Two Indicators of Cross-Correlation for the Functions from nq to 2 q

: Boolean functions play an important role in the design of secure cryptosystems and code division multiple access (CDMA) communication. Several possible generalizations of Boolean functions have been obtained in recent years. In this paper, we analyze the properties of functions from 2 o t nq q in terms of their Walsh-Hadamard transform (WHT). We provide a relationship between cross-correlation and the WHT of these functions. Also, we present a necessary and sufficient condition for the functions to have complementary autocorrelation. The Parseval’s identity for the current setup of these functions is obtained. Further, we obtained the modulus indicator (MI) and the sum-of-squares-modulus indicator (SSMI) of cross-correlation among two functions for the current setup.


Introduction
Data security has become very important for society in the present digital age.Cryptography is the art of writing secure data.The main advancement in the field of cryptography marks with the introduction of Boolean functions [1][2].Boolean bent functions were introduced by Rothaus [3] and have many applications in the fields of cryptography and coding theory.Bent functions are those Boolean functions which have the maximum distance from the set of all affine functions (equivalently, if they have a flat spectrum with respect to Walsh-Hadamard transform (WHT).In previous years, various authors [4][5][6][7][8][9][10] have presented many generalizations of Boolean functions and studied the effect of WHT on these functions.Here, in this article, we consider the generalized functions from 2 to n qq with 2 q  is a positive integer.The WHT is a crucial component in analyzing several properties of Boolean functions as well as their generalizations.The Hamming distance of Boolean functions and other coding and decoding axioms are investigated through WHT.We analyze some properties of functions from 2 to n qq with respect to the WHT.The results presented in this paper are useful in the design of secure coding and decoding algorithms.Let , and be the set of integers, real numbers, and complex numbers, respectively.Let 2 q  and n be the positive integers.Let 2 n be the vector space of dimension n over the field 2 , and q be the ring of integers
Further, Schmidt [11] has provided the construction of bent functions for the functions from n qq Let ξ be the primitive 2q-th root of unity and ζ be the q-th root of unity, then the WHT of , nq g  is given by [11] ( ) , /2 , 1 () where ,  zw is the inner product of and in .
n q zw Suppose , ,. nq gh  Then, the cross-correlation [12] of g and h at any n q gg g g g The two indicators: sum-of-squares indicator and the absolute indicator were introduced by Zhang et al. [13].These indicators of Boolean functions are referred to as the global avalanche characteristics among them.Further, corresponding to these indicators, Singh et al. [14] have given two similar indicators: the modulus indicator (MI) and the sum-of-squares-modulus indicator (SSMI) of cross-correlation among two q-ary functions.They have investigated several properties of q-ary functions with respect to MI and SSMI.Also, they have obtained lower as well as upper bounds for these two indicators.Following a similar concept, we have defined two indicators in the present setup.For possible applications in cryptographic algorithms, the functions must possess a smaller correlation.The SSMI and MI bounds obtained for the functions in the current setup show that these functions have importance in communication algorithms and code division multiple access (CDMA) systems.
The SSMI of The structure of the article is as follows: Section 2 presents the inverse WHT of the functions belonging to the present setup.Also, we present a link between the WHT and the cross-correlation of these functions.The Parseval's identity is obtained for these functions.Section 3 presents a result related to the complementary autocorrelation of these functions.It is also shown that is a bent function if and only if , where n is a positive integer.Then, we have In this section, we provide the inverse WHT and cross-correlation of the function from n q to 2q .Next, we present a relationship among the cross-correlation and WHT of these functions.Theorem 1. Suppose f is a function from n q to 2q and .
Proof.The WHT for the function f is given by /2 ( ) , .() Taking right hand side of Equation ( 1) . .
Proof.Taking right hand side of Equation ( 2) .
Suppose f is a function from n q to 2q , and . Then, we have

Complementary autocorrelation of functions
In this section, we present a result on complementary autocorrelation of the functions     .
nq fg  Let f and g on n q acquires complementary autocorrelation.Then Adding Equation ( 5) and Equation ( 6), we get which gives that for all ,   st qq

 vw
Hence, the function h is a bent.
Conversely, suppose that h is a bent.Our aim is to prove that 12 and gg are also bent functions.Suppose that the function 1 g is not a bent.Then, there must exists , if and only if .
Hence the result.
2. Using part 1, we have  , and this is a contradiction to the identity (4).Equivalently, there is not only , ( n rr  distinct ( 1) ) if and only if, for any ( ) ( ) ,?

 v
This proves that both g and h are generalized bent.

Conclusion
Boolean functions and their various existing generalizations have numerous applications in the design of cryptosystems and coding theory.In this article, the cryptographic properties of the functions from 2 to .

n qq
We have obtained the inverse WHT for the functions from n q to 2 .
q Also, we provided a relationship between the WHT and the cross-correlation of these functions.The Parseval's identity is provided in the current setup.The complementary autocorrelation of these functions is studied.The concept of the indicators (SSMI and MI) is extended to the functions of the current setup.We analyzed these indicators and provided lower as well as upper bounds for them.The results of this article are useful for their applications in wireless communication systems.

2 . 4 |
Suppose f and g are the functions from n q to 2q , Amit Paul, et al.
Now, taking left hand side of the Equation (

Theorem 4 .
Any two functions g and h have complementary autocorrelation if ( ) Any functions f and g are considered to possess complementary autocorrelation if 22 | ( ) | | ( ) | 2, a contradiction to Parseval's identity, (4).Therefore, for at least one 0 , Therefore, the functions f and g are affine.
the functions g and h are perfectly uncorrelated.
[4]lean function on n variables.Let n denote the set of all Boolean functions on n variables.Kumar et al.[4]have presented the concept of generalized bent functions which is defined as : n qq g → .The set of all such functions from to n qq is denoted by , nq .Let + denote the addition over , and , and  denote the addition over .

correlation indicators for the functions 2 n qq →
ghby using the definition of , , gh  in the following result.