# On Negabent Functions and Nega-Hadamard Transform

Authors: Zepeng Zhuo; Jinfeng Chong
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The Boolean function which has equal absolute spectral values under the nega-Hadamard transform is called negabent function. In this paper, the special Boolean functions by concatenation are presented. We investigate their nega-Hadamard transforms, nega-autocorrelation coefficients, sum-of-squares indicators, and so on. We establish a new equivalent statement onf1∥f2 which is negabent function. Based on them, the construction for generating the negabent functions by concatenation is given. Finally, the function expressed as f(Ax⊕a)⊕b·x⊕c is discussed. The nega-Hadamard transform and nega-autocorrelation coefficient of this function are derived. By applying these results, some properties are obtained.


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## Body

## 1. Introduction

Rothaus [1] introduced the class of bent functions which play an important role in cryptography and error correcting coding (where they are used to define optimum codes such as the Kerdock codes). The bent functions are those Boolean functions whose Hamming distance to the set of all affine functions is maximum. Equivalently, their spectrum with respect to the Walsh-Hadamard transform is flat (i.e., all spectral values have the same absolute value). The Walsh-Hadamard transform is an example of a unitary transformation on the space of all Boolean functions. Riera and Parker [2] considered some generalized bent criteria for Boolean functions by analyzing Boolean functions that have a flat spectrum with respect to one or more transforms chosen from a set of unitary transforms. The transforms chosen by Riera and Parker are n-fold tensor products of the identity mapping 1001, Walsh-Hadamard transformation 1/2111-1, and nega-Hadamard transformation 1/21i1-i, where i2=-1. Riera and Parker [2] mentioned that this choice is motivated by local unitary transforms that play an important role in the structural analysis of pure n-qubit stabilizer quantum states. As in the case of the Walsh-Hadamard transform, a Boolean function whose nega-Hadamard magnitude spectrum is flat is said to be negabent. Moreover, a Boolean function is called bent-negabent if it is both bent and negabent. For instance, the 6-variable function f(x)=(x1⊕x2⊕x1x2⊕x2x3)x4⊕(x3⊕x1x2⊕x2x3)x5⊕(x1⊕x3)x6 is a cubic negabent function and the 4-variable function g(y)=y1y2⊕y2y3⊕y3y4 is bent-negabent.Negabent functions and bent-negabent functions have been extensively studied during the last few years [3–11]. Parker and Pott [3] presented several constructions and classifications on bent-negabent. Schmidt et al. [4] constructed a subclass of the Maiorana-McFarland class of bent functions in which all functions are also negabent. Also, they provided an upper bound on the algebraic degree of any bent-negabent Boolean function from the Maiorana-McFarland class. Sarkar [5] studied the symmetric negabent functions and obtained that a symmetric function is negabent if and only if it is affine. Stănică et al. [6, 9] gave the detailed study of some of properties of nega-Hadamard transform and derived several results on negabentness of concatenations. They pointed out that the algebraic degree of an n-variable negabent function is at most ⌈n/2⌉. In [7], Gangopadhyay and Chaturvedi developed the technique of constructing bent-negabent functions by using complete mapping polynomials. Sarkar [8] considered negabent functions that have trace representation and completely characterized negabent quadratic monomial functions. The necessary and sufficient condition for a Maiorana-McFarland bent function to be a negabent function was presented in [8]. Su et al. [10] gave necessary and sufficient conditions for Boolean functions to be a negabent function for both an even and an odd number of variables and also determined the nega-Hadamard transform distribution of negabent functions. Further, a method to construct bent-negabent functions was provided. In [11], Zhang et al. presented two methods for constructing bent-negabent functions by using the indirect sum construction (proposed by Carlet in 2004 [12]).


## 2. Definitions and Notations

In this section we introduce a few basic concepts and notations. LetF2 denote the finite field with two elements. We denote by Bn the set of all Boolean functions of n-variable, that is, of all the functions from F2n into F2. The set of integers, real numbers, and complex numbers are denoted by Z, R, and C, respectively. The addition over Z, R, and C is denoted by +,∑i‍. The addition over F2n for all n≥1 is denoted by ⊕,⨁i. The Hamming weight wt(x) of an element x=(x1,x2,…,xn)∈F2n is the number of ones in x; that is, wt(x)=∑i=1n‍xi. We say that a Boolean function is balanced if its truth table contains an equal number of 0’s and 1’s; that is, if its Hamming weight equals wt(f)=2n-1. The Hamming distance between two functions f(x) and g(x), denoted by d(f,g), is the Hamming weight of f⊕g; that is, d(f,g)=wt(f⊕g).Any Boolean functionf(x)∈Bn, where x=(x1,x2,…,xn)∈F2n, is generally represented by its algebraic normal form (ANF): (1)fx1,x2,…,xn=⨁u∈F2nλu∏i=1nxiui,where λu∈F2 and u=(u1,u2,…,un)∈F2n. The algebraic degree of f(x), denoted by deg⁡(f), is the maximal value of wt(u) such that λu≠0. A Boolean function is affine if there exists no term of degree strictly greater than 1 in the ANF and the set of all affine functions is denoted by An. An affine function with constant term equal to zero is called a linear function. Any liner function on F2n is denoted by x·ω=x1ω1⊕x2ω2⊕⋯⊕xnωn, where x,ω∈F2n. The nonlinearity of an n-variable function f(x) is nl(f)=min⁡g∈An⁡(d(f,g)), that is, the distance from the set of all n-variable affine functions. If x=(x1,x2,…,xn)∈F2n and y=(y1,y2,…,yn)∈F2n, we define the scalar (or inner) product, respectively, as the intersection by (2)x·y=x1y1⊕x2y2⊕⋯⊕xnyn,x∗y=x1y1,x2y2,…,xnyn.In this paper, we will use the well-known identity (3)wtx⊕y=wtx+wty-2wtx∗y.The cardinality of the setA is denoted by |A|. If z=a+bi∈C, then |z|=a2+b2 denotes the absolute value of z and z¯=a-bi denotes the complex conjugate of z, where i2=-1, a,b∈R.The Walsh-Hadamard transform off∈Bn at any point ω∈F2n is denoted by (4)Hfω=2-n/2∑x∈F2n-1fx⊕ω·x.The nega-Hadamard transform of f∈Bn at any point ω∈F2n is the complex valued function:(5)NHfω=2-n/2∑x∈F2n-1fx⊕ω·xiwtx.A function f∈Bn is a bent function if |Hf(ω)|=1 for all ω∈F2n. Similarly, f is called negabent function if |NHf(ω)|=1 for all ω∈F2n. It is interesting to note that all the affine functions (both odd and even) are negabent. If f is both bent and negabent, we say that f is bent-negabent. They will be interesting as they have extreme properties in terms of two different Fourier transforms.The nega-cross-correlation coefficient off and g at ω is denoted by(6)NCf,gω=∑x∈F2n-1fx⊕gx⊕ω-1ω·x.We define the nega-autocorrelation coefficient of f at ω by (7)NCfω=∑x∈F2n-1fx⊕fx⊕ω-1ω·x.Note thatNCf,g(ω)=(-1)wt(ω)NCg,f(ω). The functions f and g are said to havecomplementary nega-autocorrelation if for all nonzero u∈F2n(8)NCfu+NCgu=0.Definition 1.
Letf(x),g(x)∈Bn, and thesum-of-squares indicator of the nega-cross-correlation between f(x) and g(x) is defined by (9)Δf,g=∑ω∈F2nNCf,g2ω.If f=g, then Δf,f is called thesum-of-squares indicator of the nega-autocorrelation of f and denoted by Δf; that is, (10)Δf=∑ω∈F2nNCf2ω.Note that NCf(0)=2n. Thus, Δf=∑ω∈F2n‍NCf2(ω)≥NCf2(0)=22n. A Boolean function f(x)∈Bn is negabent if and only if NCf(ω)=0 for all ω∈F2n-0. Hence, Δf≥22n, where the equality holds if and only if f is negabent function.


## 3. Some Cryptographic Properties of Boolean Functions by Concatenation

In this section, we will use concatenation of Boolean functions. Letf1,f2∈Bn-1 and f(x)∈Bn. We denote the concatenation of f1,f2 by f1∥f2. So, f=f1∥f2 means that in algebraic normal form (11)fx1,x2,…,xn=1⊕xnf1x1,x2,…,xn-1⊕xnf2x1,x2,…,xn-1.The concatenation simply means that the truth tables of the functions are merged. For f=f1∥f2, the upper half part of the truth table of f corresponds to f1 and the lower half part to f2. The concatenation of affine functions together with certain nonlinear function has been used in several works [13–15].In [6, 9], the function h(x,y)=f∥g=f(x)(1⊕y)⊕g(x)y was studied and the following result was obtained.Theorem 2 (see [6, 9]).
Supposeh∈Bn+1 is expressed as (12)hx,y=fx1⊕y⊕gxyfor all (x,y)∈F2n×F2, where f,g∈Bn. Then the following statements are equivalent.(i)
h is negabent.(ii)
f and g have complementary nega-autocorrelations and NCf,g(u)=0 for all u∈F2n with wt(u)≡1(mod⁡2).(iii)
|NHf(u)|2+|NHg(u)|2=2 for all u∈F2n and NHf(u)/NHg(u) is a real number whenever NHf(u)NHg(u)≠0.In the following, we establish here a new equivalent statement. Also, we give an alternate proof of Theorem13 [6, 9].Theorem 3.
Leth∈Bn+1 be expressed as (13)hx,y=fx1⊕y⊕gxyfor all (x,y)∈F2n×F2, where f,g∈Bn. Then the following statements are equivalent.(1)
h is negabent.(2)
|NHf(u)|2+|NHg(u)|2=2 for all u∈F2n and NHf(u)/NHg(u) is a real number whenever NHf(u)NHg(u)≠0.(3)
f and g are negabent functions and(14)NHfu,NHgu∈±1,±1,±i,±i,ifniseven,±1+i2,±1+i2,±1-i2,±1-i2,ifnisodd,where u∈F2n.Proof.
We first show (1)⇒(2). By using the definition of the nega-Hadamard transform, we compute that(15)NHhu,a=12NHfu+i2NHgu,ifa=0,12NHfu-i2NHgu,ifa=1.As h is negabent, NHh(u,a)=1, we have (16)12NHfu+i2NHgu=1,12NHfu-i2NHgu=1.According to (5), set (17)NHfu=2-n/2a+bi,NHgu=2-n/2c+di,a,b,c,d∈Z. Hence, (18)NHfu=2-na2+b2,NHgu=2-nc2+d2,2-n/22a-d+2-n/22b+ci=1,2-n/22a+d+2-n/22b-ci=1;that is,(19)a-d2+b+c2=2n+1,a+d2+b-c2=2n+1.From (19), we have(20)2-na2+b2+2-nc2+d2=2,ad=bc.Thus, |NHf(u)|2+|NHg(u)|2=2. Since ad=bc, suppose, for all u∈F2n, NHf(u)NHg(u)≠0; then (21)NHfuNHgu=a+bic+di=a+bic-dic+dic-di=ac+bd+bc-adic2+d2=ac+bdc2+d2∈R.
We now show (2)⇒(3). By (2), since |NHf(u)|2+|NHg(u)|2=2 for all u∈F2n, then (22)2-na2+b2+c2+d2=2;that is,(23)a2+b2+c2+d2=2n+1.
Note thata,b,c,d∈Z. There are two cases to be considered: n even and n odd.
Case  1 (n is even). By applyingJacobi’s four-square theorem, (14) has exactly 24 solutions, which are all variations in ± sign and order of (±2n/2,±2n/2,0,0). Further, it is straightforward to check that, among these 24 solutions, the eight tuples (a,b,c,d), in the list below, are also satisfying NHf(u)/NHg(u) which is a real number whenever NHf(u)NHg(u)≠0, (24)a,b,c,d∈±2n/2,0,±2n/2,0,0,±2n/2,0,±2n/2.Therefore, (25)NHfu,NHgu∈±1,±1,±i,±i.So, |NHf(u)|=|NHg(u)|=1, where u∈F2n.
Case  2 (n is odd). Similarly, fromJacobi’s four-square theorem, (14) has exactly 24 solutions, which are all variations in ± sign and order of (±2(n+1)/2,0,0,0) or (±2(n-1)/2,±2(n-1)/2,±2(n-1)/2,±2(n-1)/2). Further, it is straightforward to check that, among these 24 solutions, the eight tuples (a,b,c,d), in the list below, are also satisfying NHf(u)/NHg(u) which is a real number whenever NHf(u)NHg(u)≠0, (26)2n-1/2,2n-1/2,2n-1/2,2n-1/2,2n-1/2,2n-1/2,-2n-1/2,-2n-1/2,2n-1/2,-2n-1/2,-2n-1/2,2n-1/2,2n-1/2,-2n-1/2,2n-1/2,-2n-1/2,-2n-1/2,2n-1/2,2n-1/2,-2n-1/2,-2n-1/2,-2n-1/2,2n-1/2,2n-1/2,-2n-1/2,2n-1/2,-2n-1/2,2n-1/2,-2n-1/2,-2n-1/2,-2n-1/2,-2n-1/2.Then, (27)NHfu,NHgu∈±1+i2,±1+i2,±1-i2,±1-i2.So, |NHf(u)|=|NHg(u)|=1, where u∈F2n.
Summarizing Cases1 and 2, we conclude that f and g are negabent functions if (2) holds.
In the end, we show (3)⇒(1). According to (15), thanks to (14), (1) holds. This completes the proof.In the following, forf=f1∥f2, we discuss a connection among Δf,Δf1,Δf2, and Δf1,f2. At first, according to the proof of Theorem 3 and Corollary 2 in [6, 9], we have the following.Lemma 4 (see [6, 9]).
Letf(x,xn)=f1∥f2∈Bn, x∈F2n-1, xn∈F2, f1,f2∈Bn-1; then(28)NCfω,ωn=NCf1ω+NCf2ω,ifωn=0,NCf1,f2ω--1wtωNCf1,f2ω,ifωn=1,where ω∈F2n-1, ωn∈F2.To obtain a connection amongΔf, Δf1, Δf2, and Δf1,f2, the following lemma is needed.Lemma 5.
Letf,g∈Bn. Then(29)Δf,g=∑α∈F2nNCf,g2α=∑ω∈F2nNCfωNCgω.Proof.
According to the definition of nega-autocorrelation coefficient, we have(30)
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If we useCauchy’s inequality(31)∑iaibi2≤∑iai2∑ibi2to the sum on the right-hand side of (29), we get (32)Δf,g=∑α∈F2nNCf,g2α=∑ω∈F2nNCfωNCgω≤∑ω∈F2nNCf2ω1/2∑ω∈F2nNCg2ω1/2=Δf1/2Δg1/2=ΔfΔg;that is, Δf,g≤ΔfΔg. From Lemmas 4 and 5, we get the following.Theorem 7.
Letf(x,xn)=f1∥f2∈Bn, x∈F2n-1, xn∈F2, f1,f2∈Bn-1. Then(33)Δf=Δf1+Δf2+4Δf1,f2-2∑ω∈F2n-1-1wtωNCf1,f22ω.Proof.
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Theorem7 gives the relationship among Δf, Δf1, Δf2, and Δf1,f2. Furthermore, we have Δf≥Δf1+Δf2, where the equality holds if and only if NCf1,f2(ω)=0 for all ω∈F2n-1. By Lemma 4, we give a construction for generating negabent functions.Corollary 8.
Letf∈Bn-1. Then g∈Bn=f∥f¯ is negabent if and only if f is also negabent functions, where the notation f¯ denotes the complement function of f; that is, f¯=f⊕1.Proof.
Using (15), for any ω∈F2n-1, ωn∈F2, we have (35)NHgω,ωn=12NHfω-i2NHfω,ifωn=0,12NHfω+i2NHfω,ifωn=1,=12∓i2NHfω.Hence (36)NHgω=12∓i2NHfω=NHfω.Since f is negabent, NHfω=1 for all ωn∈F2, completing the proof.There are many ways to construct bent functions inBm+n starting from bent functions in Bm and Bn (see [16, pages 81–96]). Concatenation under certain conditions produces also bent functions of higher dimension (see [15]). In the following, we mainly consider Boolean function(37)gx,xn+1,xn+2=f1x∥f2x∥f3x∥f4x∈Bn+2;that is, the algebraic normal form of g(x,xn+1,xn+2) is (38)gx,xn+1,xn+2=f1⊕xn+1f1⊕f2⊕xn+2f1⊕f3⊕xn+1xn+2f1⊕f2⊕f3⊕f4,where fi∈Bn, i=1,2,3,4, x∈F2n, xn+1,xn+2∈F2. We first establish an important technical formula.Theorem 9.
Let functiong be defined as (37); then(39)NHgω=12NHf1u+i-1aNHf2u+i-1bNHf3u--1a⊕bNHf4u,where ω=(u,a,b)∈F2n+2, u∈F2n, a,b∈F2.Proof.
Using (5), we have (40)NHgω=2-n+2/2∑x,xn+1,xn+2∈F2n+2-1gx,xn+1,xn+2⊕x,xn+1,xn+2·ωiwtx,xn+1,xn+2=12·2-n/2∑x,xn+1,xn+2∈F2n+2-1gx,xn+1,xn+2⊕u·x⊕axn+1⊕bxn+2iwtx,xn+1,xn+2=12·2-n/2∑x∈F2n∑xn+1,xn+2∈F22-1gx,xn+1,xn+2⊕u·x⊕axn+1⊕bxn+2iwtx,xn+1,xn+2=12·2-n/2∑x∈F2n-1f1x⊕u·xiwtx+∑x∈F2n-1f2x⊕u·x⊕aiwtx+1+∑x∈F2n-1f3x⊕u·x⊕biwtx+1+∑x∈F2n-1f4x⊕u·x⊕a⊕biwtx+2=12NHf1u+i-1aNHf2u+i-1bNHf3u--1a⊕bNHf4u.This completes the proof.In (37), if f1=f4=f, f2=f3=f¯, then we obtain the following.Corollary 10.
Letf∈Bn. Then g∈Bn+2=f∥f¯∥f¯∥f is negabent if and only if f is also negabent functions.Proof.
According to (39), we have (41)NHgu,a,b=1--1a⊕b2NHfu-i-1a+-1b2NHfu=-iNHfu,a=0,b=0,NHfu,a=0,b=1,NHfu,a=0,b=1,iNHfu,a=1,b=1.Thus, NHg(u,a,b)∈{NHf(u),±iNHf(u)} for all (u,a,b)∈F2n×F2×F2. Hence, if g is negabent, then f is also negabent. Conversely, if f is negabent, then g is also negabent, completing the proof.


## 4. Nega-Hadamard Transform and Nega-Autocorrelation Coefficients of a Class of Boolean Function

In this section, we mainly study the functiong(x)∈Bn expressed as(42)gx=fAx⊕a⊕b·x⊕c,where f(x)∈Bn, a,b∈F2n, c∈F2, and A is an n×n orthogonal matrix. Here we compute the nega-Hadamard transform and nega-autocorrelation coefficient of g.Theorem 11.
Letg∈Bn, with the same data as above; then(43)NHgω=-1c⊕Ab⊕ω·aiwtaNHfAb⊕ω⊕a,(44)NCgα=-1ATa⊕b·αNCfAα.Proof.
According to (5), we have (45)NHgω=2-n/2∑x∈F2n-1fAx⊕a⊕b·x⊕c⊕ω·xiwtx=-1c2-n/2∑x∈F2n-1fAx⊕a⊕b⊕ω·xiwtx.Setting y=Ax⊕a, since A is orthogonal matrix, then ATA=AAT=I, where AT is the transpose of A and I is the identity matrix; then x=AT(y⊕a). Furthermore, when x ranges over F2n, so do Ax and Ax⊕a. Thus (46)NHgω=-1c2-n/2∑y∈F2n-1fy⊕b⊕ω·ATy+ATaiwtATy⊕a=-1c⊕b⊕ω·ATa2-n/2∑y∈F2n-1fy⊕b⊕ω·ATyiwtATy⊕a=-1c⊕Ab⊕ω·a2-n/2∑y∈F2n-1fy⊕Ab⊕ω·yiwtATy⊕a.Since (47)wtATy⊕a=ATy⊕aTIATy⊕a=y⊕aTAIATy⊕a=y⊕aTIy⊕a=wty⊕a,iwty⊕a=iwty+wta-2wty∗a=iwty+wtai-2wty∗a=iwty+wta-1-wty∗a=iwty+wta-1wty∗a=iwty+wta-1y·a,which implies that (48)NHgω=-1c⊕Ab⊕ω·a2-n/2∑y∈F2n-1fy⊕Ab⊕ω·yiwty⊕a=-1c⊕Ab⊕ω·a2-n/2∑y∈F2n-1fy⊕Ab⊕ω·yiwty+wta-1y·a=-1c⊕Ab⊕ω·aiwta2-n/2∑y∈F2n-1fy⊕Ab⊕ω⊕a·yiwty=-1c⊕Ab⊕ω·aiwtaNHfAb⊕ω⊕a.
Thus (43) holds. Next we will compute (44). Set (49)hx=gx⊕gx⊕α⊕α·x=fAx⊕a⊕b·x⊕c⊕fAx⊕α⊕a⊕b·x⊕α⊕c⊕α·x=fAx⊕a⊕fAx⊕Aα⊕a⊕b·α⊕α·x.So by using (6), we get (50)NCgα=∑x∈F2n-1gx⊕gx⊕α⊕α·x=∑x∈F2n-1fAx⊕a⊕fAx⊕Aα⊕a⊕b·α⊕α·x=-1b·α∑x∈F2n-1fAx⊕a⊕fAx⊕a⊕Aα⊕α·x.Setting y=Ax⊕a, as A is orthogonal matrix, then x=AT(y⊕a)=ATy⊕ATa. Therefore, (51)NCgα=-1b·α∑y∈F2n-1fy⊕fy⊕Aα⊕α·ATy⊕ATa=-1ATa⊕b·α∑y∈F2n-1fy⊕fy⊕Aα⊕α·ATy=-1ATa⊕b·α∑y∈F2n-1fy⊕fy⊕Aα⊕Aα·y=-1ATa⊕b·αNCfAα.This completes the proof.By Theorem11, we can easily get the following results proved in [6, 9, Theorem 1 (a) and (d)].Corollary 12.
Letf(x)∈Bn; then one obtains the following.(a)
ConsiderNHf¯(ω)=-NHf(ω), ω∈F2n.(b)
Ifg(x)=f(Ax⊕a), then NHg(ω)=(-1)Aω·aiwt(a)NHf(Aω⊕a), where A is an n×n orthogonal matrix, a∈F2n.It is known that iff(x) is a bent function in (42), then the function g(x) is also bent, where A is an n×n nonsingular matrix. The Boolean function f(x)∈Bn is a negabent function if |NHf(ω)|=1. Therefore, according to (43), we get that if f(x) is a negabent function, then g is also negabent. The following result summarizes this discussion.Corollary 13.
With the same data as in Theorem3, then if f(x) is bent-negabent, g(x) is also bent-negabent.In (42), by choosing some special cases and Corollary 13, we have the following.Corollary 14.
Letf(x)∈Bn be a bent-negabent function; then one obtains the following.(a)
f1(x)=f(x⊕ω) is bent-negabent, where ω∈F2n.(b)
f2(x)=f(x)⊕a·x⊕b is bent-negabent, where a∈F2n, b∈F2.Remark 15.
Corollary12 was mentioned in [3, Lemma 2], and Corollary 10 was proved in [4, Theorem 2] by applying [3, Lemma 2]. However, if we use Theorem 11, these results are easily obtained.


## 5. Conclusion

In this paper, the special Boolean functions by concatenation are presented. We investigate their nega-Hadamard transforms, nega-autocorrelation coefficients, sum-of-squares indicators, and so on. We establish a new equivalent statement onf1∥f2 which is negabent function. Also, we give an alternate proof of Theorem 13 [6, 9]. Based on them, the construction for generating the negabent functions by concatenation is given. Finally, the function expressed as f(Ax⊕a)⊕b·x⊕c is discussed. The nega-Hadamard transform and nega-autocorrelation coefficient of this function are derived. By applying these results, some properties are obtained. We hope that these results will be helpful in further studying of Boolean functions.



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