Measuring the information leakage is critical for evaluating the practical security of cryptographic devices against side-channel analysis. Information-theoretic measures can be used (along with Fano’s inequality) to derive upper bounds on the success rate of any possible attack in terms of the number of side-channel measurements. Equivalently, this gives lower bounds on the number of queries for a given success probability of attack. In this paper, we consider cryptographic implementations protected by (first-order) masking schemes, and derive several information-theoretic bounds on the efficiency of any (second- order) attack. The obtained bounds are generic in that they do not depend on a specific attack but only on the leakage and masking models, through the mutual information between side- channel measurements and the secret key. Numerical evaluations confirm that our bounds reflect the practical performance of optimal maximum likelihood attacks.