CertiCrypt is a framework that enables the machine-checked construction and verification of cryptographic proofs in the Coq proof assistant. CertiCrypt instruments the code-based game-based approach to cryptographic proofs, and builds upon many areas, including probability and complexity theory, algebra, semantics of programming languages, and program optimizations. In this thesis, we illustrate the application of CertiCrypt on two examples: the Hashed ElGamal encryption scheme and zero-knowledge protocols. Like previous case studies in CertiCrypt, these examples demonstrate the feasibility of formalizing complex cryptographic proofs. However, using CertiCrypt requires a high level of expertise in Coq, and is time consuming. In order to ease the adoption of formal proofs by the cryptographic community, we develop a semi-automated tool, called EasyCrypt, for elaborating security proofs of cryptographic systems from proof sketches. Proof sketches are checked automatically using SMT solvers and automated theorem provers, and then compiled into verifiable proofs in the CertiCrypt framework. We illustrate the application of EasyCrypt with two examples: the Hashed ElGamal encryption system, and the Cramer-Shoup encryption system. Finally, we extend the language of CertiCrypt with a formalization of polytime functions.
