Research On The Strict Avalanche Balanced Boolean Functions And Cryptographic Properties

Boolean functions play an important rule in the design and analysis of Stream Cipher and Block Cipher.Currently,the research on Boolean functions focuses on its expressions,constructions,realization and cryptographic properties.DNA-like learning algorithm can be used to realize and classify Boolean functions based on Neural Network.Many S-boxes in cryptography are basically multi-output Boolean functions,so their cryptographic properties can be described by Boolean functions.Besides,the complexity of series Boolean functions in Stream cipher is also an important topic for their safety research.In this paper,the classification of Boolean functions and their good cryptographic properties,including balancedness,strict avalanche criterion,nonlinearity,algebraic immunity,etc.,are discussed.Furthermore,a method to construct and enumerate balanced Boolean functions of n-variables satisfying the SAC is presented,and analysis on the expansion of balanced Boolean functions satisfying the SAC from low-dimension to high-dimension is given out.This paper is organized as follows:The first chapter introduces the background and development situation about Boolean functions.The second chapter introduces some basic concepts and cryptographic properties of Boolean functions.The third chapter gives out the minimum leap number of Boolean functions through perceptrons and DNA-Like learning algorithm,and realizes the simple classification of Boolean functions of n-variable.Chapter four discusses the balancedness and SAC of Boolean functions,and presents a condition on assuring a Boolean function satisfying balancedness and the SAC.Moreover,a method to construct Boolean functions satisfying balancedness and the SAC is presented via shift function symmetric representation matrices.The concept of strong SAC and the expression on the number of balanced Boolean functions satisfying the SSAC is also given out.The fifth chapter makes a summary and presents some prospects for future research.