On The Construction Of Boolean Functions With Optimal Algebraic Immunity And The Design Of Secure Communication Protocols

Boolean functions are widely used in secure communication, especially in cryptography. And the security of cryptographic systems and steganographic systems lies in the security of Boolean functions employed, so the Boolean function is a hotspot in cryptographic study. The immergence of powerful algebraic attacks in 2003 imposed a new design criterion, algebraic immunity, on the design of Boolean functions, and the construction of Boolean functions with high algebraic immunity is of great significance.Compared to cryptography, steganography is a newly emerging technique. And these two techniques are usually used independently, even when they are both used in a process of secure communication. In this paper, we presented several new one-stop secure communication protocols, employing Boolean functions to combine the cryptographic techniques with steganographic ones together organically. Unlike the existing methods of encrypting and hiding separately, these protocols accomplish both of them at once. To our knowledge, they are the first secure communication protocols of this kind.We presented main results of our study as follows:1. The necessary and sufficient condition of constructing Boolean functions with optimal algebraic immunity by concatenating.2. We proposed two constructions which allow us to obtain n-variable Boolean functions with good cryptographic properties from (n-1)-variable or (n+1)-variable ones.3. We studied the algebraic degree, balancedness, algebraic immunity and nonlinearity of the constructed functions.4. We put forward a new idea for ascertaining whether the algebraic immunity of a given Boolean function was optimal or not.5. We proposed several new protocols for secure communication. They are based on Boolean functions and the LSB matching method, combining steganographic and cryptographic techniques organically and possessing their advantages simultaneously. In these protocols, the employed cover media could be not only grayscale images, but also RGB color bitmaps, and the Boolean functions are used for encryption and controlling the pseudo-random increment or decrement of LSB.