Construction Of Balanced Boolean Functions Based On Bent Functions And Analysis Of Properties For K-rotation Symmetric Boolean Functions

Logical functions are widely used in cryptological systems as an important tool. With the rapid development of modern cryptography, the analysis of Boolean functions which are the representation of Logical functions has been paid more and more attention. In this paper, we propose a method to construct balanced Boolean functions based on Bent functions, and analyze the properties of k-Rotation Symmetric Boolean Functions(k-RSBFs) with the spectrum theory. The main results are as follows:1. A method to construct balanced Boolean functions is given. With directly adding the Bent functions to the indicator functions of a subset E. The analysis shows that the appropriate selection of the subset E can make the constructed functions more uniform distribution of Walsh spectrum and Auto-correlation values, and then two kinds of specific selection methods of subset E are given.2. The properties of k-RSBFs are analyzed. First of all, we present that the Walsh spectrum and Auto-correlation value are invariant when the parameters of a k-RSBFs are under k-circular shift of indices. Then the analysis of the properties shows that many properties of k-RSBFs can be described by their orbits, and the counting formulas of long cycles and short cycles on k-RSBFs'orbits are given. Specially when k =1, comparing the obtained formulas with the formulas obtained by Sarkar and Maitra , we find their formulas are not established in the case of l ? 2 and there having ai ? 2 or in the case of l ? 3 when 1n ? p1 a ? plal and the specific reasons are given.3. A preliminary study on the conjecture made by Sarkar and Maitra is done, Which is that there are no homogeneous RSBFs of degree more than 2. First of all, with the knowledge of the matrix theory, we present that a special class of homogeneous RSBFs with degree 2 are not Bent functions. Secondly, with the method of probability theory, a conlusion about the hamming weight of n variables homogeneous boolean functions,whose algebraic degree is a factor of n is given. And with the conclusion, we present that a class of homogeneous RSBFs are Bent functions if and only if their algebraic degree is 2, whose algebraic mormal form have no repeated variables.