Let be the finite field of order and be a mapping from to itself. is said to be a -to- mapping if for any . -to- mappings over finite fields in even characteristic have wide applications in symmetric cryptography, in particular in the construction of APN functions, bent functions, semi-bent functions and so on. For example, -to- mappings over finite fields with characteristic allow to construct bent Boolean functions in bivariate representation from the so-called class introduced by Carlet and Mesnager . In addition, -to- mappings over can also determine semi-bent Boolean functions in bivariate representation from the Maiorana-MaFarland class . For more applications of -to- mappings over finite fields, we refer to [13, Section 6].
Very recently, Mesnager and Qu  provided a systematic study of -to- mappings over finite fields with arbitrary characteristic including characterizations by the Walsh transform, several constructions (an AGW-like criterion, those from permutation polynomials or linear translators), some classical classes of polynomials (linearized polynomials, monomials, low degree polynomials, etc) and many explicit applications of -to- mappings. In this paper111 Some parts of the present paper have been accepted in the proceedings of the conference IWSDA 2019, mainly the determination of -to- mappings with degree and two classes of -to- trinomials and four classes of -to- quadrinomials. However, due to the limit of length, only sketch of the proofs have been included in the proceedings paper. we push further the study initiated in  and we focus on even characteristic (binary case being the most interesting for applications). More specifically, in , the authors determined completely the -to- mappings of degree at most 4 over finite fields. In this paper, we firstly consider -to- polynomials of degree over . Our method is based on the Hasse-Weil bound (see e.g. [6, 14]), which has been used recently in the study of permutation polynomials (see e.g. [1, 5]). Next, we focus on -to- mappings with few terms. The characterization of -to- monomials is trivial. The authors  presented four classes of -to- binomials of the form thanks to the known results on hyperoval sets (see Proposition 4.2). In the present paper, we investigate by MAGMA all -to- binomials of the form on , where , and . It turns out that for odd, up to the o-equivalence (see Definition 2.8), these experiment examples can be explained by the four classes of -to- binomials given in Proposition 4.2. For the case when is even, there is only one class of -to- binomials that is over . For -to- trinomials and quadrinomials, we derive, up to the QM-equivalence (see Definition 2.9), two classes of -to- trinomials of the form , which explain all the experiment examples of -to- trinomials over with . We also present twelve classes of -to- quadrinomials of the form over . Our method in this part uses the multivariate method introduced by Dobbertin  and the key point is to determine the number of solutions of equations with a high degree. In addition, we shall use an important tool (which is the resultant of two polynomials) to treat the case of quadrinomials.
The remainder of this paper is organized as follows. Section 2 introduces some basic notions which will be used in the manuscript. Based on the Hasse-Weil bound, Section 3 determines completely the -to- mappings with degree over . In Section 4, we consider -to- polynomials with few terms, including mainly experiment examples obtained by MAGMA and two classes of -to- trinomials over . In Section 5, we present twelve classes of -to- quadrinomials over . Finally, Section 6 is a conclusion. Throughout this paper, for any , we assume denotes the trace function from to , i.e., for any . Particularly, when , we use to denote the absolute trace function over , i.e., for any . The algebraic closure of is denoted by . For any sets , denotes the cardinality of .
In this section, we introduce some basic notions on the Hasse-Weil bound as well as some known results concerning the solutions of equations with low degree (quadratic, cubic and quartic). We also recall the resultant of two polynomials which will be useful in our subsequent proofs. Finally, we introduce and recall two equivalences to study -to- polynomials.
2-a The Hasse-Weil bound
In this subsection, we recall some well known results on algebraic curves and algebraic function fields, mainly the Hasse-Weil bound. These classical results can be found in most of the textbooks on algebraic curves and algebraic function fields.
Let be a function field and be perfect. Let denote the genus of . Then we have the following upper bound on the genus.
Given two plane curves and and a point on the plane, the intersection number of and at the point is defined by seven axioms. We do not include its precise and long definitions here. For more details, we refer to .
[7, Bézout’s Theorem] Let and be two projective plane curves over an algebraically closed field , having no component in common. Let and be the polynomials associated with and respectively. Then
where the sum runs over all points in the projective plane .
2-B Solutions of equations with low degree
In this subsection, we introduce some known lemmas about the solutions of some equations with low degree (quadratic, cubic, quartic), which will be used the proofs of our results.
 Let and . Then the quadratic equation has solutions in if and only if .
 Let , where . Then the cubic equation has a unique solution in if and only if .
If is a quartic polynomial over which factors as a product of two linear factors times an irreducible quadratic, we write ; if is a cubic irreducible polynomial over , we write . In , P. A. Leonard and K. S. Williams characterized the factorization of a quartic polynomial over as follows.
 Let with and . Let and denote roots of when they exist in . Set . Then the factorization of over is characterized as follows:
if and only if and ;
if and only if and , ;
if and only if ;
if and only if and ;
if and only if and .
2-C Resultant of polynomials
In this subsection, we recall some basic facts about the resultant of two polynomials. Given two non-zero polynomials of degrees and respectively
with and coefficients in a field or in an integral domain , their resultant is the determinant of the following matrix:
For a field and two polynomials , we use to denote the resultant of and with respect to . It is the resultant of and when considered as polynomials in the single variable . In this case, belongs in the ideal generated by and , and thus any satisfying and is such that (see ).
2-D O-equivalence and QM-equivalence between two 2-to-1 mappings
A permutation polynomial over is called an oval-polynomial (for short o-polynomial) if , and for each ,
is a permutation polynomial. It is well known that there is a close relation between o-polynomials and -to- mappings as follows.
A polynomial from to itself with is an o-polynomial if and only if is -to- for every .
Each o-polynomial defines an (hyper-)oval. And each hyperoval defines o-polynomials. Two o-polynomials are called (projectively) o-equivalent222Note that for a special type of bent functions, so-called Niho bent functions there is a general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials., if they define equivalent hyperovals. Hyperovals being called equivalent if they are mapped to each other by collineations (i.e. permutations mapping lines to lines). For example, for the following o-monomials
are o-equivalent to each other. We can naturally define an equivalent relation between -to- (polynomial) mappings as follows.
Let and be -to- (polynomial) mappings where . Then and are said to be o-equivalent if the corresponding o-polynomials and are equivalent.
The o-equivalence between two 2-to-1 mappings will play an important role in our classification of -to- binomials. Namely, the o-equivalence plays a major role in explaining the experiment results of -to- binomials. In particular, under the o-equivalence, Proposition 4.2 can explain all experiment results on -to- binomials over with and odd.
Next, we recall another equivalence between polynomials. Let with . As we all know, if permutes , then is a -to- polynomial over if and only if is -to- polynomial over .
It is trivial that a monomial is a permutation polynomial over if and only if Let and be two polynomials in satisfying that , where and is an integer such that . Then is -to- if and only if so is . Consequently, we recall the following notion of QM-equivalence [15, 10].
 Two polynomials and in are said to be quasi-multiplicative (QM) equivalence if there exists an integer with and for some nonzero elements in .
Using the QM-equivalence, we can simplify the experiment results on -to- trinomials (resp. quadrinomials). As indicated in Table II, there are for example only two -to- trinomials of the form over up to the QM-equivalence. In addition, we can avoid getting equivalent -to- polynomials having the same terms.
3 -to- mappings with degree over
In this section, we completely determine the -to- mappings with degree over . Clearly, for any polynomials with degree , is -to- over if and only if so is , where and . Hence, it is suffisant to consider with normalized form, i.e., is monic, , and when , the coefficient of is . That is to say, in this part, we suffice to consider where since .
Let and where . Then is not -to- over .
We assume that is -to- over . According to the definition, has exactly two solutions and for any . In addition, for any , we have
Thus for any ,
has exactly one solution in .
In the following, we assume that and . Let
where and . We also assume that . Thus Eq. (1) has exactly one solution in if and only if
for any satisfying and . Indeed,
where and . It should be noted that is equivalent to .
Assume that and . Let
Then . Together with Eq. (3) and that there exist at most elements such that , as well as that at most elements such that , we have
If is irreducible over , let be transcendentals over with . Then by Lemma 2.2, the functional fields has genus
Then by the Hasse-Weil bound, i.e., Lemma 2.1, we have
when , which is contradictory with (4).
Therefore, is not irreducible over and we assume that where , are irreducible and If , choose such that . Then . Assume that . Then or , say Then . Hence and we have
Thanks to Bézout’s Theorem, i.e., Lemma 2.3,
which is also contradictory with (4). Thus . Namely, there exists some such that
Obviously, the degree of is . Assume that After comparing the coefficients of Eq. (5), we have
When , and Eq. (2) becomes
where . Let Then if is -to-, must be irreducible from Lemma 2.6. However, it is clear that there exist some such that has or solutions in , which means can not be irreducible. Thus is not -to- in the case.
When , Eq. (1) becomes
Therefore when , is not -to- over , which completes the proof. ∎
As for the case , there exist some -to- mappings with the form of where . We obtain them by MAGMA and list them in Table I, where is a primitive element in .