Studies Of Cryptographic Properties Of Boolean Functions

Boolean functions play an important role in many cryptography systems.Especially in the design and study of stream cipher,block cipher and some symmetric cryptographic systems.The studies on the properties of Boolean functions are always one of the most significant research areas.This paper investigates several cryptographic properties of Boolean functions,and obtains the following main results:(1)The nonlinearity and correlation coefficient of a function defined on F9)2are studied.Some properties of cross-correlation coefficient and auto-correlation are given.Based on the conclusions above,the nonlinearity of Boolean functions is investigated,and derive several results on upper bound for balanced Boolean functions.An lower bound on the sum-of-squares indicator of Boolean functions is also obtained.(2)By applying autocorrelation and Walsh spectral,the resilient Boolean functions are analysed.The new lower bound for the absolute indicator of 9)variables 8)order resilient functions.Some results on Walsh spectral,degree and cross-correlation coefficient on resilient functions are summarized.(3)The additive autocorrelation of correlation immune and resilient functions is studied.Some weight divisibility results of algebraic degree for Boolean functions are derived.It has been identified that the nonlinearity and algebraic degree of the correlation immune and resilient functions are optimized simultaneously,our study shows that in such condition the sum-of-squares indicator gets its minimum value too.Based on the Walsh-Hadamard transform and cross-correlation coefficient,some known knowledge concerning the Nega-Hadamard transform and nega-crosscorrelation coefficient is summarized.