Quantum key distribution through untrusted nodes: exact solution for single-photon states

A proof of the secrecy of the quantum key distribution through an untrusted node is given. The use of entropy uncertainty relations allows us to obtain an exact solution for the length of the secret key in the single-photon case. Comparison with the exact solution for the BB84 protocol is made and the principal difference in the logical structure of the key secrecy proofs in these protocols is clearly shown, which, in our opinion, is important for the development of quantum cryptography systems. (letter)