We present a public key encryption primitive called A Posteriori Openable Public Key Encryption (APO-PKE). In addition to conventional properties of public key cryptosystems, our primitive allows each user, who has encrypted messages using different public keys, to create a special decryption key. A user can give this key to a judge to open all messages that have been encrypted in a chosen time interval with the public keys of the receivers. We provide a generic efficient construction, in the sense that the complexity of the special key generation algorithm and this key size are independent of the number of ciphertexts. We give security models for our primitive against chosen plaintext attack and analyze its security in the random oracle model.