The Squirrel Prover is a proof assistant designed for the computational verification of cryptographic protocols. It implements a probabilistic logic that captures cryptographic and probabilistic arguments used in security proofs. This logic operates in the asymptotic security setting, which limits the expressiveness of formulas and proofs. As a consequence, it can only prove security for finite interactions with a protocol, falling outside of the polynomial-level of security usually expected by cryptographers. We lift all these limitations by moving to a concrete security setting. We extend the logic with concrete security predicates, and design a corresponding proof system. We show the usefulness of these extensions on a case study, and through a novel proof-transformation result which shows that a large class of asymptotic logic security proofs can be automatically rewritten into concrete logic security proofs, improving security bounds exponentially.