Distribution of secret keys in a quantum network with trusted intermediate nodes

The basic configuration in quantum cryptography is a point-to-point configuration. This technology is not applicable to a quantum network, where long-distance distribution of keys is required. In this paper we present a procedure for sequential transfer of an external $\varepsilon_{K}$-secret key through intermediate trusted nodes using $\varepsilon_{i}$-secret keys on separate network segments. It is shown that if the external key is $\varepsilon_{K}$-secret, then after transmission through the trusted node its secrecy becomes equal to $\varepsilon_{K}+\varepsilon_{1}$. The result is generalized to the entire length of the communication line. The secrecy of the transmitted key at the final point becomes $\varepsilon_{L}=\varepsilon_{K}+\sum_{i=1}^{L}\varepsilon_{i}$, where $L$ is the number of this line segments. It is shown that for any measurements of the eavesdropper leading to classical probability distributions the complexity of revealing the transmitted key in the class of algorithms based on the consideration of the most probable keys only are determined by the secrecy parameter $\varepsilon_{L}$.