Entropic security in quantum cryptography

We present two new definitions of security for quantum ciphers which are inspired by the definitions of entropic security and entropic indistinguishability defined by Dodis and Smith. We prove the equivalence of these two new definitions. We also propose a generalization of a cipher described by Dodis and Smith and show that it can actually encrypt n qubits using less than n classical bits of key under reasonable conditions and yet be secure in an information theoretic setting. This cipher also totally closes the gap between the key requirement of quantum ciphers and classical ciphers.