Compositional Inverses of AGW-PPs

03/01/2022
by   Pingzhi Yuan, et al.
South China Normal University
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In this paper, we present two methods to obtain the compositional inverses of AGW-PPs. We improve some known results in this topic.

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1 Introduction

Let be a prime power, be the finite field of order , and be the ring of polynomials in a single indeterminate over . A polynomial is called a permutation polynomial (PP for short) of if it induces a one-to-one map from to itself.

Permutation polynomials over finite fields have been an interesting subject of study for many years, and have applications in coding theory [4], cryptography [11, 12], combinatorial design theory [3], and other areas of mathematics and engineering. Information about properties, constructions, and applications of permutation polynomials may be found in Lidl and Niederreiter [6, 7], and Mullen [8].

In 2011, Akbrary, Ghioca and Wang [1] proposed a powerful method called the AGW criterion for constructing PPs. A PP is called AGW-PP when a PP is constructed using the AGW criterion or it can be interpreted by the AGW criterion. AGW-PPs can be divided into three types: multiplicative type, additive type and hybrid type. It is difficult to obtain the explicit compositional inverse of a random PP, except for several well-known classes. Compositional inverses of different classes of PPs of special forms have been obtained in explicit or implicit forms; see [2, 5, 6, 9, 13, 14, 16, 17, 18, 19, 22, 23, 24] for more details. Recently, Niu, Li, Qu and Wang [9] obtained a general method to finding compositional inverses of AGW-PPs. They obtained the compositional inverses of all AGW-PPs of the type over , where . However there are many other classes of AGW-PPs whose compositional inverses are unknown, and it is not easy to follow the framework in [9] to obtain the compositional inverses of AGW-PPs for other types. The purpose of the present paper is to find other general methods to obtain the compositional inverses of AGW-PPs.

The rest of this paper is organized as follows. In Section 2, we prove some results related to the AGW criterion. In particular, we obtain a useful commutative diagram, which is essential for the proofs of our main theorems. In Section 3, we use the dual diagram obtained in Section 2 to finding the compositional inverses of AGW-PPs. We improve some results in [9]. In Section 4, we describe another method to compositional inverses of PPs by using the branch functions.

2 AGW criterion and the dual diagram

In this section, we present some results related to the AGW criterion, and we will give the dual diagram when the AGW criterion is applied to a bijective function .

The following lemma is taken from [1, Lemma 1.1], which is called AGW criterion now.

Lemma 2.1.

( 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:

(i) is a bijective (a permutation of ); and

(ii) is a bijective from to and if is injective on for each .

We also have other results for the bijection of a map. We have

Lemma 2.2.

Let be finite sets, a map and a surjective map. Then is a bijection if and only if

(i) is injective on each for all .

(ii) If , then .

Moreover (ii) is equivalent to for any distinct .

Proof.

If is a bijective, then (i) holds trivially. For any distinct , since , we have , and (ii) holds.

Conversely, suppose that (i) and (ii) hold, since is injective on each for all and for any distinct , we have

which implies that is a surjective, and thus is a bijection. ∎

Lemma 2.3.

Let be finite sets with , a map and a surjective map. Then is a bijection if and only if the following two conditions hold:

(i) is injective on each for all .

(ii) There exists a maps pair such that is a surjective, is a bijective and the following diagram commutes.

Proof.

(i) is obvious. If is a bijection, then, by the proof of Lemma 2.2 , is a disjointed union of , and is also a disjointed union of , i.e.

Now for any bijection , we define a map by for any . It is easy to check that is a surjective and the diagram in the theorem commutes. This proves that (ii) holds.

If (i) and (ii) holds, then by AGW criterion, is a bijection.

Remark: It is not difficult to see that there are precisely maps pairs such that (ii) holds.

We also have the following result by Lemma 2.3.

Corollary 2.1.

Let be finite sets with , a map, a surjective map and is a bijective. Then is a bijection if and only if the following two conditions hold:

(i) is injective on each for all .

(ii) There exists a unique determined surjective map such that the following diagram commutes.

Lemma 2.4.

Let be finite sets, a map and a surjective map. Suppose that there is a set and a map such that for any with . Then there exists a unique map such that the following diagram commutes.

Furthermore, is a bijection if the following two conditions hold

(i) is injective on each for all .

(ii) is an injection.

Proof.

Since is surjective, so it is easy to check that

satisfies for all , so the above diagram commutes.

If (i) and (ii) hold, then for any with , then , and so , which implies that . By Lemma 2.2, we conclude that is a bijection.

Lemma 2.5.

Let be finite sets, a map and a surjective map. Then is a bijection if and only if the following conditions hold

(i) is injective on each for all .

(ii) there exist a set , an injective and a map such that the following diagram commutes.

Proof.

Suppose that is a bijection, then we take

for any and . Since is a bijection, so is a well-defined surjective map. Moreover, if and .

The other direction follows from Lemma 2.2.∎

Lemma 2.6.

Let be finite sets, a map and a surjective map. Then is a bijection if and only if is a surjection.

Proof.

Obviously.∎

The following result is essential in this paper, which will be used in the sequel.

Theorem 2.1.

Let the notations be defined as in Lemma 2.1. If is a bijection, and are the compositional inverses of and , respectively, then the have , i.e. the following diagram commutes

Proof.

By assumption, we have , hence

which yields . This completes the proof.∎

We call the diagram in Theorem 2.1 the dual diagram of the AGW criterion.

3 Compositional inverses of AGW-PPs

In this section, we present one approach to finding the compositional inverses of AGW-PPs by using the dual diagram of the AGW criterion. Our first result is as follows.

Theorem 3.1.

Let be a prime power, and subsets of with . Let , , and be maps such that both and are surjective maps and .

Let be a PP and a AGW-PP over , and let and be the compositional inverses of and , respectively. Then we have

Proof.

By assumption and Theorem 2.1, we have the following commutative diagram

F_q^∗[d]_λ [r]^f & F_q^∗[d]_¯λ [r]^f^-1 & F_q^∗[d]^λ

S [r]^g & ¯S [r]^g^-1 & S

Hence . Since is the compositional inverse of , we have , that is

It follows that , which implies that

This completes the proof.∎

Remark: In Theorem 3.1, we use to avoid the case of . If we use , then we must consider the case of independently.

For AGW-PPs in the hybrid case, we have

Lemma 3.1.

([1, Theorem 6.3]) Let be any power of the prime number , let be any positive integer, and let be any subset of containing . Let be any polynomials such that and , and let be any polynomial satisfying

(1) ; and

(2) for all and all . Then the polynomial is a PP for if and only if induces a permutation of .

We apply Theorem 3.1 to to obtain the following corollary, which improves Theorem 22 in [9].

Corollary 3.1.

Let the symbols be defined as in Lemma 3.1. Let permute and be the compositional inverse of over . Then the compositional inverse of is given by

The following result was discovered independently by several authors, and it can be proved by AGW criterion.

Lemma 3.2.

([10, Theorem 2.3] [15, Theorem 1] [25, Lemma 2.1])Let be a prime power and , where and is an integer. Then permutes if and only if

(1) and

(2) permutes .

Applying Theorem 3.1 to , we obtain the following result, which is a same as in [5, Theorem 2.3].

Corollary 3.2.

Let defined in Lemma 3.2 be a permutation over and be the compositional inverse of over . If and and are two positive integers satisfying . Then the compositional inverse of in is given by

Proof.

Since and and are two positive integers satisfying , the compositional inverse of is . Applying Theorem 3.1 to , we obtain the desired result. ∎

For the compositional inverses of AGW-PPs in the additive case, we have

Theorem 3.2.

Let be a prime power, and let be subsets with . Let , , and be maps such that both and are surjective maps and .

Let be a PP and a AGW-PP over , and let and be the compositional inverses of and , respectively. Then we have

Proof.

By assumption and Theorem 2.1, we have the following commutative diagram

F_q [d]_λ [r]^f & F_q [d]_¯λ [r]^f^-1 & F_q [d]^λ

S [r]^g & ¯S [r]^g^-1 & S

Hence . Since is the compositional inverse of , we have , that is

It follows that , which implies

This completes the proof.∎

Lemma 3.3.

([20, Theorem 6.1]): Assume that is a finite field and are finite subsets of with such that the maps and are surjective and is additive, i.e.,

Let , and be maps such that

and for every . Then the map permutes if and only if permutes .

We apply Theorem 3.2 to , we obtain Theorem 16 in [9].

Corollary 3.3.

([9, Theorem 16]): Let the symbols be defined as in Lemma 3.3. Let be a permutation over and be the compositional inverse of over . Then the compositional inverse of is given by

For , and a map , is called a -linear translator [1] of with respect to if for all and .

Lemma 3.4.

([1, Theorem 6.4]): Let and be a surjective map. Let be a -linear translator with respect to for the map . Then for any which maps into , we have that is a PP of if and only if permutes .

Applying Theorem 3.2 to , we get the following corollary, which improves Theorem 27 in [9].

Corollary 3.4.

Let defined as in Lemma 3.4 be a PP on and be the compositional inverse of . Then the compositional inverse of is given by

We end this section with another AGW-PP over finite fields, which is the corrected version of Theorem 3.13 in [21].

Lemma 3.5.

Let be a positive integer, a linearized polynomial such that , where is the associated polynomial of . Let be a solution of and is a polynomial with . Let be a linearized polynomial. Then for every , the polynomial

permutes if and only if permutes . Moreover, if permutes , then

Proof.

By assumption, we have since , hence . It follows that the following diagram commutes

where and . By Lemma 3.3, permutes if and only if permutes . Further, if permutes , by Theorem 3.2 and the following diagram

F_q^n [d]_L(x)+δ [r]^f & F_q^n [d]_L(x) [r]^f^-1 & F_q^n [d]^L(x)+δ

S [r]^L_1(x-δ) & ¯S [r]^L_1^-1(x)+δ & S

we get

4 Branch PPs and their compositional inverses

For a permutation polynomial by AGW criterion, we have

As is a surjective map, we have that is a disjointed union of , i.e.

Let ,

Then can be viewed as a branch function of . Let be the local inverse of , i.e. . For any non-empty subset of , let

be the characteristic function on

, i.e.

Then we have

Theorem 4.1.

Let be a non-empty subset of , and is a branch bijection defined as above, then we have

Proof.

For , we have , hence

which implies that

This completes the proof.∎

For any branch bijection, it is not easy to give the characteristic functions . However, it is easy for the branch functions when we use the cyclotomic cosets as branches.

Let be a fixed primitive element of , . The integer is called the index of . Let be the set of all non-zero -th powers, i.e. . is a subgroup of of index . The cosets of are the cyclotomic cosets

Let denote the set of -th roots of unity in , i.e.

We have

Lemma 4.1.

Let be a fixed primitive element of , and . Let , and

Let . Then we have

for .

Proof.

The result follows from the following well-known result that

Now we will the branch function to give two proofs of Theorem 11 in [9].

Proposition 4.1.

([9, Theorem 11]) Let defined in Lemma 3.2 be a permutation over and be the compositional inverse of over . Suppose and are two integers satisfying . Then the compositional inverse of in is given by

Proof.

The first proof: View as a branch function of . Since is a constant for any , it is easy to check that satisfy . Here . Since , we have

and

Hence , where . Since is a bijection, we have and , which implies that

Hence , and we are done.∎

Second proof: From the first proof we have , so it suffices to prove that . We use the following commutative diagram to prove this.

By the diagram, we have

(4.1)

Since , we obtain

i.e. , hence

By (4.1), we have

It follows that , and we are done.

We end the paper with two results on branch functions.

Lemma 4.2.

Let

be an odd prime power,

a fixed primitive element of , . Let and be two positive integers and let

be defined as a branch function, i.e.

Then is a PP over if and only if and .

Proof.

Since

thus we have the following commutative diagram

where . Observe that permutes if and only if , and permutes if and only if . Therefore by AGW criterion is a PP over if and only if and . This completes the proof. ∎

Corollary 4.1.

Let be a AGW-PP defined as in Lemma 4.2. We have

(i) If is odd, then is a PP if and only if and is a square. (ii) If is even, then is a PP if and only if and is not a square. Moreover, the number of such PPs is , where is the Euler function.

Proof.

By Lemma 4.2, the proofs of (i) and (ii) are obvious. For the number of such PPs, it is easy to see that are distinct bijections on if and only if