We consider a novel security requirement of encryption schemes that we call “key-privacy” or “anonymity”. It asks that an eavesdropper in possession of a ciphertext not be able to tell which specific key, out of a set of known public keys, is the one under which the ciphertext was created, meaning the receiver is anonymous from the point of view of the adversary. We investigate the anonymity of known encryption schemes. We prove that the El Gamal scheme provides anonymity under chosen-plaintext attack assuming the Decision Diffie-Hellman problem is hard and that the Cramer-Shoup scheme provides anonymity under chosen-ciphertext attack under the same assumption. We also consider anonymity for trapdoor permutations. Known attacks indicate that the RSA trapdoor permutation is not anonymous and neither are the standard encryption schemes based on it. We provide a variant of RSA-OAEP that provides anonymity in the random oracle model assuming RSA is one-way. We also give constructions of anonymous trapdoor permutations, assuming RSA is one-way, which yield anonymous encryption schemes in the standard model.