Generalized Algebraic Immunity Analysis And Cycle Distribution Of A Class Of Bch Codes

Recently,algebraic attack has proved to be a powerful class of attacks which might threaten to almost all types of cryptographic systems.Since the complexity of the attack depends on the AI (Algebraic Immunity)of cryptographically significant Boolean function, the research on AI has become a hot topic in cryptography.In addition, how to generate good key stream sequence is always at the centre of the research on stream ciphers.We mainly investigate the EAI(Extended Algebraic Immunity)of Boolean function and the period distribution of a class of q-ary BCH codes.Firstly,we research some relations between Boolean function f and its algebraic complement f~e under different indices,and get the relations of f and f under the weight and annihilators set AN.The relation between AN(f)and AN(f~e)allows us to find them together in a fast way whose computational quantities is about 1/2 of previous method's provided algebraic normal form of the function fis known.The relation between the weights of f and f~e shows that if f is balanced,f~e must not be balanced.Following,we prove that a class of functions is weak to resist algebraic attack from EAI,and analyze the Boolean functions with 1-form linear structure,which indicates that the resistance against algebraic attack of the functions with linear structure is not good.We also study the properties of AI and EAI,and present a necessary condition for n-variable Boolean function to be of the optimal EAI. Then we analyze the Boolean functions with optimal AI constructed by some known existing constriction methods for their EAI,and find out those functions have not optimal EAI.We give two necessary conditions for Boolean functions with strong resistance against algebraic attack.Secondly,the calculation formula for period distribution of q-ary BCH codes with designed distance 7 are obtained,which is also used to count the non-periodic cyclic equivalence classes with Mobius formula.