the form .
Let be a prime power and , where . Assume that is a polynomial on , where . Then, is an -cycle permutation over if and only if
for all .
First of all, . Then we only need to consider the non-zero case below.
Assume that and for all . For any , one can find a such that . Then we have
which indicates that is an -cycle permutation over .
Assume that is an -cycle permutation over . For each and any such that , we have
Suppose that there is an such that , then we have
Hence, holds for any , which implies that . Therefore,
The last equation holds since is an -cycle permutation over . ∎
The following result is a direct consequence of the above for , which was given in  very recently.
[37, Theorem 1] Let be a prime power and , where . Assume that is a polynomial on , where . Then, is a triple-cycle permutation over if and only if
for all .
Theorem 3.1 provides a useful tool for both determining and constructing -cycle permutations of the form . In order to construct -cycle permutations more efficiently, we further look for more tools in different forms for -cycle permutations. A consequence of Theorem 2.8 is the following. It is a necessary condition for triple-cycle permutations.
[37, Corollary 1] Assume that is a triple-cycle permutation over . Then is a triple-cycle permutation on .
Thus, to obtain -cycle permutations of the form over , it is crucial to find a suitable . According to Theorem 2.8, is necessary to be an -cycle permutation on the subgroup of for being an -cycle permutation on
Below we represent approaches to obtain many new -cycle permutations reductively over finite fields from the known ones on their subfields. They are obtained by assuming that is a subfield of .
Let be a prime power. Assume that such that holds for any , where . Then is an -cycle permutation over if and only if .
Let be a prime power and , where , , and . Let satisfy . Then is an -cycle permutation on if and only if is an -cycle permutation on .
If we choose a special in Theorem 3.5 such that is exactly a multiplicative group of a finite field, then the following corollary is obtained.
Let be a prime power and be positive integers with and . Let Then is an -cycle permutation on if and only if is an -cycle permutation on .
A consequence of Corollary 3.6 for is the following.
[37, Corollary 4] Let be a prime power and be positive integers with and . Let Then is a triple-cycle permutation on if and only if is a triple-cycle permutation on .
Corollary 3.6 allows us to construct new explicit -cycle permutations over finite fields from the known ones on their subfields. Examples will be given in the next section (see Corollarys 4.5 and 4.8 ).
In the perspective of cyclotomic, we propose a general method to determine the coefficients of from a given -cycle permutation over , which can be seen as a generalization of [40, Theorem 6] and [37, Theorem 3].
Let be a primitive element of and be a generator of the subgroup . For any , , and , let be rearrangements of , satisfying and . Assume that are integers for , and is a reduced polynomial modulo , such that
Then is an -cycle permutation over if and only if
According to and for , . One can obtain that
Thus is an -cycle permutation on , i.e., for , we have
Together with Theorem 3.1, is an -cycle permutation over if and only if
which is equivalent to
After simplifying Eq. (5), one can obtain that
by applying . Thus, is an -cycle permutation if and only if Eq. (4) holds. ∎
In particular, we have the following consequence if is the identity map.