Abstract: Boolean functions with optimal algebraic immunity play an important role in stream ciphers. Based on the univariate polynomial representation of Boolean functions, a construction of Boolean functions with maximum extended algebraic immunity (EAI) is proposed, some previous results are special cases of our construction. The EAI of symmetric Boolean functions which have maximum algebraic immunity (AI) are analyzed by using the properties of symmetric Boolean functions. The result shows that there are only -variable ( even) symmetric Boolean functions achieve maximum EAI.