  We leverage the Gibbs inequality and its natural generalization to R\'enyi
entropies to derive closed-form parametric expressions of the optimal lower
bounds of $\rho$th-order guessing entropy (guessing moment) of a secret taking
values on a finite set, in terms of the R\'enyi-Arimoto $\alpha$-entropy. This
is carried out in an non-asymptotic regime when side information may be
available. The resulting bounds yield a theoretical solution to a fundamental
problem in side-channel analysis: Ensure that an adversary will not gain much
guessing advantage when the leakage information is sufficiently weakened by
proper countermeasures in a given cryptographic implementation. Practical
evaluation for classical leakage models show that the proposed bounds greatly
improve previous ones for analyzing the capability of an adversary to perform
side-channel attacks.
