Security of counterfactual quantum cryptography

Recently, a ``counterfactual'' quantum-key-distribution scheme was proposed by T.-G. Noh [Phys. Rev. Lett. 103, 230501 (2009)]. In this scheme, two legitimate distant peers may share secret keys even when the information carriers are not traveled in the quantum channel. We find that this protocol is equivalent to an entanglement distillation protocol. According to this equivalence, a strict security proof and the asymptotic key bit rate are both obtained when a perfect single-photon source is applied and a Trojan horse attack can be detected. We also find that the security of this scheme is strongly related to not only the bit error rate but also the yields of photons. And our security proof may shed light on the security of other two-way protocols.