Diffusion Properties Of Boolean Functions Analysis And Saturation Optimal Boolean Function Of The Structure

This paper mainly deals with the propagation criteria of resilient functions and partially-bent functions and the construction of saturated best functions. First we present a lower bound of nonlinearity of resilient functions with known algebra number and propagation criteria. Furthermore we give the form and extent of nonlinearity of these functions. Then, we consider the suppose of J.Seberry, X.M.Zhang å'ŒY.Zheng about the set R_f^c, during the research we found a counter example to the suppose, See example 3.1.1, which shows that the suppose is not true. Further research we proved the suppode is true for a kind of given boolean functions viz partially-bent functions, see theorem 3.1.1. Based on this character we give some special propagation criteria and correlation immune characteristics of partially-bent functions.concretely we study two vector sets of partially-bent functions, that is, the set R^c satisfying the propagation criteria and the set ζ^c whose Walsh spectrums equal to zero, are discussed. If R^c(ζ^c) is a non-empty set, then 2^n-1≤ < |R^c| < 2ⁿ (2^n-1≤ |ζ^c| < 2ⁿ). Under the condition of nondegenerate linear transformation, a partially-bent function f(x) on V_n satisfies the SAC if and only if there exists a vector a∈ V_n such that â–³_f(a) = 0. Under the condition of |R| > 2, we present an upper bound of the degree of partially-bent functions satisfy the propagation criteria. Finally we study the construction of saturated best function and obtain a sufficient and necessary condition about Boolean function to be Saturated Best function.