Another Class of Perfect Nonlinear Polynomial Functions

Abstract

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 𝑥^𝑝𝑘+1+𝑢𝑥²+𝑣𝑥^𝑘2𝑝 over GF(𝑝^2𝑘) are presented. This class of PN functions are all EA-equivalent to 𝑥².

1. Introduction

Let𝑝be a prime andGF(𝑝^𝑛)a finite field with𝑝^𝑛elements. Let𝑓be a mapping fromGF(𝑝^𝑛)to itself. Let𝑁(𝑎,𝑏)denote the number of solutions𝑥∈GF(𝑝^𝑛)of𝑓(𝑥+𝑎)−𝑓(𝑥)=𝑏, where𝑎,𝑏∈GF(𝑝^𝑛), and letΔ_𝑓=max{𝑁(𝑎,𝑏)∣𝑎,𝑏∈GF(𝑝^𝑛),𝑎≠0}. Nyberg [1] defined a mapping𝑓to be differentially𝑘-uniform ifΔ_𝑓=𝑘. For applications in cryptography, one would like to employ functions for whichΔ_𝑓is as small as possible. The differentially 2-uniform function is called APN function. And we know that APN functions are optimal overGF(2^𝑛). 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 that𝑓(𝑥+𝑎)−𝑓(𝑥)is a permutation polynomial for all𝑎∈GF(𝑝^𝑛)^∗=GF(𝑝^𝑛)⧵{0}. Planar functions were introduced in [2] 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 on𝑝^𝑛(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 [11], Coulter and Matthews introduced the first family of PN polynomials. Ding and Yuan [5] generalized their results and presented a new skew Hadamard difference set. Helleseth et al. [12] showed a family of PN binomials𝑢𝑥^𝑝𝑘+1+𝑥²overGF(𝑝^2𝑘)which is equivalent to the monomial𝑥². 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 [18] 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 [19] and introduced a character theoretic approach to prove the planarity of a function in [20]. Recently, they presented new commutative semifields with two parameters and then get new PN functions [21]. In their paper [22], Kyureghyan and Özbudak constructed some new PN functions by the products of two linearized polynomials.

In [12], the binomial composed with inequivalent monomials𝑥²and𝑥^𝑞+1was shown to be equivalent to the monomial𝑥²overGF(𝑞²). What about the planarity of trinomial composed with monomials𝑥²,𝑥^2𝑞and𝑥^𝑞+1? 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 trinomials𝑥^𝑝𝑘+1+𝑢𝑥²+𝑣𝑥^2𝑝𝑘overGF(𝑝^2𝑘). These PN trinomials are shown to be equivalent to monomial𝑥²in Section 4. We then conclude this paper in Section 5 with some future work.

2. Preliminaries

Let𝑝be an odd prime, and let𝑛be a positive integer. Let𝜒(𝑥)be a function onGF(𝑝^𝑛)defined by𝜒(𝑥)=𝑥^{(𝑝𝑛−1)/2}. Then, we get that𝜒(𝑥)=0when𝑥=0,𝜒(𝑥)=1when𝑥is a square inGF(𝑝^𝑛)⧵{0}, and𝜒(𝑥)=−1when𝑥is a nonsquare inGF(𝑝^𝑛)⧵{0}.

The𝑝-weight of a nonnegative integer𝑚is the sum of the digits in its𝑝-adic representation; that is, if𝑚=∑_𝑖𝑏_𝑖𝑝^𝑖with0≤𝑏_𝑖<𝑝, then the𝑝-ary weight of𝑚is∑_𝑖𝑏_𝑖∈ℤ. Recall that any mapping of GF(𝑝^𝑛)can be represented by a polynomial overGF(𝑝^𝑛)of degree less than𝑝^𝑛. Moreover, different such polynomials define different mappings. This allows us to identify the set of mappings ofGF(𝑝^𝑛)with the set of polynomials overGF(𝑝^𝑛)with degree less than𝑝^𝑛. The algebraic degree of a polynomial overGF(𝑝^𝑛)is the maximal𝑝-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 ^𝑛−1_𝑖,𝑗=0𝑎_𝑖,𝑗𝑥^{𝑝𝑖+𝑝𝑗},𝑎_𝑖,𝑗∈GF𝑝^𝑛 (1) are called Dembowski-Ostrom (DO) polynomials in [2].

Let𝑐_𝑖∈GF(𝑝^𝑛). A polynomial of the form𝐿(𝑥)=∑^𝑛−1_𝑖=0𝑐_𝑖𝑥^𝑝𝑖is called linearized or𝑝-polynomial overGF(𝑝^𝑛). The sum of a linear mapping and a constant inGF(𝑝^𝑛)is called an affine mapping.

Two functions𝑓,𝑔∶GF(𝑝^𝑛)→GF(𝑝^𝑛)are called extended affine (EA) equivalent, if𝑔=𝐿₁∘𝑓∘𝐿₂+𝐿₃for some affine permutations𝐿₁,𝐿₂and affine function𝐿₃. The functions𝑓and𝑔are called Carlet-Charpin-Zinoviev (CCZ) equivalent if the graphs of𝑓and𝑔are affine equivalent [23]. CCZ-equivalent functions have the same differential uniformity and the same extended Walsh spectrum. It is showed in [24] that the CCZ-equivalence coincides with the EA-equivalence for PN functions. For planar DO polynomials, CCZ-equivalence coincides with linear equivalence [14].

In [3], 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)𝑥²overGF(𝑝^𝑛)(folklore);(b)𝑥^𝑝𝑘+1overGF(𝑝^𝑛), where𝑘≤𝑛/2and𝑛/(𝑘,𝑛)is odd ([2, 11]);(c)𝑥¹⁰±𝑥⁶−𝑥²overGF(3^𝑛), where𝑛≥5is odd ([5, 11]);(d)𝑥^𝑝𝑠+1−𝑣𝑥^{𝑝2𝑘+𝑝𝑘+𝑠}overGF(𝑝^3𝑘), where𝑘₁=𝑘/gcd(𝑘,𝑠),𝑠₁=𝑠/gcd(𝑘,𝑠),𝑘₁is odd, ord(𝑣)=𝑞²+𝑞+1, and at least one of the following conditions hold:𝑘₁+𝑠₁≡0mod 3,𝑝^𝑘≡𝑝^𝑠≡1mod 3 ([13, 17]);(e)𝑥^𝑝𝑠+1−𝑣𝑥^{𝑝3𝑘+𝑝𝑘+𝑠}overGF(𝑝^4𝑘), where2𝑘/gcd(2𝑘,𝑠)is odd, ord(𝑣)=𝑞³+𝑞²+𝑞+1, and𝑝^𝑘≡𝑝^𝑠≡1mod 4 ([13]);(f)𝑥^(3𝑘+1)/2overGF(3^𝑛), where𝑘≥3is odd and(𝑘,𝑛)=1([11]);(g)𝑥^𝑞+1+𝜔𝛽𝑥^𝑝𝑠+1+𝜔𝛽^𝑞𝑥^{𝑞(𝑝𝑠+1)}overGF(𝑞²), where𝑞=𝑝^𝑚,𝜔,𝛽∈GF(𝑞²),𝜔+𝜔^𝑞=0,𝑠is a positive integer,𝛽is not agcd(𝑞+1,𝑝^𝑠+1)th power, and there is no0≠𝑎∈GF(𝑞²)such that𝑎^𝑞+𝑎=0and𝑎^𝑝𝑠=−𝑎([14, 18]);(h)𝑥²+𝑥^2𝑞𝑚+𝐺(𝑥^𝑞2+1)overGF(𝑞^2𝑚), where𝑞is a power of an odd prime𝑝,𝑚=2𝑘+1,  and𝐺(𝑥)=ℎ(𝑥−𝑥^𝑞𝑚)withℎ(𝑥)=∑^𝑘_𝑖=0(−1)^𝑖𝑥^𝑞2𝑖+∑^𝑘−1_𝑗=0(−1)^𝑘+𝑗𝑥^𝑞2𝑗+1([18, 27]);(i)𝑥²+𝑥⁹⁰overGF(3⁵)([28]).

Below, we always let𝑝be an odd prime and𝑞, 𝑛, 𝑘positive integers with𝑛=2𝑘and𝑞=𝑝^𝑘.

3. A New Family of PN Trinomials over  𝐺𝐹(𝑝^2𝑘)

In this section, we propose a new family of PN trinomials overGF(𝑝^2𝑘)which are composed of inequivalent monomials𝑥²and𝑥^𝑝𝑘+1.

Theorem 1. Let𝑢,𝑣,𝜃∈𝐺𝐹(𝑞²)with𝜃=(𝑣^𝑞+1−𝑢^𝑞+1)²−(𝑢−𝑣^𝑞)^𝑞+1, and let𝑓∶𝐺𝐹(𝑞²)→𝐺𝐹(𝑞²)be given by𝑓(𝑥)=𝑥^𝑞+1+𝑢𝑥²+𝑣𝑥^2𝑞. Then,𝑓(𝑥)is PN if and only if𝜃^(𝑞−1)/2=1.

Proof. We need to count the number of solutions of𝑓(𝑥+𝑎)−𝑓(𝑥)=𝑏under the conditions defined above for any𝑎≠0,𝑏inGF(𝑞²). The equation𝑓(𝑥+𝑎)−𝑓(𝑥)=𝑏can be written as (𝑥+𝑎)^𝑞+1+𝑢(𝑥+𝑎)²+𝑣(𝑥+𝑎)^2𝑞−𝑥^𝑞+1−𝑢𝑥²−𝑣𝑥^2𝑞=𝑏. (2) LetΔ=𝑏−𝑓(𝑎). Then, (2) turns to 𝑎𝑥^𝑞+𝑎^𝑞𝑥+2𝑢𝑎𝑥+2𝑣𝑎^𝑞𝑥^𝑞=Δ. (3) As (3) is affine, we just need to consider the caseΔ=0. WhenΔ=0, for the function𝑓(𝑥)to be PN, it is necessary and sufficient that𝑎𝑥^𝑞+𝑎^𝑞𝑥+2𝑢𝑎𝑥+2𝑣𝑎^𝑞𝑥^𝑞=0has𝑥=0as its only solution for any nonzero𝑎∈GF(𝑞²). That is, say𝑥^𝑞−1=−(𝑎^𝑞+2𝑢𝑎)/(𝑎+2𝑣𝑎^𝑞)≠0has no solution overGF(𝑞²). Therefore,𝑓(𝑥)is PN if and only if the equation(𝑎^𝑞+2𝑢𝑎)^𝑞+1=(𝑎+2𝑣𝑎^𝑞)^𝑞+1is not true. It can be written as 𝑢−𝑣^𝑞𝑎²+2𝑢^𝑞+1−𝑣^𝑞+1𝑎^𝑞+1+𝑢^𝑞−𝑣𝑎^2𝑞=0. (4) If𝑢−𝑣^𝑞=0, (4) is always true. We assume that𝑢−𝑣^𝑞≠0. Let𝑡=𝑎^𝑞−1. Since𝑎≠0, we can get 𝑢^𝑞−𝑣𝑡²+2𝑢^𝑞+1−𝑣^𝑞+1𝑡+𝑢−𝑣^𝑞=0 (5) from (4). Then, we get𝑡(𝑣^𝑞+1−𝑢^𝑞+1±𝜃^1/2)/(𝑢^𝑞−𝑣)with𝜃=(𝑣^𝑞+1−𝑢^𝑞+1)²−(𝑢−𝑣^𝑞)^𝑞+1. As we know𝑡^𝑞+1=1, then we have 𝑣^𝑞+1−𝑢^𝑞+12±𝜃^1/2+𝜃^𝑞/2𝑣^𝑞+1−𝑢^𝑞+1+𝜃^(𝑞+1)/2(𝑢−𝑣^𝑞)^𝑞+1=1. (6)
Since𝜃^𝑞=𝜃, then we get𝜃^𝑞/2=±𝜃^1/2. If𝜃^(𝑞−1)/2=0or −1, we can obviously find that (6) is true. If𝜃^(𝑞−1)/2=1, then we get 𝑣^𝑞+1−𝑢^𝑞+12±2𝜃^1/2𝑣^𝑞+1−𝑢^𝑞+1=+𝜃𝑢−𝑣^𝑞𝑞+1 (7) from (6). It leads to𝜃^1/2=±(𝑣^𝑞+1−𝑢^𝑞+1)and𝑢−𝑣^𝑞=0, which contradicts the first assumption𝑢−𝑣^𝑞≠0. Therefore,𝑓(𝑥)is PN if and only if𝜃^(𝑞−1)/2=1.

We can get the following corollary from Theorem 1.

Corollary 2. Let𝑓∶𝐺𝐹(𝑞²)→𝐺𝐹(𝑞²)be given by𝑓(𝑥)=𝑥^𝑞+1+𝑢𝑥²+𝑣𝑥^2𝑞, where𝑢^1+𝑞=𝑣^1+𝑞. Then,𝑓(𝑥)is PN if and only if𝑞≡1 (mod 4),𝜒(𝑢^𝑞−𝑣)=1or𝑞≡3 (mod 4),𝜒(𝑢^𝑞−𝑣)=−1.

Proof. From Theorem 1, we get that𝑓(𝑥)is PN if and only if((𝑣^𝑞+1−𝑢^𝑞+1)²−(𝑢−𝑣^𝑞)^𝑞+1)^(𝑞−1)/2=1. Since𝑢^1+𝑞=𝑣^1+𝑞, we just need(−(𝑢−𝑣^𝑞)^𝑞+1)^(𝑞−1)/2=1;thatis,0≠−(𝑢−𝑣^𝑞)^𝑞+1is a2(𝑞+1)th power. When𝜒(𝑢^𝑞−𝑣)=1,−(𝑢−𝑣^𝑞)^𝑞+1is a2(𝑞+1)th power if and only if −1 is a2(𝑞+1)th power, which is equivalent to𝑞≡1 (mod 4). When𝜒(𝑢^𝑞−𝑣)=−1, we get that(𝑢−𝑣^𝑞)^𝑞+1is a(𝑞+1)th power and not2(𝑞+1)th power. In this case,−(𝑢−𝑣^𝑞)^𝑞+1is a2(𝑞+1)th power if and only if𝑞≡3 (mod 4).

Remark 3. If𝑓(𝑥)is PN in Corollary 2, we have𝑢^(1+𝑞)/2=−𝑣^(1+𝑞)/2for accuracy. Since((𝑢−𝑣^𝑞)/(𝑢^𝑞−𝑣))⋅(𝑣/𝑢)=(𝑢𝑣−𝑣^𝑞+1)/(𝑢^𝑞+1−𝑢𝑣)=−1, then we get(𝑢^𝑞−𝑣)^𝑞−1(𝑣/𝑢)=−1. Whether𝑞≡1 (mod 4),𝜒(𝑢^𝑞−𝑣)=1or𝑞≡3 (mod 4),𝜒(𝑢^𝑞−𝑣)=−1, we get that𝑣/𝑢is a(𝑞−1)th power and not a2(𝑞−1)th power, which implies𝑢^(1+𝑞)/2=−𝑣^(1+𝑞)/2.

The PN functions defined in Corollary 2 exist. For example, the function𝑥^3𝑘+1−𝑥²+𝑥^2⋅3𝑘is PN overGF(3^2𝑘), where𝑘is 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 monomial𝑥². First, we give a simple proof to show that the PN functions defined in Corollary 2 are equivalent to𝑥².

Theorem 4. The PN function𝑓(𝑥)defined in Corollary 2 is linear equivalent to𝑥².

Proof. Since𝑢^(1+𝑞)/2=−𝑣^(1+𝑞)/2, we obtain(𝑣/𝑢)^1+𝑞=1and𝑣/𝑢is a(𝑞−1)th power. Let𝜃²=𝑣/𝑢, and let𝐿₁(𝑥)=((1−2𝜃𝑣^𝑞)/(𝑢−𝑣^𝑞))𝑥+((2𝜃𝑢−1)/(𝑢−𝑣^𝑞))𝑥^𝑞,𝐿₂(𝑥)=𝑥+𝜃𝑥^𝑞be linear polynomials onGF(𝑞²). We can get𝐿₁(𝑓(𝑥))=𝐿₂(𝑥)².
Assume that 𝐿₂(𝑥+𝑎)=𝐿₂(𝑥), we obtain𝑎+𝜃𝑎^𝑞=0, which implies that𝑎=0or𝑎^𝑞−1=−𝜃. If𝑎^𝑞−1=−𝜃, we get𝑎^2(𝑞−1)=𝑣/𝑢. Since𝑣/𝑢is not a2(𝑞−1)th power in Corollary 2 then we have𝑎^𝑞−1≠−𝜃and𝑎=0. Thus,𝐿₂(𝑥)=𝑥+𝜃𝑥^𝑞is a linear permutation.
If one of1−2𝜃𝑣^𝑞and2𝜃𝑢−1equals to 0, we can get that𝐿₁(𝑥)is a linear permutation. If both of1−2𝜃𝑣^𝑞and2𝜃𝑢−1equal 0, we can get𝑢^𝑞=𝑣. This is not true for𝜒(𝑢^𝑞−𝑣)≠0. Otherwise, we assume𝐿₁(𝑥+𝑎)=𝐿₁(𝑥). Then, we get𝑎=0or𝑎^𝑞−1=(2𝜃𝑣^𝑞−1)/(2𝜃𝑢−1). If𝑎≠0, we get(2𝜃𝑣^𝑞−1)^1+𝑞=(2𝜃𝑢−1)^1+𝑞. We can deduce that𝜃^𝑞(𝑢^𝑞−𝑣)=−𝜃(𝑢−𝑣^𝑞). It leads to 𝑣𝑢^(𝑞+1)/2=𝜃^𝑞+1=−𝑢−𝑣^𝑞𝑢^𝑞⋅𝑣−𝑣𝑢=𝑣^𝑞+1−𝑢𝑣𝑢^𝑞+1−𝑢𝑣=1. (8) Then, we get that𝑣/𝑢is a2(𝑞−1)th power which contradicts the known result𝑢^(1+𝑞)/2=−𝑣^(1+𝑞)/2. Therefore,𝐿₁(𝑥)=((1−2𝜃𝑣^𝑞)/(𝑢−𝑣^𝑞))𝑥+((2𝜃𝑢−1)/(𝑢−𝑣^𝑞))𝑥^𝑞is 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 function𝑓(𝑥)defined in Theorem 1 is linear equivalent to𝑥².

Proof. If the PN function𝑓(𝑥)=𝑥^𝑞+1+𝑢𝑥²+𝑣𝑥^2𝑞is linear equivalent to𝑥²overGF(𝑞²), there must exist linear permutations𝐿₁(𝑥)=∑^𝑛−1_𝑗=0𝑏_𝑗𝑥^𝑝𝑗and𝐿₂(𝑥)=∑^𝑛−1_𝑗=0𝑐_𝑗𝑥^𝑝𝑗such that ^𝑛−1_𝑗=0𝑏_𝑗𝑥^1+𝑞+𝑢𝑥²+𝑣𝑥^{2𝑞𝑝𝑗}=^𝑛−1_𝑖,𝑗=0𝑐_𝑖𝑐_𝑗𝑥^{𝑝𝑖+𝑝𝑗}. (9) When𝑖−𝑗≠0,𝑘∈ℤ_2𝑘, there is no item of the type𝑥^{𝑝𝑖+𝑝𝑗}on the left side of (9). Then, we can get that the coefficient of𝑥^{𝑝𝑖+𝑝𝑗}equals to 0. It shows that2𝑐_𝑖𝑐_𝑗=0. Assume that 𝑐_𝑙≠0for some𝑙, we obtain that𝑐_𝑗=0when𝑗≠𝑙,𝑙+𝑘∈ℤ_2𝑘. Then, (9) can be written as ^𝑛−1_𝑗=0𝑏_𝑗𝑥^1+𝑝𝑘+𝑢𝑥²+𝑣𝑥^{2𝑝𝑘𝑝𝑗}=𝑐²_𝑙𝑥^2𝑝𝑙+𝑐²_𝑙+𝑘𝑥^{2𝑝𝑙+𝑘}+2𝑐_𝑙𝑐_𝑙+𝑘𝑥^{𝑝𝑙+𝑝𝑙+𝑘}. (10) When𝑗≠𝑙, 𝑙+𝑘, we can see that there is no item of type𝑥^{𝑝𝑗(1+𝑝𝑘)}and𝑥^2𝑝𝑗on the right side of (10). Then, we have𝑏_𝑗+𝑏_𝑗+𝑘=0and𝑏_𝑗𝑢^𝑝𝑗+𝑏_𝑗+𝑘𝑣^{𝑝𝑗+𝑘}=0. This leads to𝑏_𝑗=−𝑏_𝑗+𝑘and𝑏_𝑗(𝑢^𝑝𝑗−𝑣^{𝑝𝑗+𝑘})=0. Since𝑢−𝑣^𝑝𝑘≠0, then we obtain that𝑏_𝑗=0when𝑗≠𝑙, 𝑙+𝑘. Therefore, (10) can be written as 𝑏_𝑙𝑥^1+𝑝𝑘+𝑢𝑥²+𝑣𝑥^{2𝑝𝑘𝑝𝑙}+𝑏_𝑙+𝑘𝑥^1+𝑝𝑘+𝑢𝑥²+𝑣𝑥^{2𝑝𝑘𝑝𝑙+𝑘}=𝑐²_𝑙𝑥^2𝑝𝑙+𝑐²_𝑙+𝑘𝑥^{2𝑝𝑙+𝑘}+2𝑐_𝑙𝑐_𝑙+𝑘𝑥^{𝑝𝑙+𝑝𝑙+𝑘}. (11) Comparing the coefficients of𝑥^2𝑝𝑙,𝑥^{2𝑝𝑙+𝑘}, and𝑥^{𝑝𝑙(1+𝑝𝑘)}of (11), we can get the following equations: 𝑏_𝑙+𝑏_𝑙+𝑘=2𝑐_𝑙𝑐_𝑙+𝑘,𝑏_𝑙𝑢^𝑝𝑙+𝑏_𝑙+𝑘𝑣^{𝑝𝑙+𝑘}=𝑐²_𝑙,𝑏_𝑙𝑣^𝑝𝑙+𝑏_𝑙+𝑘𝑢^{𝑝𝑙+𝑘}=𝑐²_𝑙+𝑘. (12)(13)(14) From (12) and (13), we obtain𝑏_𝑙+𝑘=(𝑐²_𝑙−2𝑐_𝑙𝑐_𝑙+𝑘𝑢^𝑝𝑙)/(𝑣^{𝑝𝑙+𝑘}−𝑢^𝑝𝑙). We can also obtain𝑏_𝑙+𝑘=(𝑐²_𝑙+𝑘−2𝑐_𝑙𝑐_𝑙+𝑘𝑣^𝑝𝑙)/(𝑢^{𝑝𝑙+𝑘}−𝑣^𝑝𝑙)from (13) and (14). Then, we have 𝑢^{𝑝𝑙+𝑘}−𝑣^𝑝𝑙𝑐_𝑙𝑐_𝑙+𝑘²+2𝑣^{𝑝𝑙(1+𝑝𝑘)}−𝑢^{𝑝𝑙(1+𝑝𝑘)}𝑐_𝑙𝑐_𝑙+𝑘+𝑢^𝑝𝑙−𝑣^{𝑝𝑙+𝑘}=0. (15) We note that𝐿₂(𝑥)=𝑐_𝑙𝑥^𝑝𝑙+𝑐_𝑙+𝑘𝑥^{𝑝𝑙+𝑘}is a permutation if and only if𝑐_𝑙/𝑐_𝑙+𝑘is not a(𝑝^𝑘−1)th power. Comparing (5) and (15), we have𝑐_𝑙/𝑐_𝑙+𝑘=−𝑡^𝑝𝑙. Under the conditions of Theorem 1,𝑡^𝑞+1=1is not true. Then, we get that𝑐_𝑙/𝑐_𝑙+𝑘is not a(𝑝^𝑘−1)th power and𝐿₂(𝑥)is a permutation.
From (12)–(14), we get 𝑏_𝑙+𝑘=(𝑐²_𝑙−2𝑐_𝑙𝑐_𝑙+𝑘𝑢^𝑝𝑙)/(𝑣^{𝑝𝑙+𝑘}−𝑢^𝑝𝑙) and 𝑏_𝑙=(2𝑐_𝑙𝑐_𝑙+𝑘𝑣^{𝑝𝑙+𝑘}−𝑐²_𝑙)/(𝑣^{𝑝𝑙+𝑘}−𝑢^𝑝𝑙). If one of𝑏_𝑙and𝑏_𝑙+𝑘equals to 0, then𝐿₁(𝑥)=𝑏_𝑙𝑥^𝑝𝑙+𝑏_𝑙+𝑘𝑥^{𝑝𝑙+𝑘}is a monomial permutation. If both of𝑏_𝑙and𝑏_𝑙+𝑘equal to 0, we get𝑢=𝑣^𝑝𝑘, which leads to a contradiction. Now, we consider the case that both𝑏_𝑙and𝑏_𝑙+𝑘are not equal to 0. In this case,𝐿₁(𝑥)=𝑏_𝑙𝑥^𝑝𝑙+𝑏_𝑙+𝑘𝑥^{𝑝𝑙+𝑘}is a permutation if and only if𝑏_𝑙/𝑏_𝑙+𝑘is not a(𝑝^𝑘−1)th power. We assume that𝑏_𝑙/𝑏_𝑙+𝑘=(2𝑐_𝑙+𝑘𝑣^{𝑝𝑙+𝑘}−𝑐_𝑙)/(𝑐_𝑙−2𝑐_𝑙+𝑘𝑢^𝑝𝑙)is a(𝑝^𝑘−1)th power. Then, we have(2𝑐_𝑙+𝑘𝑣^{𝑝𝑙+𝑘}−𝑐_𝑙)^𝑝𝑘+1=(𝑐_𝑙−2𝑐_𝑙+𝑘𝑢^𝑝𝑙)^𝑝𝑘+1, which implies 𝑢^𝑝𝑙−𝑣^{𝑝𝑙+𝑘}𝑐_𝑙𝑐_𝑙+𝑘^𝑝𝑘+𝑢^{𝑝𝑘+𝑙}−𝑣^𝑝𝑙𝑐_𝑙𝑐_𝑙+𝑘+2𝑣^{𝑝𝑙(1+𝑝𝑘)}−𝑢^{𝑝𝑙(1+𝑝𝑘)}=0. (16) From (15), we can get𝑐_𝑙/𝑐_𝑙+𝑘=(𝑢^{𝑝𝑙(1+𝑝𝑘)}−𝑣^{𝑝𝑙(1+𝑝𝑘)}±𝜃^𝑝𝑙/2)/(𝑢^{𝑝𝑘+𝑙}−𝑣^𝑝𝑙)with𝜃defined in Theorem 1. Substituting the value of𝑐_𝑙/𝑐_𝑙+𝑘into (16), we have 𝑢^{𝑝𝑙(1+𝑝𝑘)}−𝑣^{𝑝𝑙(1+𝑝𝑘)}±𝜃^{𝑝𝑘+𝑙/2}+𝑢^{𝑝𝑙(1+𝑝𝑘)}−𝑣^{𝑝𝑙(1+𝑝𝑘)}±𝜃^𝑝𝑙/2+2𝑣^{𝑝𝑙(1+𝑝𝑘)}−𝑢^{𝑝𝑙(1+𝑝𝑘)}=0. (17) Since𝜃^{(𝑝𝑘−1)/2}=1, from (17), we get𝜃^𝑝𝑙/2=0, which is a contradiction. Therefore, we get that𝑏_𝑙/𝑏_𝑙+𝑘is not a(𝑝^𝑘−1)th power and𝐿₁(𝑥)is a permutation. The proof is completed.

5. Conclusion

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 function𝑥²by 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 terms𝑥^{𝑞𝑖+𝑞𝑗}(0≤𝑖,𝑗<𝑛) overGF(𝑞^𝑛), where𝑛≥3and𝑞is an odd prime power. However, it may be possible to determine them in some special cases (e.g., see [15]). 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.

Acknowledgments

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).