The algebraic immunity AI(f) of a Boolean function f is defined as the minimum degree of all annihilators of f. The high value of algebraic immunity consists a necessary condition for Boolean functions used in stream ciphers to resist algebraic attacks. of the two values. In this paper, we introduce the notion of extended algebraic immunity (f) defined as the maximum of pAI(f) and pAI(f ⌖ 1), where pAI(f) is the minimum degree of all annihilators of f (pAI(f ⌖ 1) of f ⌖ 1 respectively). We introduce a lower bound of the r-th order nonlinearity of a Boolean function f with given AI(f) and (f). The bound is tighter than all known lower bounds, where only the algebraic immunity AI(f) is used. The value of (f) can be computed as part of the calculation of AI(f), with no extra computational cost.