Unguessable Atoms: A Logical Foundation for Security

We show how a type of atoms, which behave like urelements, and a new proposition that expresses the independence of a term from an atom can be added to any logical system after imposing minor restrictions on definitions and computations. Working in constructive type theory, we give rules for the independence proposition and show how cryptographic protocols can be modeled as automata exchanging atoms. This model provides a unifying framework for reasoning about security and allows us to combine a general model of computation with a simple model of acquisition of secret information. As an application, we prove a fundamental property of nonces that justifies the axioms for nonces used in the protocol composition logic (PCL) of Datta, Derek, Mitchell and Roy. The example shows that basic security properties are naturally expressed in terms of independence and the causal ordering of events. The rules and example proofs are fully implemented in the Nuprl proof development system.