Remotely keyed encryption schemes (RKESs), introduced by Blaze 6], support high-bandwidth cryptographic applications (such as encrypted video conferences) in which long-lived secrets (such as users' private keys) never leave lower-bandwidth environments such as secure smart-cards. We provide a formal framework in which to study the security of RKESs and give RKESs that satisfy our formal security requirements. Our RKESs are e cient in that the amount of communication and computation required of the smart-card is independent of the input size. In one proof of security, we use the pseudorandom permutation framework of Naor and Reingold 18] in an essential way.