Research Article | Open Access
Another Class of Perfect Nonlinear Polynomial Functions
Perfect nonlinear (PN) functions have been an interesting subject of study for a long time and have applications in coding theory, cryptography, combinatorial designs, and so on. In this paper, the planarity of the trinomials over GF() are presented. This class of PN functions are all EA-equivalent to .
Letbe a prime anda finite field withelements. Letbe a mapping fromto itself. Letdenote the number of solutionsof, where, and let. Nyberg  defined a mappingto be differentially-uniform if. For applications in cryptography, one would like to employ functions for whichis as small as possible. The differentially 2-uniform function is called APN function. And we know that APN functions are optimal over. This concept is of interest in cryptography since differential and linear cryptanalysis exploit the uniform property of the functions which are used in many block ciphers, such as DES. The differentially 1-uniform function is called PN function. It is interesting to observe that PN functions have also been studied under the name of planar functions which are functions such thatis a permutation polynomial for all. Planar functions were introduced in  to describe projective planes with certain properties. In recent papers [3, 4], PN functions were used to describe new finite commutative semifields of odd order. In [5, 6], it was shown that a PN function yields either a skew Hadamard difference set or a Paley type partial difference set depending on(mod 4). PN function is one of the most important cryptographic functions [7, 8] and has extensive applications in cryptography and communication. For example, PN and APN functions were used to construct optimal constant-composition codes and signal sets [9, 10].
Since PN functions have many applications in coding theory, cryptography, combinatorial designs, and so on, it is interesting to find new PN functions. We call a function “new” only if it is CCZ-inequivalent to the old ones. As we know, there are only three classes of PN monomials. Whether there exists another class of PN monomials is an open problem. In , Coulter and Matthews introduced the first family of PN polynomials. Ding and Yuan  generalized their results and presented a new skew Hadamard difference set. Helleseth et al.  showed a family of PN binomialsoverwhich is equivalent to the monomial. After that, some new methods were used to construct new PN functions (please see [13–17] and references therein), but it is still difficult to find more new PN functions. Many constructions of PN functions are used of the link between quadratic PN functions and commutative semifields. Bierbrauer  introduced a general projection method to construct commutative semifields and generalized the known PN functions. Pott and Zhou presented a switching construction of PN functions in  and introduced a character theoretic approach to prove the planarity of a function in . Recently, they presented new commutative semifields with two parameters and then get new PN functions . In their paper , Kyureghyan and Özbudak constructed some new PN functions by the products of two linearized polynomials.
In , the binomial composed with inequivalent monomialsandwas shown to be equivalent to the monomialover. What about the planarity of trinomial composed with monomials,and? In this paper, we will answer this question. In Section 2, we recall some definitions and tools used later in the paper. In Section 3, we characterize the planarity of the trinomialsover. These PN trinomials are shown to be equivalent to monomialin Section 4. We then conclude this paper in Section 5 with some future work.
Letbe an odd prime, and letbe a positive integer. Letbe a function ondefined by. Then, we get thatwhen,whenis a square in, andwhenis a nonsquare in.
The-weight of a nonnegative integeris the sum of the digits in its-adic representation; that is, ifwith, then the-ary weight ofis. Recall that any mapping of can be represented by a polynomial overof degree less than. Moreover, different such polynomials define different mappings. This allows us to identify the set of mappings ofwith the set of polynomials overwith degree less than. The algebraic degree of a polynomial overis the maximal-weight of the exponents of its nonzero terms. A polynomial is called quadratic if it has algebraic degree 2. The following polynomials of algebraic degree 2 are called Dembowski-Ostrom (DO) polynomials in .
Let. A polynomial of the formis called linearized or-polynomial over. The sum of a linear mapping and a constant inis called an affine mapping.
Two functionsare called extended affine (EA) equivalent, iffor some affine permutationsand affine function. The functionsandare called Carlet-Charpin-Zinoviev (CCZ) equivalent if the graphs ofandare affine equivalent . CCZ-equivalent functions have the same differential uniformity and the same extended Walsh spectrum. It is showed in  that the CCZ-equivalence coincides with the EA-equivalence for PN functions. For planar DO polynomials, CCZ-equivalence coincides with linear equivalence .
In , Coulter and Henderson proved that planar DO polynomials are equivalent to commutative semifields with odd characteristic. Many new PN functions are defined by corresponding commutative semifields with no explicit function expressions, such as Dickson semifields and Cohen-Ganley semifileds [4, 25, 26]. In the following, we just list the known EA-inequivalent PN functions which have explicit function expressions:(a)over(folklore);(b)over, whereandis odd ([2, 11]);(c)over, whereis odd ([5, 11]);(d)over, where,,is odd, ord, and at least one of the following conditions hold:mod 3,mod 3 ([13, 17]);(e)over, whereis odd, ord, andmod 4 ();(f)over, whereis odd and();(g)over, where,,,is a positive integer,is not ath power, and there is nosuch thatand([14, 18]);(h)over, whereis a power of an odd prime,, andwith([18, 27]);(i)over().
Below, we always letbe an odd prime and, , positive integers withand.
3. A New Family of PN Trinomials over
In this section, we propose a new family of PN trinomials overwhich are composed of inequivalent monomialsand.
Theorem 1. Letwith, and letbe given by. Then,is PN if and only if.
Proof. We need to count the number of solutions ofunder the conditions defined above for anyin. The equationcan be written as
Let. Then, (2) turns to
As (3) is affine, we just need to consider the case. When, for the functionto be PN, it is necessary and sufficient thathasas its only solution for any nonzero. That is, sayhas no solution over. Therefore,is PN if and only if the equationis not true. It can be written as
If, (4) is always true. We assume that. Let. Since, we can get
from (4). Then, we getwith. As we know, then we have
Since, then we get. Ifor −1, we can obviously find that (6) is true. If, then we get from (6). It leads toand, which contradicts the first assumption. Therefore,is PN if and only if.
We can get the following corollary from Theorem 1.
Corollary 2. Letbe given by, where. Then,is PN if and only if (mod 4),or (mod 4),.
Proof. From Theorem 1, we get thatis PN if and only if. Since, we just need;is ath power. When,is ath power if and only if −1 is ath power, which is equivalent to (mod 4). When, we get thatis ath power and notth power. In this case,is ath power if and only if (mod 4).
Remark 3. Ifis PN in Corollary 2, we havefor accuracy. Since, then we get. Whether (mod 4),or (mod 4),, we get thatis ath power and not ath power, which implies.
The PN functions defined in Corollary 2 exist. For example, the functionis PN over, whereis even.
4. The Linear Equivalence of the New PN Trinomials
In this section, we will discuss the linear equivalence between our new PN functions and the known PN monomial. First, we give a simple proof to show that the PN functions defined in Corollary 2 are equivalent to.
Theorem 4. The PN functiondefined in Corollary 2 is linear equivalent to.
Proof. Since, we obtainandis ath power. Let, and let,be linear polynomials on. We can get.
Assume that , we obtain, which implies thator. If, we get. Sinceis not ath power in Corollary 2 then we haveand. Thus,is a linear permutation.
If one ofandequals to 0, we can get thatis a linear permutation. If both ofandequal 0, we can get. This is not true for. Otherwise, we assume. Then, we getor. If, we get. We can deduce that. It leads to Then, we get thatis ath power which contradicts the known result. Therefore,is also a linear permutation. This completes the proof.
Inspired by the proof of Theorem 4, we get a generalized result in the following theorem.
Theorem 5. The PN functiondefined in Theorem 1 is linear equivalent to.
Proof. If the PN functionis linear equivalent toover, there must exist linear permutationsandsuch that
When, there is no item of the typeon the left side of (9). Then, we can get that the coefficient ofequals to 0. It shows that. Assume that for some, we obtain thatwhen. Then, (9) can be written as
When, , we can see that there is no item of typeandon the right side of (10). Then, we haveand. This leads toand. Since, then we obtain thatwhen, . Therefore, (10) can be written as
Comparing the coefficients of,, andof (11), we can get the following equations:
From (12) and (13), we obtain. We can also obtainfrom (13) and (14). Then, we have
We note thatis a permutation if and only ifis not ath power. Comparing (5) and (15), we have. Under the conditions of Theorem 1,is not true. Then, we get thatis not ath power andis a permutation.
From (12)–(14), we get and . If one ofandequals to 0, thenis a monomial permutation. If both ofandequal to 0, we get, which leads to a contradiction. Now, we consider the case that bothandare not equal to 0. In this case,is a permutation if and only ifis not ath power. We assume thatis ath power. Then, we have, which implies From (15), we can getwithdefined in Theorem 1. Substituting the value ofinto (16), we have Since, from (17), we get, which is a contradiction. Therefore, we get thatis not ath power andis a permutation. The proof is completed.
In this paper, we present a family of PN trinomials and determine the necessary and sufficient conditions which assure their planarity. All these PN functions are shown to be equivalent to the known PN functionby using the definition of linear equivalence. Our results give an answer for the question presented in the introduction. It seems hard to determine the planarity of the linear combination of terms over, whereandis an odd prime power. However, it may be possible to determine them in some special cases (e.g., see ). We will continue this study and try to find more new PN functions in the future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
We would like to thank the anonymous reviewers for their invaluable suggestions and helpful comments, which greatly improved the paper. We sincerely thank the Editor for the kind help provided. This project was supported by NSFC-Union Science Foundation of Henan (no. U1304103), National Nature Science Foundation of China (no. 11201214), Natural Science Foundation of Henan Province (no. 122300410261), and Foundation of Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education (no. 93K172012K07).
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