Construction of the optimal algebraic immunity Boolean functions with correlation immunity from transformation RSBFs

In this paper, we discuss the effect of the cryptographic properties of translation transformation rotation symmetric Boolean functions. On construction of rotation symmetric Boolean functions with the optimal algebraic immunity, we construct correlation immunity Boolean functions with the optimal algebraic immunity by translation transformation and concatenation transformation.


Introduction
The algebraic immunity, optimal algebraic immunity, nonlinearity, diffusion and correlation immunity are all important security properties of cryptographic functions and research contents of cipher security.The algebraic immunity and optimal algebraic immunity of algebraic attacks are the hot spots in the current research.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 Bent function and the rotational symmetric Boolean functions(RSBFs) are important functions in cryptography.If there is a Bent function in the RSBFs, and how to construct the optimal algebraic immune function with the Bent function are important problems to be studied [8~14].In this paper, we discuss the existence and structure of rotational symmetric Bent function using derivative, translation and cascade, and the problem of the optimal algebraic immune function of the immune system, which is based on the Bent function of the rotational symmetry.It is more convenient to judge the diffusion by the derivative calculation than by the diffusion.

Preliminaries
To study cryptographic properties of Boolean functions, we proposed the concept of the e-derivative [15~17].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

If 1 r
, (1) turns into the e-derivative of .As a result, the simplified form below can be easily derived.

Definition 4:
For any arbitrary ) ( ) ( , , , ) ), then Boolean functions ( ) f x was called a m-order correlation immune function, m was called order.The correlation immunity of order m and the correlation immunity order are both written as ( ) CI m .

Definition 5:
. The algebraic degree of annihilators of the lowest algebraic degree in all nonzero annihilators of ( ) f x and 1 ( ) f x are called algebraic immunity order which is written as ( ( )) AI f x or ( ) AI f .( ) f x are optimal algebraic immunity functions when ( ) 2 , and 2 ) x GF " is defined as According to the above Definitions, we can get Lemmas easily.
Lemma: For any arbitrary Boolean function ( ) f x , the following equations are true: ( )

The impact of Translation transform on the properties of RSBFs
In this part, we first discusses the influence of translation transformation on the cryptographic properties of the symmetric Boolean functions.In this paper, we take (0,0, ,0) 5 " .Proof: of 0 ( ) r f x 5 , the results own the same highest degree terms with 0 ( ) r f x .And among the newly generated item whose degree is less than 0 deg ( ) r f x 's, the items with even number will be fully offset, and items with odd number will be retained.

Construction of higher-order algebraic immunity functions with correlation immunity
We know fully rotational symmetry Boolean function under the translational transform remained a lot of good cryptographic properties.Theorem 4 first looking for higher-order algebraic immune completely rotation symmetric Boolean function, on the basis of this again by translational transform to construct high order algebraic immunity related to immune function.
Theorem 4: Let x be a vector whose dimension is not less than 2n , and n is an odd.Since , and are n-order algebraic immunity functions.And when the dimension of x is equal to 2n , 2 0 0 ( ) ( ) n f x f x which meet the aforementioned conditions are optimal algebraic immunity functions. Proof: , each n-degree item is gated by the addition of items with a number of 1 2 1 must be an even number.Known n is an odd number, therefore ( 1)/2 n must be an even number in ( 1)/2 n n .So, must both be even numbers, must be an odd number (otherwise the product of 2 is an odd, we can get .So Clearly, when ( ) g x x x x x x x x x .At the same time, 3 0 ( ) ( ) 0 f x g x and 5 0 ( ) ( ) 0 f x g x establish.

Conclusions
Translation transformation makes transformed function remain the original diffusion times, nonlinearity, correlation immunity order, algebraic immunity, and the original rotational symmetry.But after the cascade transformation, some properties of the function will remain, while other properties, such as the nature of diffusion, correlation immunity and rotational symmetry will lost.The characteristic of translation transformation makes it possible to construct optimal algebraic immunity function by cascade conversion and optimal algebraic immunity rotationally symmetric correlation immune Boolean function.
In this paper, the optimal Algebraic Immune Function of a rotational symmetric Boolean function is found.And the translational transformation and cascade is used to calculate the optimal algebra are not related immune rotation symmetric Boolean function transform as the optimum algebraic immunity related immune function, increase the ability to resist related attacks.Results and methods are very significant.
In this paper, we use the derivative to prove the correlation immunity and the diffusion property.

5 "
, if ( ) g x are annihilators of the lowest algebraic degree of 0 ( ) r f x , ( ) g & must be annihilators of the lowest algebraic degree of 0 ( ) r f & , so ( ) g x 5 must be annihilators of the lowest algebraic degree of 0 ( ) r f x 5 .Known by the proof of Theorem 1, we can get deg ( ) deg ( )