Encryption that is only semantically secure should not be used on messages that depend on the underlying secret key; all bets are off when, for example, one encrypts using a shared key K the value K. Here we introduce a new notion of security, KDM security, appropriate for key-dependent messages. The notion makes sense in both the publickey and shared-key settings. For the latter we show that KDM security is easily achievable within the random-oracle model. By developing and achieving stronger notions of encryption-scheme security it is hoped that protocols which are proven secure under “formal” models of security can, in time, be safely realized by generically instantiating their primitives.