Algebraic immunity is a new cryptographic criterion. In order to resist algebraic attacks, Boolean functions used in cryptosystem should have high algebraic immunity. This paper presents a new recursive construction of Boolean function with maximum algebraic immunity. The constructed Boolean functions are pointed out to be balanced.

Algebraic immunity is a cryptographic criterion for Boolean functions used in cryptosystem to resist algebraic attacks. They usually should have high algebraic immunity. Chen proposed a first order recursive construction of Boolean functions and checked that they had optimum algebraic immunity for n<8.This paper gives a detail proof of having optimum algebraic and being balanced for all n>0.