Cryptographic Properties Of The Three Types Of Non-linear Combination Of Functions

This dissertation discusses the properties of strict maximum logic functions, maximum logic functions with even number of variables, and select logic functions. The main results are as follows:1. Using the theory of probability, this paper discusses the autocorrelation properties of SML (strict maximum logic) functions completely. It gives the value distributions of the autocorrelation function of SML functions. Then it obtains the conclusion that SML functions can't resist towards differential cryptanalysis. By a linear transformation of coordinates, it discusses the construction for functions being correlation immune and provides an explicit construction for functions having good properties of SML functions and satisfying the strict avalanche criterion.2. Relative to SML functions, this paper proposes the conception of maximum logic functions with even number of variables. Using the theories of probability and combinatorial mathematics, it studies the Walsh transform and the algebraic structure of a part of them. Then it proves that these functions with large number of variables have perfect stability and shows that the algebraic structure of these functions is also perfect.3. This paper is concerned with the Walsh transform and the autocorrelation function of select logic functions. Select logic functions with large number of variables have perfect stability and can resist towards cryptanalysis of best affine approximation, but they can't resist towards differential cryptanalysis efficiently because of weak propagation property. By a linear transformation of coordinates, this paper also provides an explicit construction for functions satisfying the strict avalanche criterion or being correlation immune.