Distributed Storage Over a Public Channel: Trade-off between Privacy and Shared Key Lengths

Consider a user who wants to store a file across multiple servers such that at least t servers are needed to reconstruct the file and any z colluding servers cannot learn more than a fraction α of the file. Unlike traditional secret-sharing models that assume the availability of secure channels at no cost, we assume that the user can only transmit data to the servers through a public channel and that the user and each server share an individual secret key of length n bits. For a fixed key length n, we characterize the fundamental trade-off between the privacy leakage parameter α and the file length that the user can store in the servers. Furthermore, for the optimal trade-off between α and the file length, we determine (i) the minimum amount of local randomness needed by the user, (ii) the minimum amount of public communication from the user to the servers, and (iii) the minimum storage requirement at the servers.