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Mathematical Problems in Engineering
Volume 2015, Article ID 959617, 8 pages
http://dx.doi.org/10.1155/2015/959617
Research Article

On Negabent Functions and Nega-Hadamard Transform

School of Mathematical Science, Huaibei Normal University, Huaibei, Anhui 235000, China

Received 29 June 2015; Revised 23 October 2015; Accepted 26 October 2015

Academic Editor: Laura Gardini

Copyright © 2015 Zepeng Zhuo and Jinfeng Chong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 on which is negabent function. Based on them, the construction for generating the negabent functions by concatenation is given. Finally, the function expressed as is discussed. The nega-Hadamard transform and nega-autocorrelation coefficient of this function are derived. By applying these results, some properties are obtained.

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 -fold tensor products of the identity mapping , Walsh-Hadamard transformation , and nega-Hadamard transformation , where . 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 -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 is a cubic negabent function and the 4-variable function is bent-negabent.

Negabent functions and bent-negabent functions have been extensively studied during the last few years [311]. 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 -variable negabent function is at most . 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. Let denote the finite field with two elements. We denote by the set of all Boolean functions of -variable, that is, of all the functions from into . The set of integers, real numbers, and complex numbers are denoted by , , and , respectively. The addition over , , and is denoted by . The addition over for all is denoted by . The Hamming weight of an element is the number of ones in ; that is, . We say that a Boolean function is balanced if its truth table contains an equal number of ’s and ’s; that is, if its Hamming weight equals . The Hamming distance between two functions and , denoted by , is the Hamming weight of ; that is, .

Any Boolean function , where , is generally represented by its algebraic normal form (ANF): where and . The algebraic degree of , denoted by , is the maximal value of such that . 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 affine function with constant term equal to zero is called a linear function. Any liner function on is denoted by , where . The nonlinearity of an -variable function is , that is, the distance from the set of all -variable affine functions. If and , we define the scalar (or inner) product, respectively, as the intersection by In this paper, we will use the well-known identity

The cardinality of the set is denoted by . If , then denotes the absolute value of and denotes the complex conjugate of , where , .

The Walsh-Hadamard transform of at any point is denoted by The nega-Hadamard transform of at any point is the complex valued function:A function is a bent function if for all . Similarly, is called negabent function if for all . It is interesting to note that all the affine functions (both odd and even) are negabent. If is both bent and negabent, we say that is bent-negabent. They will be interesting as they have extreme properties in terms of two different Fourier transforms.

The nega-cross-correlation coefficient of and at is denoted byWe define the nega-autocorrelation coefficient of at by

Note that . The functions and are said to have complementary nega-autocorrelation if for all nonzero

Definition 1. Let , and the sum-of-squares indicator of the nega-cross-correlation between and is defined by If , then is called the sum-of-squares indicator of the nega-autocorrelation of and denoted by ; that is, Note that . Thus, . A Boolean function is negabent if and only if for all . Hence, , where the equality holds if and only if is negabent function.

3. Some Cryptographic Properties of Boolean Functions by Concatenation

In this section, we will use concatenation of Boolean functions. Let and . We denote the concatenation of by . So, means that in algebraic normal form The concatenation simply means that the truth tables of the functions are merged. For , the upper half part of the truth table of corresponds to and the lower half part to . The concatenation of affine functions together with certain nonlinear function has been used in several works [1315].

In [6, 9], the function was studied and the following result was obtained.

Theorem 2 (see [6, 9]). Suppose is expressed as for all , where . Then the following statements are equivalent.(i) is negabent.(ii) and have complementary nega-autocorrelations and for all with .(iii) for all and is a real number whenever .

In the following, we establish here a new equivalent statement. Also, we give an alternate proof of Theorem   [6, 9].

Theorem 3. Let be expressed as for all , where . Then the following statements are equivalent.(1) is negabent.(2) for all and is a real number whenever .(3) and are negabent functions andwhere .

Proof. We first show (1)(2). By using the definition of the nega-Hadamard transform, we compute thatAs is negabent, , we have According to (5), set Hence, that is,From (19), we haveThus, . Since , suppose, for all , ; then We now show (2)(3). By (2), since for all , then that is,Note that . There are two cases to be considered: even and odd.
Case  1 ( is even). By applying Jacobi’s four-square theorem, (14) has exactly 24 solutions, which are all variations in sign and order of . Further, it is straightforward to check that, among these 24 solutions, the eight tuples , in the list below, are also satisfying which is a real number whenever , Therefore, So, , where .
Case  2 ( is odd). Similarly, from Jacobi’s four-square theorem, (14) has exactly 24 solutions, which are all variations in sign and order of or . Further, it is straightforward to check that, among these 24 solutions, the eight tuples , in the list below, are also satisfying which is a real number whenever , Then, So, , where .
Summarizing Cases and , we conclude that and 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, for , we discuss a connection among , and . At first, according to the proof of Theorem and Corollary in [6, 9], we have the following.

Lemma 4 (see [6, 9]). Let , , , ; thenwhere , .

To obtain a connection among , , , and , the following lemma is needed.

Lemma 5. Let . Then

Proof. According to the definition of nega-autocorrelation coefficient, we have

Remark 6. If we use Cauchy’s inequalityto the sum on the right-hand side of (29), we get that is, . From Lemmas 4 and 5, we get the following.

Theorem 7. Let , , , . Then

Proof. Applying (28) and (29), we have Theorem 7 gives the relationship among , , , and . Furthermore, we have , where the equality holds if and only if for all . By Lemma 4, we give a construction for generating negabent functions.

Corollary 8. Let . Then is negabent if and only if is also negabent functions, where the notation denotes the complement function of ; that is, .

Proof. Using (15), for any , , we have Hence Since is negabent, for all , completing the proof.

There are many ways to construct bent functions in starting from bent functions in and (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 functionthat is, the algebraic normal form of is where , , , . We first establish an important technical formula.

Theorem 9. Let function be defined as (37); thenwhere , , .

Proof. Using (5), we have This completes the proof.

In (37), if , , then we obtain the following.

Corollary 10. Let . Then is negabent if and only if is also negabent functions.

Proof. According to (39), we have Thus, for all . Hence, if is negabent, then is also negabent. Conversely, if is negabent, then 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 function expressed aswhere , , , and is an orthogonal matrix. Here we compute the nega-Hadamard transform and nega-autocorrelation coefficient of .

Theorem 11. Let , with the same data as above; then

Proof. According to (5), we have Setting , since is orthogonal matrix, then , where is the transpose of and is the identity matrix; then . Furthermore, when ranges over , so do and . Thus Since which implies that Thus (43) holds. Next we will compute (44). Set So by using (6), we get Setting , as is orthogonal matrix, then . Therefore, This completes the proof.

By Theorem 11, we can easily get the following results proved in [6, 9, Theorem (a) and (d)].

Corollary 12. Let ; then one obtains the following.(a)Consider , .(b)If , then , where is an orthogonal matrix, .

It is known that if is a bent function in (42), then the function is also bent, where is an nonsingular matrix. The Boolean function is a negabent function if . Therefore, according to (43), we get that if is a negabent function, then is also negabent. The following result summarizes this discussion.

Corollary 13. With the same data as in Theorem 3, then if is bent-negabent, is also bent-negabent.

In (42), by choosing some special cases and Corollary 13, we have the following.

Corollary 14. Let be a bent-negabent function; then one obtains the following.(a) is bent-negabent, where .(b) is bent-negabent, where , .

Remark 15. Corollary 12 was mentioned in [3, Lemma ], and Corollary 10 was proved in [4, Theorem ] by applying [3, Lemma ]. 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 on which is negabent function. Also, we give an alternate proof of Theorem [6, 9]. Based on them, the construction for generating the negabent functions by concatenation is given. Finally, the function expressed as 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Natural Science Foundation of Anhui Higher Education Institutions of China (nos. KJ2014A220 and KJ2014A231).

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