In this paper, we are interested in the Computationally Complete Symbolic Attacker (CCSA) approach to security protocol verification, which relies on probabilistic logics to represent and reason about the interaction traces between a protocol and an arbitrary adversary. The proof assistant SQUIRREL implements one such logic. The CCSA logics come with cryptographic axioms whose soundness derives from the security of standard cryptographic games, e.g. PRF, EUF, IND-CCA. Unfortunately, these axioms are complex to design and implement; so far, these tasks are manual, ad-hoc and error-prone. We solve these issues in this paper, by providing a formal and systematic method for deriving axioms from cryptographic games. Our method relies on synthesizing (parts of) an adversary w.r.t. some cryptographic game, through the notion of bi-deduction. Concretely, we define a rich notion of bi-deduction, justify how to use it do derive cryptographic axioms, provide a proof system for bi-deduction, and an automatic proof-search method which we implemented in SQUIRREL.