Spectral Theory And Its Applications In Communication Security Techniques

This dissertation mainly discusses spectral theory and its applications in communicationsecurity techniques, the main results that the author obtained are as follows,(1) By using the first order wash spectrum, the relationship between Correlation-immu-nity and nonlinearity of Boolean functions is given; the relationship among nonlinearity, thenumber that autoeorrelation value is zero and the nurnber that spectral value is zero of Booleanfunctions is discussed; spectral properties of Boolean functions satisfying high order strictavalanche criterion and high degree propagation criteria respectively are presented and the con-eept of the propagation criteria is generalized; It is proved that some dasses of Boolean func-tions that are researched in design and analysis of the digital devices are equvalent to someclasses of Boolean functions that are discussed in cryptology and the structures of these func-tions are further characterized; nonlinearity and propagation property of a family of correla-tion-immune functions is developed and the methods constructing high nonlinearity and Bal-maced correlation-immune functions is described ; Basing on the idea of ref. [47], and an itera-tire method constructing highly nonlinearity and balanced Boolean functions from Bent func-tions is obtained, the shortcoming of ref. [47] is analyzed and it is pointed that conjectures ofboth ref. [21] and ref. [49] are wrongs It is proved that balanced Boolean functions satisfying2k—1 degree propagation criteria over F₂²⁴ are not existent; the correlations between the outputsequences and the input sequences of combiners with 1 bit memory as well as the effect ofmemory on the correlations are investigated.(2) The notation of the generalized first order Walsh spectrum is introduced, Some cryp-tographic properties of Multi-output functions are characterized by the spectrum; Some permu-tations with low difference uniform and high nonlinearity are constructed, and it is pointedthat the conjecture of ref. [57] is wrong, the concept of the ruth order walsh spectrum is pre-sented and the applications of the spectrum is investigated ; the correlations between Bent func-tions and their variables are discussed. Additionally, the eryptographic properties of Muhi-val-ued logical functions over any finite field are investigated by using spectral transform over finitefield.(3) The cryptographic properties of Multi-Valued Logical functions over residue rings arecharacterized by Chrestenson Spectrum; spectral representation of correspondence and nonlin-earity of multi-valued logical functions is given;the spectral Characterization of the degenera-tion and linear structures of multi-valued logical functions is discussed; the spectral characteri-zation of multi-valued Correlation-Immune logical functions is developed and some methodsjudging multi-vlaued Correlation-Immune Logical functions are described; Some conditionsthat a group of multi-valued logical functions forms a orthogonal system are revealed and somemethods constructing the orthogonal system are given.