Characterizations Of The Matrix Of The Cryptographic Properties Of Boolean Functions

Using characteristic matrix, this dissertation gives matrix description on propagation characteristic of Boolean function and the existence & construction of a kind of CI-SAC(k) function; then it analyzes spectrum characteristic of a m-valued logical function which fulfills the SAC of higher order. The main results are as follows: Using characteristic matrix, it is the first time that a necessary and sufficient condition is given on a Boolean function satisfying the propagation criterion at some vector. On the basis of it, the necessary and sufficient conditions are given on a Boolean function satisfying strict avalanche criterion (SAC) and on a balanced SAC function which achieves the maximum degree and has not nonzero linear structure. Finally, we present two special "filling vacancies in the proper order" methods of construction and "solving the equations" method of construction. The Boolean function is called CI-SAC(k) function if it is correlation immune and satisfies strict avalanche criterion of order k . Using characteristic matrix, a necessary and sufficient condition is given on the function(x₁,+… +x_n)(x_n+1, +…+ x_n+k+1) + g(x₁,…,x_n) being balanced CI-SAC(k) function andachieving the maximum degree. At the same time, a simple and feasible method is presented for constructing this excellent function. We analyze spectrum characteristic of a m-valued logical function which fulfills the SAC of higher order. Then two recursion methods are presented for Boolean function satisfying strict avalanche criterion of order k.