We present an analysis of key wrapping APIs with generic policies. We prove that certain minimal conditions on policies are sufficient for keys to be indistinguishable from random in any execution of an API. Our result captures a large class of API policies, including both the hierarchies on keys that are common in the scientific literature and the non-linear dependencies on keys used in PKCS#11. Indeed, we use our result to propose a secure refinement of PKCS#11, assuming that the attributes of keys are transmitted as authenticated associated data when wrapping and that there is an enforced separation between keys used for wrapping and keys used for other cryptographic purposes. We use the Computationally Complete Symbolic Attacker developed by Bana and Comon. This model enables us to obtain computational guarantees using a simple proof with a high degree of modularity.