1. Introduction
Let be a power of a prime, , be the finite field with elements and . A function is called planar if for each , the difference function
is a permutation on . Functions satisfying the previous condition exist only for odd, since for even holds. In the even characteristic case, functions such that is a to map are called almost perfect nonlinear (APN) and they are connected with the construction of Sboxes in block ciphers [21, 22]. In the odd characteristic case, planar functions, also called perfect nonlinear functions (or 1differentiable), are never bijections, since the corresponding derivative is bijective and there exists a unique such that . Planar functions in odd characteristic achieve the best possible differential properties, which makes them useful in the construction of cryptographic protocols, in particular for the design of Sboxes in block chypers; see [21, 22]. Such functions have close connections with projective planes and have been investigated since the seminal paper [10] where the authors showed that, considering two groups and of order , every planar function from into gives rise to an affine plane of order . Finally, these functions are strictly related to the construction of relative difference set [14], optimal constantcomposition codes [12], secret sharing schemes arising from certain linear codes [6], signal sets with good correlation properties [11], finite semifields [7]. More recently, Zhou [26] defined a natural analogue of planar functions for even characteristic: If is even, a function is planar if, for each nonzero , the function is a permutation on . As shown by Zhou [26] and Schmidt and Zhou [25], such planar functions have similar properties and applications as their odd characteristic counterparts.
In the past years, many papers have been devoted to existence and nonexistence results for planar mappings, using a variety of methods; see [8, 15, 17, 18, 9, 4, 23].
In this paper we investigate the planarity of , where is a linearized polynomial. In this direction, the following results provides a necessary condition.
Proposition 1.1.
[18, Proposition 1] Let be linear mappings. If the mapping is planar then necessarily the mappings and are bijective on .
Polynomial of the type belong to the family of so called DembowskiOstrom polynomials, that is polynomials of the type
(1) 
see [10]. It has been conjectured for many years that all planar polynomials belong to such family. Counterexamples have been provided in [8], where the authors prove that the monomial is planar over if and only if . In the monomial case, the DembowskiOstrom conjecture remains still open for .
There are different possibile definitions of equivalence between planar functions. Two functions and from to itself are called extended affine equivalent (EAequivalent) if , where the mappings are affine, and , are permutations of . The following theorem lists the currently known families of EAinequivalent planar functions.
Theorem 1.2.
Apart from those listed in Theorem 1.2, a number of other constructions are known in the literature. The following theorems colletect some of them.
Theorem 1.3.
In this paper we investigate the planarity of on , a particular example of DembowskiOstrom polynomials. We make use of the study of curves associated with polynomials . The main result of this paper is the following.
Theorem.
Let be a power of an odd prime and consider the function of the type
(2) 
where . Then, is planar if and only if one of the following condition holds

and ;

, .

, .
2. Construction of planar polynomials of type
In this section we provide the complete classification of planar polynomial of the type , where .
Note that is planar if and only if for each the linearized polynomial
is a permutation polynomial. This happens if and only if the following matrix
(3) 
has rank for each choice of . In the following we consider a polynomial
Let be the homogenization of . Then it is readily seen that
(5) 
Our main approach involves specific algebraic curves of degree three, whose reducibility is investigated in the following propositions.
Proposition 2.1.
Let be a curve defined by , , with
If has a line of the form as a component, then one of the following holds

or , or

and , or

, and , or

, or

and .
Proof.
First we deal with the case . If such a line is a component, by direct computations either or . On the other hand, it easily seen that if or then is a factor of .
By direct computations, if and only if
We distinguish two cases.

.

If then or . In this last case, either (already done) or (and then ) and .

If then or . In this last case otherwise .


.

If then or .

If then (already done), or , or or . Note that implies , with . Since would imply (already done) then and .

∎
Proposition 2.2.
Suppose that , , contains a line of the form as a component with . Then one of the following cases occurs.

or

, and

and

, and
where .

, , , and
where .
Proof.
Proposition 2.1 applies here. The first three cases are just easy computations.
Now suppose that and . Then , where and vanishes.
If , , and then , where . It is easily seen that vanishes. ∎
Proposition 2.3.
The curve , decomposes in . The curve is reducible only if . In this case .
Proof.
If , then there is nothing to prove. Suppose now and then . Then reads
If is not absolutely irreducible then it contains a line of type , , or for some . By direct computations, the first two cases imply , a contradiction. The case gives and . ∎
Finally, we need this easy results on linearized polynomials.
Proposition 2.4.
A linearized polynomial , with has nonzero roots in if and only if .
Proof.
The polynomial has a nonzero root if and only if the rank of the matrix
(6) 
is smaller than three, that is, . ∎
We are now in a position to prove our main results concerning planar polynomials of type on ; see (2).
Theorem 2.5.
Let be a power of an odd prime and consider the function of the type
where . Then, is planar if and only if one of the following condition holds

and ;

, .

, .
Proof.
Recall that is planar if and only if has no roots . We consider first the cases .
By Propositions 2.1 and 2.2 if has a linear factor but and then such a factor corresponds to a linearized factor of which is either or , with and . It can be easily verified, using Proposition 2.4, that such a factor has a nonzero root in and therefore is not planar.
We deal now with the case .

If , by Proposition 2.3 , that is and its unique root is and thus is planar.
We are left with the cases in which is absolutely irreducible. Now consider the polynomial as in (5). Let us fix a normal basis of over . Then each element can be written in a unique way as , with , . The curves defined by and are isomorphic. Since and therefore are absolutely irreducible, so it is . Since , by HasseWeil Theorem the curve defined by has rational points which correspond to nonzero roots of . Thus, in all these cases is not planar. ∎
Remark 2.6.
By Theorem 2.5 the number of pairs such that is a planar polynomial in is .
3. Acknowledgments
The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA  INdAM).
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