Research On Cryptographic Properties Of Rotation Symmetric Boolean Functions

Boolean functions play an important role in the design and analysis of moderncryptographic systems. Boolean functions can be used to construct some cryptographic systemswidely. Their securities have close relation to the cryptographic properties of the chosen Booleanfunctions. Hence researches on the cryptographic properties of Boolean functions and theBoolean functions with special cryptographic properties are always the hot issues in cryptology.Rotation symmetric Boolean function is a class of Boolean functions with good cryptographicproperties, it has been received great attentions of the researchers since it was proposed. Westudy the cryptophic properties of rotation symmetric Boolean functions in this paper, and themain results are outlined as follows.(1) The properties of a class of quadratic rotation symmetric Boolean functions are studied.Algorithm to display the truth table of this class of functions has been given and the recursions oftheir Hamming weights are also proposed. The properties of this class of rotation symmetricBoolean functions are explained in the point of view of recursion.(2)The cryptographic properties of a class of quartic rotation symmetric Boolean functionwhose algebraic normal form is with single orbit are studied. Firstly the recursion of Hammingweights of this class of functions is given. Then, we factor the function into several sub-functions.Finally, we prove that the nonlinearity of the function is the same as its Hamming weight withthe recursion about Fourier-transform values of the sub-functions.(3) The cryptographic properties of a class of cubic rotation symmetric Boolean function arestudied. Tighter lower bounds of second order nonlinearities of cubic single orbit rotationsymmetric Boolean functions are presented.