Construction Of Rotation Symmetric Boolean Functions With Optimal Algebraic Immunity

In recent years,with the emergence of a new cryptanalysis technique-algebraic attack,many cryptographic algorithms such as block cipher,public-key cipher,stream cipher and even Hash functions are seriously threatened.Algebraic immunity becomes an important property for choosing Boolean functions,which is used to measure the resistance of Boolean functions to algebraic attack.Therefore,Boolean functions used in cryptosystem should have high algebraic immunity,even the optimum algebraic immunity.Among the various Boolean functions,the rotation symmetric Boolean function is a hot topic in current research..It not only provides efficient computation,but also has good cryptographic propertiesIn this dissertation,the constructions of two class of rotation symmetric Boolean functions with optimal algebraic immunity are given,and their nonlinearity and algebraic degree are analyzed.The specific results are as follows1.The construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity is given.By the construction of sets T and U,the newly constructed n-variable functions are not only achieve the optimal algebraic immunity,but also have the best nonlinearity in current similar constructions when n≥25.In addition,it is proved that such functions have the optimal algebraic degree,if n ≠ 2^m+1.2.The construction of even-variable rotation symmetric Boolean functions with optimal algebraic immunity is given.By the construction of sets T,U,V,S,the n-variable functions not only achieve optimal algebraic immunity,but also have very high nonlinearity.In addition,it is proved that such functions have the optimal algebraic number when n = 2^m,and have suboptimal when 2^m + 1 ≤n≤2^m+1-1.