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CHARACTERIZATION OF GENERALIZED NEGABENT FUNCTIONS Rashmeet Kaur¹

Department of Mathematics , National Institute of Technology, Raipur, Raipur, 492010, Chhattisgarh, India

Deepmala Sharma²

Department of Mathematics, National Institute of Technology, Raipur, Raipur, 492010, Chhattisgarh, India

Abstract In this article, we study generalized negabent Boolean functions from . We present characterization of generalized negabent Boolean functions in terms of negabent functions. The results are analogous to generalized bent functions.

1 INTRODUCTION

Boolean bent functions introduced by Rothaus [4] serves as an important combinatorial objects having wide range of applications in coding theory, the difference set theory, and cryptography.

Even though, the researchers have obtained few classes of bent functions, their complete characterization is still difficult. On the other hand, researchers have focused on generalization of Boolean functions which was introduced much later in [7]. Schmidt [2] considered the function from and these functions have drawn more attention. In [3] Pott and Parker introduced negabent functions by considering the nega hadamard transform. Negabent functions have flat spectra under nega hadamard transform. The functions which are both bent and negabent are called bent- negabent functions. The generalization of negabent function is called generalized negabent function [1]. Generalized negabent functions are the functions having flat spectrum with respect to generalized nega hadamard transform.

In this article, we study generalized negabent function and present characterization of generalized negabent functions on Further we give a construction of generalized negabent functions on

2 PRELIMINARIES

We denote the set of integers by Z and Zr represents the ring of integers modulo r.

The addition over Z is represented by +, whereas ⊕ represents the addition over Boolean function is a function

from where is an n-

dimensional vector space over The set of all n-variable Boolean functions is denoted by Bn. Support of a Boolean is

defined by The

Hamming weight of a Boolean function is defined by the set The Hamming distance between two Boolean functions is defined by the set The Algebraic Normal form of a Boolean function is defined as

where

The algebraic degree of f is defined by the number of variables in highest order monomial with non-zero coefficient.

Affine Boolean functions are the functions with algebraic degree at most one. The set of all n-variable affine Boolean functions is denoted by If the constant term of affine Boolean function is zero, then it is called linear Boolean function

If and

are in then the scalar product is defined by

The Walsh transform of f at any point is defined by

A Boolean function with n even is said to be a bent function if and only if

for all

Generalized Boolean function on n- variables is defined as a function from

a positive integer) [5]. The set of all n-variable generalized Boolean function is denoted by

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2 The generalized walsh hadamard transform of at any point u is defined by

where is the q-th primitive root of unity

A function is said to be generalized bent function if

The nega hadamard transform of f at is given by

A function f is said to be negabent if and only if

The generalized nega hadamard transform of any Boolean function f at any point u is defined by

A Boolean function

for all is said to be a generalized negabent function.

3 RESULTS ON GENERALIZED NEGABENT FUNCTIONS

In this section, we present some results on characterization of generalized negabent functions.

Theorem 1.

then

roof. We compute

Now, by puttin

we get

we may rewrite this equality

Theorem 2

then

Proof. We compute

)

Now, by putting

and

we obtain

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3 from which we derive our resu

Theorem 3 Z be

defined

Proof. We compute generalized negahadamard coefficie

Now, by taking square of norm, we obain

Since and ⊕ r

are all negabent functio

By using imposed conditions, the remaining coefficients are all zero

So, f is generalized negabent function n.

4 CONCLUSION

In this article, we have presented results on characterization of generalized negabent functions defined . Moreover, we present a construction of generalized negabent function

REFERENCES

1. Chaturvedi and A. K. Gangopadhyay, ”On Generalized NegaHadamard Transform,” in Quality, Reliability, Security and Robustness in Heterogeneous Networks (Lectures Notes of the Institute for Computer Sciences, Social Informatics and Telecommunication Engineering), vol. 115, Springer, Berlin, Heidelberg, 2013, pp.

771-777.

2. K. Schmidt, ”Quaternary constant- amplitude codes for multicode CDMA,”

IEEE International Symposium on Information Theory, vol. 55, no. 4, pp.

1824-1832, 2009.

3. M. G. Parker and A. Pott, ”On Boolean functions which are bent and negabent,” in Sequences, Subsequences, and Consequences (Lecture Notes in Computer Science), vol. 4893. Berlin, Germany:

Springer-Verlag, 2007, pp. 9-23.

4. O. S. Rothaus, ”On bent functions,” J.

Combinat. Theory, Ser. A, vol. 20, no. 3, pp. 300-305, 1976.

5. P. Sole and N. Tokareva, ”Connections between quaternary and binary bent functions,”

http://eprint.iacr.org/2009/544.pdf, 6. P. Stanica, T. Martin, S. Gangopadhyay

and B. K. Singh, ”Bent and generalized bent Boolean functions,” Des. Codes Cryptogr., vol. 69, pp. 77-94, 2013.

7. P.V Kumar, R.A Scholtz and L.R Welch,

”Generalized bent functions and their properties,” J. Combinat. Theory, Ser. A, vol. 40, no. 1, pp. 90- 107, 1985.