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Two-way quantum key distribution in a uniformly distributed quantum space via a special mapping and its analytical security proofs

Within the field of quantum cryptography, two-way direct quantum secure communication is a relatively new proposal for sharing secret information that is not fully explored yet. We propose a general model for two-way deterministic quantum key distribution, and a special mapping from integers to a uniformly distributed quantum space is introduced for qubit pad preparation. A reduced model, more practical, is also proposed by relaxing the security assumption of the qubit pad preparation. The main work of this paper focuses on the security proofs of these two models. Under collective attacks, we fully analyze the security basis of the general model, and, in a relatively simple way, we provide an analytical security proof for the reduced model in a depolarization quantum channel. Our results show exactly the analytical upper-bound of the amount of useful key that a powerful eavesdropper could extract in both models, and the security of the reduced model is not reduced. One advantage of this work is that the special mapping guarantees the random distribution property of the qubit pad in a uniformly distributed quantum space, thus guaranteeing the security of the two models, and the other advantage is the simplicity of the analytical derivations of security proofs.