On the Optimal Communication Complexity of Error-Correcting Multi-server PIR

An ℓ -server Private Information Retrieval (PIR) scheme enables a client to retrieve a data item from a database replicated among ℓ servers while hiding the identity of the item. It is called b-error-correcting if a client can correctly compute the data item even in the presence of b malicious servers. It is known that b-error correction is possible if and only if ℓ> 2 b. In this paper, we first prove that if error correction is perfect, i.e., the client always corrects errors, the minimum communication cost of b-error-correcting ℓ -server PIR is asymptotically equal to that of regular (ℓ- 2 b) -server PIR as a function of the database size n. Secondly, we formalize a relaxed notion of statistical b-error-correcting PIR, which allows non-zero failure probability. We show that as a function of n, the minimum communication cost of statistical b-error-correcting ℓ -server PIR is asymptotically equal to that of regular (ℓ- b) -server one, which is at most that of (ℓ- 2 b) -server one. Our main technical contribution is a generic construction of statistical b-error-correcting ℓ -server PIR for any ℓ> 2 b from regular (ℓ- b) -server PIR. We can therefore reduce the problem of determining the optimal communication complexity of error-correcting PIR to determining that of regular PIR. In particular, our construction instantiated with the state-of-the-art PIR schemes and the previous lower bound for single-server PIR result in a separation in terms of communication cost between perfect and statistical error correction for any ℓ> 2 b.