 # Several classes of bent functions over finite fields

Let 𝔽_p^n be the finite field with p^n elements and Tr(·) be the trace function from 𝔽_p^n to 𝔽_p, where p is a prime and n is an integer. Inspired by the works of Mesnager (IEEE Trans. Inf. Theory 60(7): 4397-4407, 2014) and Tang et al. (IEEE Trans. Inf. Theory 63(10): 6149-6157, 2017), we study a class of bent functions of the form f(x)=g(x)+F(Tr(u_1x),Tr(u_2x),⋯,Tr(u_τx)), where g(x) is a function from 𝔽_p^n to 𝔽_p, τ≥2 is an integer, F(x_1,⋯,x_n) is a reduced polynomial in 𝔽_p[x_1,⋯,x_n] and u_i∈𝔽^*_p^n for 1≤ i ≤τ. As a consequence, we obtain a generic result on the Walsh transform of f(x) and characterize the bentness of f(x) when g(x) is bent for p=2 and p>2 respectively. Our results generalize some earlier works. In addition, we study the construction of bent functions f(x) when g(x) is not bent for the first time and present a class of bent functions from non-bent Gold functions.

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## 1 Introduction

Boolean bent functions were first introduced by Rothaus in 1976  as an interesting combinatorial object with maximum Hamming distance to the set of all affine functions. Over the last four decades, bent functions have attracted a lot of research interest due to their important applications in cryptography , sequences  and coding theory [2, 6]. Kumar, Scholtz and Welch in  generalized the notion of Boolean bent functions to the case of functions over an arbitrary finite field.

Given a function mapping from to , the Walsh transform of is defined by

 ˆf(b)=∑x∈Fpnωf(x)−Tr(bx),b∈Fpn,

where is a complex primitive -th root of unity. According to , is called a -ary bent function if all its Walsh coefficients satisfy . A -ary bent function is called regular if holds for some function mapping to , and it is called weakly regular if there exists a complex having unit magnitude such that for all . The function is called the dual of and it is also bent.

An interesting class of bent functions over finite fields with the form

 f(x)=g(x)+F(Tr(u1x),Tr(u2x),⋯,Tr(uτx)) (1)

was studied in the past years, where is a function from to , is an arbitrary reduced polynomial in , is an integer and for all . The initial work on this issue is due to Mesnager  who studied the case , and is a bent function whose dual function has a null second order derivative. This motivated Xu et al. to construct bent functions by using some known bent functions via the cases for  and for [18, 19], respectively. Later, Wang et al. characterized the bentness of when is bent and for and consequently constructed bent functions of the form (1) from some bent functions whose dual functions are already known . Meanwhile, also for , Tang et al.  investigated the bentness of with the form (1) for an arbitrary reduced polynomial and a bent function whose dual satisfies

 ˜g(x−∑τi=1uiti)=˜g(x)+∑τi=1gi(x)ti, (2)

where and is a function from to for each . The analogues of the results in 

for an odd prime

were obtained in  where was required to be a homogeneous quadratic bent function and its dual also satisfies (2). In 2019, Zheng et al.  showed that for , of the form (1) is bent for any reduced polynomial if and only if is bent whose dual satisfies (2).

Inspired by the above works, in this paper, we further study the construction of bent functions with the form (1). We first derive a generic result on Walsh transform of the function of the form (1) in which is not necessarily bent. Then we characterize the bentness of in (1) when is bent whose dual satisfies

 ˜g(x−∑τi=1uiti)=˜g(x)+∑1≤i≤j≤τAijtitj+∑τi=1gi(x)ti (3)

for some . Our results generalize some earlier works in this direction. In addition, we attempt to construct bent functions having the form (1) from non-bent functions and consequently obtain a class of such bent functions by using non-bent Gold functions and . To the best of our knowledge, the construction of bent functions with the form (1) from either bent functions satisfying (3) for some or non-bent functions is studied in this paper for the first time in the literature.

The rest of this paper is organized as follows. Section 2 gives some preliminaries. Section 3 provides some results on the Walsh transform of functions with the form (1) and characterizes the bentness of such functions when satisfies (3) for some nonzero . Section 4 proposes a class of bent functions of the form (1) in which is a non-bent Gold function and . Section 5 concludes this paper.

## 2 Preliminaries

Throughout this paper, let denote the finite field with elements, where is a positive integer and is a prime. The trace function from to its subfield is defined by . In particular, when , we use the notation instead of .

### 2.1 Algebraic degree

A function is often represented by its algebraic normal form:

 F(x1,⋯,xn)=∑e=(e1,⋯,en)∈Fnpa(e)(n∏i=1xeii),a(e)∈Fp. (4)

A polynomial in with the form (4) is called a reduced polynomial. The algebraic degree of , denoted by , is defined as , where . The following lemma will be used to determine the algebraic degree of some reduced functions, which is a direct generalization of the result proposed in [15, Lemma 2.1].

###### Lemma 1.

Let be linearly independent over where is an integer with Let be a reduced polynomial in of algebraic degree . Then the following univariate function

 F(Tr(u1x),Tr(u2x),⋯,Tr(uτx))

has algebraic degree d.

The algebraic degree of a -ary bent function has been characterized as follows.

###### Lemma 2.

([8, Propositions 4.4 and 4.5]) Let be a bent function from to , then the algebraic degree of satisfies , and if is weakly regular bent, then .

### 2.2 Certain exponential sums

For each , the function defines an additive character for . The character is called the canonical additive character of .

###### Lemma 3.

([10, Theorems 5.15 and 5.33]) Let be an odd prime, be the quadratic multiplicative character of and . Then

 ∑x∈Fpχ(ax2+bx)=η(a)√p∗ω−b24a,a∈F∗p,b∈Fp.
###### Lemma 4.

Let be an odd prime and for . Denote

 H=∑(x,y)∈F2pωa1x2+a2y2+a3xy+a4x+a5y.

If , then

 H=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩p2,ifa1=0,a2=a4=a5=0,η(a2)p√p∗ω−a254a2,ifa1=0,a2≠0,a4=0,η(a1)p√p∗ω−a244a1,ifa1≠0,a5=a3a42a1,0,otherwise,

and if , then

 H=η(a23−4a1a2)pωa2a24+a1a25−a3a4a5a23−4a1a2.
###### Proof.

We first consider the case . If , then and Lemma 3 yields

 H=∑(x,y)∈F2pωa4x+a2y2+a5y=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩p2,ifa2=a4=a5=0,p√p∗η(a2)ω−a254a2,ifa2≠0,a4=0,0,otherwise.

If , again by Lemma 3 one obtains

 H=√p∗η(a1)∑y∈Fpωa2y2+a5y−(a3y+a4)24a1=√p∗η(a1)∑y∈Fpω(a5−a3a42a1)y−a244a1

due to . This implies that

 H=p√p∗η(a1)ω−a244a1

if and otherwise.

Next, we calculate for the case . If , then and it can be readily verified that

 H=∑(x,y)∈F2pω(a3y+a4)x+a2y2+a5y=pωa2(−a4a3)2+a5(−a4a3).

If , then by Lemma 3, one gets

 H=√p∗η(a1)∑y∈Fpωa2y2+a5y−(a3y+a4)24a1=p∗η(4a1a2−a23)ωa1a23−4a1a2(a5−a3a42a1)2−a244a1.

Then the result follows from and . This completes the proof. ∎

### 2.3 Walsh transform of Gold functions

In this subsection, we present some known results on the Walsh transform of Gold functions which will be used to construct bent functions of the form (1) from non-bent functions in Section 4.

Coulter studied the Walsh transform of Gold functions for odd  and  respectively. Let be positive integers with and , where , then the Walsh transform of has been determined as below.

###### Lemma 5.

([5, Theorem 4.2]) Suppose is odd. Then for ,

where is the unique element satisfying .

###### Lemma 6.

([3, 4, Theorems 1 and 2] and [5, Theorem 5.3]) Suppose is even, and is a prime. Then for ,

 ˆg(b)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩(−1)m/dpm¯¯¯¯χ(axpk+10),ifapn−1pd+1≠(−1)md,(−1)m/d+1pm+d¯¯¯¯χ(axpk+10),ifapn−1pd+1=(−1)md,Trn2d(b/acpk)=0,0,otherwise,

where satisfies and is the solution of .

## 3 Constructions of bent functions of the form (1)

In this section, we first derive a generic result on the Walsh transform of with the form (1) in which is not necessarily bent. Then, we characterize the bentness of in (1) for a bent function whose dual satisfies (3) with some for and respectively.

The Walsh transform of a multivariate function over is

 ˆF(b1,⋯,bn)=∑(x1,⋯,xn)∈FnpωF(x1,⋯,xn)−∑ni=1bixi, (5)

where . Then the inverse Walsh transform of is given by

 ωF(x1,⋯,xn)=1pn∑(b1,⋯,bn)∈Fnpω∑ni=1bixiˆF(b1,⋯,bn). (6)

Using (10) and (6), the Walsh transform of in (1) can be expressed as below.

###### Theorem 1.

Let be defined as (1), then for any ,

 ˆf(b)=1pτ∑(t1,⋯,tτ)∈FτpˆF(t1,⋯,tτ)ˆg(b−∑τi=1tiui).

In particular, if , then

 ˆf(b)=1pτ−1τ−1∑i,j=1p−1∑ti=0p−1∑xj=0ω−∑τ−1i=1xitiˆg(b−∑τ−1i=1tiui−∏τ−1i=1xiuτ).
###### Proof.

According to (6), for any , one obtains

 ˆf(b) =∑x∈Fpnωg(x)+F(Tr(u1x),Tr(u2x),⋯,Tr(uτx))−Tr(bx) =1pτ∑(t1,⋯,tτ)∈Fτp∑x∈Fpnωg(x)−Tr(bx)+∑τi=1Tr(uix)tiˆF(t1,⋯,tτ) =1pτ∑(t1,⋯,tτ)∈FτpˆF(t1,⋯,tτ)ˆg(b−∑τi=1tiui).

If , then by (10), one gets

 ˆf(b) =1pτ∑(t1,⋯,tτ)∈Fτp∑(x1,⋯,xτ)∈Fτpωx1⋯xτ−∑τi=1tixiˆg(b−∑τi=1tiui) =1pττ−1∑i,j=1p−1∑ti=0p−1∑xj=0p−1∑tτ=0p−1∑xτ=0ω(∏τ−1i=1xi−tτ)xτ−∑τ−1i=1tixiˆg(b−∑τi=1tiui) =1pτ−1τ−1∑i,j=1p−1∑ti=0p−1∑xj=0ω−∑τ−1i=1xitiˆg(b−∑τ−1i=1tiui−∏τ−1i=1xiuτ)

due to the fact that if . This completes the proof. ∎

###### Remark 1.

Note that Theorem 1 holds for an arbitrary prime and is a function from to which is not necessary to be bent. It generalizes some earlier works:

1. Theorem 1 is a generalization of Lemma 1 in .

2. If one takes , and , where , then our result gives for any . When is a Boolean bent function with the dual , it can be verified that is bent if and only if . This is Corollary 5 in .

3. If one takes , and , where , then our result gives for any . This is Lemma 1 in .

4. If one takes , and for , then by Theorem 1 one obtains for any , where is a primitive -rd root of unity. This is exactly Lemma 4 in .

### 3.1 Bent functions of the form (1) for p=2

Let and be defined as (1). Tang et al.  proved that is bent for any reduced polynomial in if is a bent function whose dual satisfies (2). Later, Zheng et al.  showed that this condition is also necessary. In this section, we consider the bentness of for a more general bent function , i.e., satisfies (3) for some , where and is a positive integer.

Suppose that is a bent function over and its dual satisfies (3), then for , and , where , one gets

 ˆg(b−∑τi=1tiui)=2n2(−1)˜g(b−∑τi=1tiui),

where is the dual of . This together with (10) and Theorem 1 gives

 ˆf(b) = 12τ∑(t1,⋯,tτ)∈Fτ2ˆF(t1,⋯,tτ)ˆg(b−∑τi=1tiui) (7) = 2n/2−ττ∑i,j=11∑xi=01∑tj=0(−1)F(x1,⋯,xτ)+˜g(b)+∑1≤i

where is a function from to for each .

Now assume that the nonzero elements in are , where is a positive integer, for and . For simplicity, denote and define

 Γ={i1,j1,⋯,iℓ,jℓ}. (8)

Then, we discuss (7) as the following two cases:

Case I: , where .

For this case, we have for and otherwise. Then (7) becomes

 ˆf(b) = 2n/2−τ(−1)˜g(b)τ∑i,j=11∑xi=01∑tj=0(−1)F(x1,⋯,xτ)+∑ℓs=1tistjs+∑τi=1(hi+xi)ti = 2n/2−2ℓ(−1)˜g(b)∑i,j∈Γ1∑xi=01∑tj=0(−1)F(x1,⋯,xτ)|xi=hi,i∉Γ+ℓ∑s=1(tjs+(his+xis))tis+ℓ∑s=1(hjs+xjs)tjs = 2n/2−ℓ(−1)˜g(b)∑i∈Γ1∑xi=0(−1)F(x1,⋯,xτ)|xi=hi,i∉Γ+ℓ∑s=1(his+xis)(hjs+xjs),

where the second identity holds due to if and otherwise for any .

Then, in this case, defined by (1) can be bent for certain special .

###### Theorem 2.

Let and be pairwise distinct elements in , where . Let be a bent function over whose dual satisfies (3) with for and otherwise, where is defined by (8). If and satisfies

 F(x1,⋯,xτ)|xi=hi,i∉Γ=∑ℓs=1(Fisxis+Fjsxjs)+F0

for some , where , then defined by (1) is bent and its Walsh transform at point is

 ˆf(b)=2n/2(−1)˜g(b)+∑ℓs=1[(his+Fjs)(hjs+Fis)+hishjs]+F0.
###### Example 1.

Let , , be a primitive element of and . From  we know that the dual of satisfies (3) with , for and . Taking , , , , and , then and . It indicates and satisfies

 F(x1,x2,x3,x4)|x1=Tr(b)+1,x4=Tr(ξ16b)=(Tr(b)+1)x2+(Tr(b)+1)Tr(ξ16b)x3.

Magma shows that