The Algebraic Immunity Of Boolean Functions

Boolean functions play an important role in digital communication, especially in designing the system of private key in communication security. Recent years, the research of the algebraic attack of stream ciphers has become a hot topic. Algebraic attack has successfully applied to the system of stream ciphers based on linear feedback shift registers, which has made great effect on the system of stream ciphers. Many cryptograpgy researchers have made great effort to solving a large system of multivariate algebraic equations to recover the secret key by algebraic attack and constructing the annihilators. In order to resist algebraic attack, a new cryptographic property of Boolean functions which is known as algerbraic immunity has been proposed by Meier et al. The propose of algerbraic immunity had made a new task for the analysis and construction of cryptographic functions. Algerbraic immunity also becomes one of the standard to measure the cryptographic properties of Boolean functions. So the construction of the Boolean functions with maximum algerbraic immunity is very important.In chapter 1, both the history and the new advance in cryptography and Boolean functions are mainly introduced, which include the applied background. In chapter 2, some knowledge of cryptology based on algebra is briefly given, and discuss the algebra properties of Boolean functions and the indicators of security. In chapter 3, analyze the property of algerbraic immunity and the annihilators in detail, and study the relation between the other properties. In chapter 4, we find a sufficient condition when Boolean functions have maximum algebraic immunity, we also analyze two other constructions of Boolean functions with maximum algebraic immunity.