The algebraic immunity and the optimal algebraic immunity functions of a class of correlation immune H Boolean functions

. We put forward an efficient method to study the algebraic immunity of H Boolean functions with Hamming weight of , getting the existence of the higher-order algebraic immunity functions with correlation immunity. We also prove the existing problem of the above 2-order algebraic immunity functions and the optimal algebraic immunity functions. Meanwhile, we solve the compatibility of algebraic immunity and correlation immunity. What is more, the main theoretical results are verified through the examples and are revealed to be correct. Such researches are important in cryptographic primitive designs, and have significance and role in the theory and application range of cryptosystems.


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 [1], 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. Among them, the algebraic immunity of Boolean functions is current central issues [2~6].
The H Boolean functions have the propagation property. When we study the correlation immunity and algebraic immunity of the H Boolean functions, it is equivalent to study the compatibility of the propagation property, algebraic immunity and correlation immunity of Boolean functions. It is the basis of studying the correlation immunity and algebraic immunity of the H Boolean functions to study the correlation immunity and algebraic immunity of the H Boolean functions with the Hamming weight of 1 2 2 2 n n . In this paper, using the derivative and e-derivative [7~9] as research tools, we study the importance of study cryptographic properties of H Boolean function with Hamming weight of 1 2 2 2 n n .

Preliminaries
To study cryptographic properties of H Boolean functions, we proposed the concept of the e-derivative. The ederivative and derivative are defined here as Definition 1&2.

Definition 1:
The e-derivative (e-partial derivative) of n-dimensional Boolean functions for r variables 1 2 , , , is defined as for a single variable, which is denoted by ( ) / i ef x ex ( 1,2, , ) i n " . As a result, the simplified form below can be easily derived.

Definition 2:
The derivative (partial derivative) of ndimensional Boolean functions for r variables 1 2 , , , is defined as If 1 r , (2) turns into the derivative of 1 2 ( ) ( , , , ) for a single variable, which is denoted by . As a result, the simplified form below can be easily derived.

3:
The cascade function 2) If 1 2 m m m ,and there doesn't exist ( ) , and x g is an annihilator of the lowest algebraic degree of a. If m-degree terms of 1 ( ) g x and 2 ( ) g x are unequal, or at least one m-degree term is unequal in many mdegree terms, b. If all the m-degree terms of 1 ( ) g x and 2 ( ) g x are equal, but the second high-degree terms are unequal, or at least one of many second high-degree terms is unequal, ( ( )) AI f x m .

Remark:
The results of Theorem 1 are based on the Definition 3 of cascade in this paper.
For the proof of Theorem 1, attention should only be paid that we should take Proofs of other clauses in Theorem 1 are easy, so we omit it here. Theorem 2 will further discuss the existence of correlation immunity H Boolean functions whose algebraic immunity is not less than 2 (that ( ( )) 2 AI f x t ), and the existence of the optimal algebraic immunity H Boolean functions with correlation immunity.
then we can get 1) There are ( )