The Walsh Transform of a Class of Boolean Functions

: The Walsh transform is an important tool to investigate cryptographic properties of Boolean functions. This paper is devoted to study the Walsh transform of a class of Boolean functions defined as


Introduction
Boolean functions are important objects in discrete mathematics. They play a role in symmetric cryptography and error-correcting coding theory, and they also have a significant influence on the design and analysis of cryptographic algorithms. The Walsh transform is a vital tool to investigate cryptographic properties of Boolean functions. Some important properties of cryptographic functions, such as resiliency and nonlinearity can be characterized by their Walsh transform [1][2][3] . An interesting problem is to find Boolean functions with few Walsh transform values and determine their distributions.
Bent functions introduced by Rothaus [4] in 1976 are interesting combinatorial objects with maximum Hamming distance to the set of all affine functions, but they cannot be used in cryptography directly since they exist only in an even number of variables and are not be balanced. Such functions have been extensively studied because of their important applications in coding theory [5,6] , cryptography [7] , and sequence designs [8] . To get balanced functions with high nonlinearity in odd or even number of variable, Carlet [9] generalized the bent functions to plateaued functions and they take Walsh transform values 0, 2 k  for a fixed positive integer k . Semi-bent, as a particular case, is an important kind of Boolean functions with three Walsh transform values. In Ref. [10], some classes of Boolean functions with four-valued Walsh spectra are presented by complementing the values of bent functions at two points, one of which is zero and the other is nonzero, and their Walsh spectrum distributions are determined finally. Inspired by this work, recently Jin et al [11] presented three classes of Boolean functions with six-valued Walsh spectra, which were derived from bent functions by complementing their values at the zero and another two nonzero points, and determined their Walsh spectrum distributions with a similar method. In Ref. [12], some classes of Boolean functions with five Walsh transform values were presented by adding the product of two or three linear functions into some known bent functions, and their Walsh spectrum distributions were determined finally. In Ref. [13], Tang et al gave a generic construction of bent functions defined as , ( ) g x is any known bent function over 2 n F satisfying some conditions, 1 2 ( , , , ) F X X X   is an arbitrary polynomial in 2 1 2 [ , , , ] F X X X   . In particular, the cases of ( , , ) F X X X X X X  and 1 2 ( , ) F X X  1 2 X X have been studied by Xu et al [12] and Mesnager [14] , respectively. The purpose of this paper is to present the Walsh transform of the Boolean function defined as 1 1 where ( ) f x and ( ) h x are Boolean functions over 2 n F and 2 n F   . In particular, the case of has been studied by Pang et al [15] . This paper is organized as follows. In Section 1, we give some basic concepts and results. In Section 2, we present the Walsh transform of the Eq. (1). In Section 3, we conclude this paper.  For any positive integer | k n , the trace function from 2 n F to 2 k F is the mapping defined as

Preliminaries
is called the absolute trace function. Some important and useful properties of the trace function are provided in the following: 1) 3) For any Tr ( ) n  satisfies the transitivity property, that is, 1 1 Tr ( ) Tr (Tr ( )) n m n m    .

5) For any
Let f be a Boolean function from 2 n F to 2 F , and the set of which is denoted by n B . The Walsh The nega-Hadamard transform of ( )

Main Results
Let n be a positive integer and f be a Boolean function from 2 n F to 2 F . For any The relationship between ( ) is given in Theorem 1.
Then, the Walsh transform of ( ) g x at ( ) 2) The proof of 2) is obvious from 1).
In Ref. [16], first they presented the necessary and sufficient conditions for the functions 2    it is the one discussed in Ref. [16].
Further, by using some permutation trinomials over 2 n F , they presented some classes of negabent functions of the form

Conclusion
In this paper, we proposed the Walsh transform of a class of Boolean functions by using the properties of the Walsh transform and the trace function. Then, we hope that we can deduce the Walsh spectrum distributions of ( ) g x defined as Eq. (1) by suitable choices of ( ) f x and ( ) h x . Further, several new classes of Boolean functions with few Walsh transform values are obtained.