Construction And Analysis Of Boolean Functions With Optimal Algebraic Immunity

Because algebraic attack is proposed by Courtios and Meier, in stream ciphers, algebraic immunity is now an important property for Boolean functions. As a result, constructing Boolean functions with optimal algebraic immunity got extensive attention by Scholars at home and abroad.In this paper, we study the method of constructing Boolean functions with optimal algebraic immunity, analyze properties of tow class of functions and obtain the following main results:(1) Firstly, according to Sihong Su and Xiaohu Tang’s paper, which determine the concrete coefficients in the linear expression of the column vectors with respect to a given basis of the generator matrix of Reed Muller code (Designs Codes Cryptography Published online:01February2013), a new method of constructing Boolean functions with optimal algebraic immunity based on the generator matrix of Reed Muller code is given. Secondly, we provide a simper and direct proof for this construction. Last, Boolean functions are changed into1-resilient functions by LT method.(2) The constructions of M-M and PS Bent functions are introduced in the text. By analyzing, we get some result. M-M functions with optimal algebraic immunity and high nonlinearity only by a modification under limit of some conditions, but unknown M-M Boolean functions are not optimal. As PSap Boolean functions, have good algebraic structure, the nonlinearity and algebraic immunity are good by modifications.