Private Entanglement over Arbitrary Distances, Even Using Noisy Apparatus

We give a security proof of quantum cryptography based entirely on entanglement purification. Our proof applies to all possible attacks (individual and coherent). It implies the security of cryptographic keys distributed with the help of entanglement-based quantum repeaters. We prove the security of the obtained quantum channel which may not be used only for quantum key distribution, but also for secure, albeit noisy, transmission of quantum information.