The Nonlinearity of Sum and Product for Boolean Functions

Abstract. In this paper, we study the relationship between the nonlinearity of Boolean function and the nonlinearity of the sum and product of Boolean function, while derivative and e-derivative are used to study the problem further . We obtain that the sum of two functions’ nonlinearity is not less than the nonlinearity of the sum of two functions. The relationship between the nonlinearity of function and the nonlinearity of the sum and product of two functions are also obtained. Furthermore, we also get the relationship between the nonlinearity of the product of functions, and the derivative and e-derivative of function. Moreover, we also deduced some important applications on the basis of the above work.


Introduction
The cipher security is the core of the cryptosystem, and only a cryptosystem with good security has an existing significance.Boolean functions with a variety of secure cipher properties are the key factors to design the cryptosystem with the ability to resist multiple cipher attacks and good safety performance.It is of great importance for a security cryptosystem to study some properties of Boolean functions, which make the cryptosystem resist various attacks, such as high algebraic degree, high nonlinearity, the strict avalanche criterion and propagation, higher-order correlation immunity and higher-order algebraic immunity.Therefore, there are some important research problems, such as the existence, the feature, the design, the construction and the count of Boolean functions with some kind of secure cryptographic property [1~7].
The nonlinearity of Boolean functions is a measure of the ability to resist the affine approximation attack.The higher the nonlinearity of Boolean functions, the stronger the ability to resist the affine approximation attacks.It is an important task to study the nonlinearity of Boolean functions [8~9].
With the sum or product of two Boolean functions, the Boolean functions with good cryptographic properties can be obtained.As the sum of a 2 time homogeneous H Boolean function and a / 2 n monomial can be a Bent function.Therefore, it is also necessary to study the cryptographic properties of the sum of the function or the product function of two Boolean functions, such as nonlinearity.
In this paper, we will discuss whether there is a triangle inequality between the nonlinearity of Boolean functions and the sum of two functions.At the same time, the derivative and e-derivative of the Boolean function are used to reveal the relationship between the nonlinearity of Boolean function and the sum or the product of the two functions.

Preliminaries
To study cryptographic properties of H Boolean functions, we proposed the concept of the e-derivative [10~12].The e-derivative and derivative are defined here as Definition 1&2.
Definition 1: The e-derivative (e-partial derivative) of n-dimensional Boolean functions , , , is defined as .As a result, the simplified form below can be easily derived.
, , ,0, , , ) , , , is defined as for a single variable, which is denoted by . As a result, the simplified form below can be easily derived.

Definition 3:
The cascade function ) x GF " is defined as According to Definition 1~3, we can get Lemmas 1 easily.
Lemma 1: For any arbitrary Boolean function ( ) f x , the following equations are true: ( )

The triangle inequality for the nonlinearity of the sum of two Boolean functions
We are always wondering if there exists a relationship among the nonlinearity of two Boolean functions and the sum function ,which can be represented by a triangle inequality.Theorem 1 just reveals the triangle inequality.Theorem 1: Suppose Boolean functions N , the nonlinearity of the sum N are contacted with the following triangle inequality : And there exists 0 ( ) [ ] There are , there must be .

Proof
) The proof ends.Theorem 1&2 reveal that the nonlinearity of the sum of the two Boolean functions is not more than the sum of nonlinearity of two functions, and nonlinearity of the sum of the two functions and the nonlinearity of product of two functions.For the nonlinearity of the sum of two functions with two functions of the relationship between the nonlinearity of a single function, need to use ederivative and derivative to discuss, back to do the work.

The nonlinearity of Boolean functions with the derivative and e-derivative
In Theorem 3, we will discuss the relationship between the nonlinearity of the sum of two Boolean functions, and the nonlinearity of a single function.