Properties of security protocols such as authentication and secrecy can be verified by explictly generating an operational model of the protocol and then seeking for insecure states. However, message exchange between the intruder and the honest participants induces a form of state explosion that makes the model infinite in principle. Building on previous work on symbolic semantics, we propose a general framework for automatic analysis of security protocols that make use of a variety of crypto-functions. We start from a base language akin to the spi-calculus, equipped with a set of generic cryptographic primitives. We propose a symbolic operational semantics that relies on unification and provides finite and effective protocol models. Next, we give a method to carry out trace analysis directly on the symbolic model. Under certain conditions on the given cryptographic primitives, our method is proven complete for the considered class of properties.