A general theory of public-key cryptography is developed that is based on the mathematical framework of complexity theory. Two related approaches are taken to the development of this theory, and these approaches correspond to different but equivalent formulations of the problem of cracking a public-key cryptosystem (PKCS). The first approach is to model the cracking problem as a partial decision problem called a ``promise problem.'' Every NP-hard promise problem is shown to be uniformly NP- hard, and a number of results and a conjecture about promise problems are shown to be equivalent to separability assertions for sets in NP that are the natural analogues of well-known results in classical recursion theory. The conjecture, if it is true, implies nonexistence of PKCS having NP-hard cracking problems. The second approach represents the cracking problem of a PKCS as a partial computational problem directly. Using this approach, it is shown that one-way functions exist if and only if \(P\neq UP\) and that one-way functions with greater cryptographic significance exist if and only if NP contains disjoint P-inseparable sets. The paper concludes with a discussion of almost-everywhere security measures for PKCS.