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In this paper we take a closer look at the security and efficiency of public-key encryption and signature schemes in public-key infrastructures (PKI). Unlike traditional analyses which assume an "ideal" implementation of the PKI, we focus on the security of joint constructions that consider the certification authority (CA) and the users, and include a key-registration protocol and the algorithms of an encryption or a signature scheme. We therefore consider significantly broader adversarial capabilities. Our analysis clarifies and validates several crucial aspects such as the amount of trust put in the CA, the necessity and specifics of proofs of possession of secret keys, and the security of the basic primitives in this more complex setting. We also provide constructions for encryption and signature schemes that provably satisfy our strong security definitions and are more efficient than the corresponding traditional constructions that assume a digital certificate issued by the CA must be verified whenever a public key is used. Our results address some important aspects for the design and standardization of PKIs, as targeted for example in the standards project ANSI X9.109.