Security in distributed storage systems by communicating a logarithmic number of bits

We investigate the problem of maintaining an encoded distributed storage system when some nodes contain adversarial errors. Using the error-correction capabilities that are built into the existing redundancy of the system, we propose a simple linear hashing scheme to detect errors in the storage nodes. Our main result is that for storing a data object of total size M using an (n, k) MDS code over a finite field Fq, up to t1 = ⌊(n - k)/2⌋ errors can be detected, with probability of failure smaller than 1/M, by communicating only O(n(n-k) logM) bits to a trusted verifier. Our result constructs small projections of the data that preserve the errors with high probability and builds on a pseudorandom generator that fools linear functions. The transmission rate achieved by our scheme is asymptotically equal to the min-cut capacity between the source and any receiver.