We propose an information-theoretic framework that aims at unifying and optimizing several previous works on the side-channel security of masked im- plementations of any order d in some Abelian group: Duc at al.at EURO- CRYPT2015, Dziembowski et al.at TCC2016, Ch ́erisey et al.at CHES2019, Prest et al.at CRYPTO2019, Masure et al.at CARDIS2022, Ito et al.at CCS2022, Liu et al.at ITW2023, and B ́eguinot et al.at COSADE2023 and ISIT2023. In this general framework, two theoretical ingredients are system- atically leveraged: (i) a variation of a Fano inequality relating the attack performance (success rate) to a measure of information between the sensi- tive variable and the leakage; (ii) a variation of a Mrs. Gerber Lemma lower bounding a statistical measure of the sensitive variable by the product of sim- ilar measures for its d + 1 masking shares. Depending on the choice of the information measure and of the statistical measure, and possibly on Pinsker- type inequalities relating these measures, one can establish anew all previously published lower bounds on the number of queries necessary to achieve a given attack success rate. These results make progress on the evaluation of the secu- rity guarantees of higher order masking, and stimulate further research on best possible bounds & possible application to other types of masking schemes.