At CHES 2009, Renauld, Standaert and Veyrat-Charvillon introduced a new kind of attack called Algebraic Side-Channel Attacks (ASCA). They showed that side-channel information leads to effective algebraic attacks. These results are mostly experiments strongly based on a the use of a SAT-solver. This arti- cle presents a theoretical study in order to explain and to characterize the algebraic phase of these attacks. We study more general algebraic attacks based on Gro ̈bner methods. We show that the complexity of the Gr ̈obner basis computations in these attacks depends on a new notion of algebraic immunity defined in this paper, and on the distribution of the leakage information of the cryptosystem. We also study two examples of common leakage models: the Hamming weight and the Hamming distance models. For instance the study in the case of the Hamming weight model gives that the probabil- ity of obtaining at least 64 (resp. 130) linear relations is about 50% for the substitution layer of PRESENT.