On two-to-one mappings over finite fields

06/27/2019 ∙ by Sihem Mesnager, et al. ∙ 0

Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of x^rh(x^(q-1)/d), those from permutation polynomials, from linear translators and from APN functions. Then we present 2-to-1 polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of 2-to-1 mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.

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1 Introduction

Permutation mappings (or -to- mappings) over finite fields have been extensively studied for their applications in cryptography, coding theory, combinatorial design, etc. For recent advances on permutation polynomials over finite fields, we refer to the excellent survey [17] and the references therein. For a detailed study of involutions over finite fields (in characteristic 2), we send the reader to [9]. Two-to-one (-to-) mappings are involved in several criteria in particular to design special important primitives in symmetric cryptography such as APN functions, bent functions and more general plateaued functions. Despite their importance, they have never been studied in the literature. The objective of this paper is to provide a systematic study of two-to-one mappings over finite fields including characterizations, criteria and methods for handling and designing such functions as well as effective constructions.

The paper is organized as follows. Section 2 gives preliminaries and fixes the notation. In Section 3, we first present the definitions of -to- mappings over finite fields as well as basic properties, and then provide a characterization of -to- mappings by means of the Walsh transforms. Section 4 is devoted to the constructions of -to- mappings. We shall present several constructions. First, an AGW-like criterion for -to- mappings is given. Next, constructions of -to- polynomial mappings with the form of are provided. Furthermore, constructions of -to- mappings from permutation polynomials, from linear translators and from APN functions respectively are given. In Section 5 we present -to- polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. In Section 6, we are interested in applications of -to- mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. It should be noted that this section is not only an application of the obtained results, but also a motivation to study 2-to-1 mappings. In all those sections, we shall review what is known and provide several new results.

2 Notation and Preliminaries

For a set , will denote the cardinality of . For any field , . Let , and be respectively the set of all natural, real and complex numbers. Let be a prime number and be a positive integer. The finite field with elements is denoted by or , which can be viewed as an

-dimensional vector space over

, and it is denoted by . Denote by the algebraic closure of . The trace function is defined as

which is called the absolute trace of . More general, the trace function is defined as

A linearized polynomial (or additive polynomial), is a polynomial of the shape . A polynomial is called an affine polynomial if it equals to the summation of a linearized polynomial and a constant term.

Let be a function from to . We can give a corresponding complex-valued function from to defined as for all where is a complex primitive -th root of unity. The Walsh transform of

is the Fourier transform

from to of defined as for all , where denotes an inner product (for instance, the usual inner product) in . We can take if is identified with . Note that if then and a function from to is said to be a Boolean function.

3 Definitions and a characterization of -to- mappings over finite fields

3.1 Definitions of -to- mappings

Firstly, we give the definition of -to- mappings over any finite set.

Definition 1.

Let and be two finite sets, and let be a mapping from to . Then is called a -to- mapping if one of the following two cases hold:

  1. is even, and for any , it has either or preimages of ;

  2. is odd, and for all but one

    , it has either or preimages of , and the exception element has exactly one preimage.

Throughout this paper, we mainly focus on the mappings over finite fields. Let and be two finite fields of order and , respectively. Let be a mapping from to . Then according to the above definition, if , then is a -to- mapping if and only if the equation has either zero or two solutions in for any , or equivalently, for all . While for an odd prime , a mapping is -to- if and only if all but one elements in the image set of have two preimages and the exceptional element has one preimage, or equivalently, there exists a unique such that and , for all . Without loss of generality, we can assume that the exceptional element of is . Moreover, if its unique preimage is also the zero element, then we have the following remark.

Remark 2.

Let with if and only if , where is odd. Then is a -to- mapping if and only if, has either zero or two solutions in for any .

In the end of this subsection, we calculate the number of all -to- mappings over . It seems to be a huge number.

Proposition 3.

Denote by the number of all -to- mappings . Then

Proof.

Let be a -to- mapping over . Then the size of its image set is . For the first element of the image set, its preimage have choices, while for the second element, it has choices, so on and so forth, the last element has choices. Hence we have

Then the result follows from the well-known String formula. ∎

It is well known that the number of all mappings (resp. bijective mappings) from to itself is (resp. ) . Denote the latter number by . Then we have

We list the ratio of these two numbers for in the following table. The values are rounded to three significant figures.

It seems from the above table that the number of all -to- mappings over is much greater than that of all bijective mappings over .

3.2 A characterization of -to- mappings over by means of the Walsh transforms

In this subsection we present a characterization of -to- mappings over by means of the Walsh transforms. The main idea goes back to Carlet [4] who has characterized the differential uniformity of vectorial functions by the Walsh transform. Let be a polynomial over . Recall that is -to- if and only if, for every in , the equation has 0 or 2 solutions. Let be a vectorial Boolean function. The Walsh transform of at equals by definition the Walsh transform of the so-called component function at , that is:

Let be any polynomial over such that for and for every . Hence for any and , we have

and is a two-to-one function if and only if this inequality is an equality for any . Furthermore, for any , we have

and is -to- if and only if this inequality is an equality.

We shall now characterize this condition by means of the Walsh transform. We have:

and therefore, for :

Hence we have the following characterization of -to- mappings over by the Walsh transform.

Theorem 4.

Let be a vectorial Boolean function. Then

and is -to- if and only if this inequality is an equality.

Now, let us consider the polynomial over equal to . It takes value 0 when equals 0 or 2 and takes strictly positive value when is in . We have then the following corollary.

Corollary 5.

Let be a vectorial Boolean function. Then

and this inequality is an equality if and only if is -to-.

4 Constructions of -to- mappings

In this section, we present different methods to construct -to- mappings over finite fields.

4.1 AGW-like criterion for -to- mappings

The criterion, discovered by Akbary, Ghioca and Wang [1], is a simple and effective method that establishes the permutation property of a mapping through a commutative diagram. The significance of the AGW criterion resides in the fact that it not only provides a unified interpretation for many previous constructions of permutations polynomials but also facilitates numerous new discoveries. In this subsection we will generalize AGW criterion to construct -to- mappings over finite fields.

We give a brief description of this subsection for the readers’ convenience. First, the AGW criterion is generalized to construct -to- mappings over finite sets (Proposition 6). Second, three general constructions (Theorem 8, Theorem 9, and Proposition 10) are given by applying this generalized AGW criterion. Then several explicit -to- polynomials over finite fields are constructed from Proposition 10, and most of the constructions are divided into two cases.

Proposition 6.

Let be a finite set, be two finite sets such that . Let be four mappings defined as the following diagram such that . If is bijective from to , is -to- for any , and there is at most one such that is odd, then is a -to- mapping over .

Proof.

Let . Assume that there exists an element in such that . Let . Then

Since is bijective from to , there exists a unique element such that . Hence . If is even, then has exactly two solutions in (one is ) since is -to- for any . If is odd, then with one exception, has exactly two preimages of in . Further, since at most one of is odd for all , we know that is a -to- mapping over . ∎

Remark 7.

If is -to- from to , and is injective for any , then one can only deduce that for any , it has at most two preimages. Similarly, let and assume that there exists an element in such that . Let . Then

Since is -to-, there exist exactly two elements in such that with at most one exception. Hence or . Then it follows from the assumptions that is -to- for any that there exist at most two elements in such that . It seems not easy to add a condition such that is a -to- mapping over . We leave this problem to interested readers.

By applying Proposition 6, we can give the following two general constructions.

Theorem 8.

Consider any polynomial , any additive polynomials , any -linear polynomial satisfying , and any polynomial such that . Let

and

If is bijective from to , is -to- for any , and there is at most one such that is odd, then is a -to- mapping over .

Proof.

We have

the second equality holds since , is -linear and . Hence we get the following commutative diagram:

Then the result follows directly from Proposition 6. ∎

Theorem 9.

Let be an even prime power, let be a positive integer, and let be -linear polynomials over seen as endomorphisms of the -module . Let be such that . Assume

and

For any , let . If , for any , where is a nonzero element of , and is a permutation over , then is -to- over .

Proof.

We apply Proposition 6 with , , , and . Since , and and are -linear polynomials over , one can easily verified that . For any , is linearized. It is -to- over if and only if . Hence the result follows from Proposition 6. ∎

The above two constructions are quite general and can be used to construct more explicit -to- polynomials. Due to the space limit, we will only take the first one as an example and give several explicit constructions. The interested readers are cordially invited to apply the second one to construct more -to- polynomials.

The following proposition follows from Theorem 8, and is the foundation of later constructions in this subsection.

Proposition 10.

Let , and be two -linear polynomials over seen as endomorphisms of the -module , and let such that If for some , and permutes , then

is -to- over .

Proof.

In Theorem 8, let , then since both and are -linear polynomials over . Further, is -to- for any since . The result then follows from Theorem 8. ∎

By applying Proposition 10, we have the following theorem.

Theorem 11.

Let , , and let . Let and be -linear polynomials over . Let be such that . Let

and

If for some , and permutes , then is -to- over .

Proof. In Proposition 10, we let , , and . For any , since and is a -linear polynomial, we obtain

and thus

as in Proposition 10.

Next we study in detail some of the consequences of Proposition 10 (or alternatively of Theorem 8 when ) for two specific choices of -linear polynomials. First we consider the case and next we study the case .

Case 1.

The first result in this case follows directly from Proposition 10.

Proposition 12.

Let , be a -linear polynomial over seen as an endomorphism of the -module and be the trace function from to . Let be such that Assume

and

If for some , and permutes , then is -to- over .

By applying Proposition 12, we get the following construction.

Theorem 13.

Let , be a -linear polynomial over , let , and let such that . Assume . If for some , and permutes , then is -to- over .

Case 2.

Similarly, we have the following two results.

Proposition 14.

Let , be a -linear polynomials over seen as an endomorphism of the -module . Let be such that for all . Assume

and

If is -to- over and permutes over , then is -to- over .

Proof.

In Theorem 8, let . For any , is -to- if and only if is -to- over since . Hence the result follows. ∎

Theorem 15.

Let , be a -linear polynomials over seen as an endomorphism of the -module . Let be such that for all . Assume

and

If is -to- over and permutes over , then both and are -to- over .

Proof.

We only prove for as the other case can be proved similarly. In Proposition 14, let . Then

since . Hence the result follows. ∎

4.2 -to- polynomial mappings with the form of

In this subsection, we construct two-to-one polynomial mappings with the form of . We need to be even. Hence it is assumed that is odd throughout this subsection.

Proposition 16.

Let be an odd prime power, be positive integers such that . Let , where such that if , and let and . Let . If is -to- from to and , then is a -to- mapping over .

Proof.

Since , we know that is -to- for any . Then the result follows directly from the fact that and Proposition 6. ∎

Then we have the following result.

Corollary 17.

Suppose that there exists such that for all . If and , then is -to- over .

Theorem 18.

Suppose that , where and , has no roots in , and . Then is -to- over .

Proof.

For any , we have