Adaptively-secure encryption schemes ensure secrecy even in the presence of an adversary who can corrupt parties in an adaptive manner based on public keys, ciphertexts, and secret data of already-corrupted parties. Ideally, an adaptively-secure encryption scheme should, like standard public-key encryption, allow arbitrarily-many parties to use a single encryption key to securely encrypt arbitrarily-many messages to a given receiver who maintains only a single short decryption key. However, it is known that these requirements are impossible to achieve: no noninteractive encryption scheme that supports encryption of an unbounded number of messages and uses a single, unchanging decryption key can be adaptively secure. Impossibility holds even if secure data erasure is possible
