Lifting Private Information Retrieval from Two to any Number of Messages

Abstract

We study private information retrieval (PIR) on coded data with possibly colluding servers. Devising PIR schemes with optimal download rate in the case of collusion and coded data is still open in general. We provide a lifting operation that can transform what we call one-shot PIR schemes for two messages into schemes for any number of messages. We apply this lifting operation on existing PIR schemes and describe two immediate implications. First, we obtain novel PIR schemes with improved download rate in the case of MDS coded data and server collusion. Second, we provide a simplified description of existing optimal PIR schemes on replicated data as lifted secret sharing based PIR.

On the Capacity of Private Information Retrieval from Coded, Colluding, and Adversarial Servers

“…The main contribution of this paper is the proof for the capacity of MDS-coded, linear, symbol-separated PIR with coded and colluding servers. The converse (upper bound) is given by Theorem 1, and an achievable scheme can be found by combining [12] and [15,Thm. 3], where a lift operation is introduced, which together with the star product scheme [12] achieves the derived capacity upper bound.…”

Private Polynomial Computation for Noncolluding Coded Databases

“…Note that the curve for the systematic scheme follows a staircase in which there arek points on each horizontal line of the staircase. This follows directly from the term ñ k in the definition ofn in (29).…”

“…Assume that n −k ≤ k. Then, 1 ≤ ñ k ≤ 1 + k k ≤ 2. If ñ k = 1, then it follows directly from (29) and Lemma 6 that n =n and ν = k + n −k = k + min{k, n −k}. Otherwise, if ñ k = 2, then k =k, 3k > n ≥ 2k, and from (29), we haven = k +k = 2k.…”

“…Proof: Letδ ñ k and Γ n −δk. From our choices ofn in (29), one can verify that Γ ≤ k and Γ is well-defined. Accordingly, construct a matrix A k×n as in Definition 8 with…”

“…This difference is due to the simplified rate definition used in [22]. Moreover, our proposed scheme cannot readily be obtained using the concept of refinement and lifting of so-called one-shot schemes as introduced for PIR in [29], since this concept cannot readily be applied to the function computation case.…”