Hierarchical and compositional verification of cryptographic protocols

Two important approaches to the verification of security protocols are known under the general names of symbolic and computational, respectively. In the symbolic approach messages are terms of an algebra and the cryptographic primitives are ideally secure; in the computational approach messages are bitstrings and the cryptographic primitives are secure with overwhelming probability. This means, for example, that in the symbolic approach only who knows the decryption key can decrypt a ciphertext, while in the computational approach the probability to decrypt a ciphertext without knowing the decryption key is negligible. Usually, the cryptographic protocols are the outcome of the interaction of several components: some of them are based on cryptographic primitives, other components on other principles. In general, the result is a complex system that we would like to analyse in a modular way instead of studying it as a single system. A similar situation can be found in the context of distributed systems, where there are several probabilistic components that interact with each other implementing a distributed algorithm. In this context, the analysis of the correctness of a complex system is very rigorous and it is based on tools from information theory such as the simulation method that allows us to decompose large problems into smaller problems and to verify systems hierarchically and compositionally. The simulation method consists of establishing relations between the states of two automata, called simulation relations, and to verify that such relations satisfy appropriate step conditions: each transition of the simulated system can be matched by the simulating system up to the given relation. Using a compositional approach we can study the properties of each small problem independently from the each other, deriving the properties of the overall system. Furthermore, the hierarchical verification allows us to build several intermediate refinements between specification and implementation. Often hierarchical and compositional verification is simpler and cleaner than direct one-step verification, since each refinement may focus on specific homogeneous aspects of the implementation. In this thesis we introduce a new simulation relation, that we call polynomially accurate simulation, or approximated simulation, that is compositional and that allows us to adopt the hierarchical approach in our analyses. The polynomially accurate simulations extend the simulation relations of the distributed systems context in both strong and weak cases taking into account the lengths of the computations and of the computational properties of the cryptographic primitives. Besides the polynomially accurate simulations, we provide other tools that can simplify the analysis of cryptographic protocols: the first one is the concept of conditional automaton, that permits to safely remove events that occur with negligible probability. Starting from a machine that is attackable with negligible probability, if we build an automaton that is conditional to the absence of these attacks, then there exists a simulation. And this allows us to work with the simulation relations all the time and in particular we can also prove in a compositional way that the elimination of negligible events from an automaton is safe. This property is justified by the conditional automaton theorem that states that events are negligible if and only if the identity relation is an approximated simulation from the automaton and its conditional counterpart. Another tool is the execution correspondence theorem, that extends the one of the distributed systems context, that allows us to use the hierarchical approach. In fact, the theorem states that if we have several automata and a chain of simulations between them, then with overwhelming probability each execution of the first automaton is related to an execution of the last automaton. In other words, we have that the probability that the last automaton is not able to simulate an execution of the first one is negligible. Finally, we use the polynomially accurate simulation framework to provide families of automata that implement commonly used cryptographic primitives and to prove that the symbolic approach is sound with respect to the computational approach.