Analysis On Properties Of Some Hot Boolean Functions

The investigation of Boolean functions is very important in cryptological system designsand analysis. Applying the theories of number theory, algebra, analysis and probability and basedon construction of Boolean functions with optimum properties, analysis on properties of Booleanfunctions and applying of Boolean functions, this paper is devoted to the research of constructionand properties of concatenating Boolean functions, properties of symmetric Boolean functionsand linear approximation of2-round Trivium algorithm. The main results are as follows:1. In the field of constructions of Boolean functions, we analyze properties of one class ofconcatenating Boolean functions. We also construe one class of multioutput Boolean functions,propose the general Walsh spectrum of the class of multioutput Boolean functions based onKrawtchouk polynomial and Krawtchouk matrix, and analyze correlation immunity, propagation,nonlinearity and algebraic immunity of this multioutput Boolean functions. All the results showthat the class of multioutput Boolean functions has good properties when base functions havegood properties, which provide references for constructing multioutput Boolean functions withgood cryptological properties.2. In the field of properties analysis of Boolean functions, three aspects of work areresearched. Firstly, cryptological properties of one class of symmetric Boolean functions （called） are researched, including algebraic degree, linear structure and correlation immunity. Weprove a subclass of to have optimum algebraic immunity, give necessary condition for having optimum algebraic immunity, and give a lower bound of the count of Boolean functionssatisfying the necessary condition. Secondly, for special n, d and elementary symmetric Booleanfunctions X （d, n）with n variables and degree d, we propose that major of X （d, n）areunbalanced. These results partially improve the proof of Cusick conjecture. We obtain thestructure and count of symmetric Boolean functions with extended algebraic immunity and showthat n-error algebraic immunity of f can’t research an upper bound k when f is symmetricBoolean function, n=2k and k≠2^m+1. Thirdly, we give necessary and sufficient conditionssuch that one class of Boolean functions f （x）satisfies correlation immunity based on theisomorphism between F₂ⁿ and F_2ⁿ. We give some necessary conditions such that one class ofBoolean functions with maximum algebraic immunity satisfies correlation immunity and alsogive an upper bound for the count of the class of Boolean functions satisfying correlationimmunity.3. In the field of applications of Boolean functions, we attack2-round Trivium algorithmthrough linear approximation by applying the spectrum characteristic and the best affineapproximation theorem. We propose a linear approximation with deviation2-29and8linear approximations with deviation2-30by changing clock number and linear approximations of thefirst round. In order to identify a secret key given, the results show that we can supply thesuccess rate with1/16of data amount comparing with primary data amount required.