Abstract
A (K, N, T, K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) instance of private information retrieval from MDS coded data with colluding servers (in short, MDS-TPIR), is comprised of K messages and N distributed servers. Each message is separately encoded through a (K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , N) MDS storage code. A user wishes to retrieve one message, as efficiently as possible, while revealing no information about the desired message index to any colluding set of up to T servers. The fundamental limit on the efficiency of retrieval, i.e., the capacity of MDS-TPIR is known only at the extremes where either T or K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> belongs to {1, N}. The focus of this work is a recent conjecture by Freij-Hollanti, Gnilke, Hollanti, and Karpuk which offers a general capacity expression for MDS-TPIR. We prove that the conjecture is false by presenting as a counterexample a PIR scheme for the setting (K, N, T, K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) = (2, 4, 2, 2), which achieves the rate 3/5, exceeding the conjectured capacity, 4/7. Insights from the counterexample lead us to capacity characterizations for various instances of MDS-TPIR, including all cases with (K, N, T, K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) = (2, N, T, N -1), where N and T can be arbitrary.
Highlights
Private Information Retrieval (PIR) is the problem of retrieving one out of K messages from N distributed servers in such a way that any individual server learns no information about which message is being retrieved
Each message is encoded through a (Kc, N ) MDS storage code, so that the information stored at any Kc servers is exactly enough to recover all K messages
We settle a conjecture on the capacity of MDS-TPIR by Freij-Hollanti et al [6] by constructing a scheme that beats the conjectured capacity for one particular instance of MDS-TPIR
Summary
Private Information Retrieval (PIR) is the problem of retrieving one out of K messages from N distributed servers (each stores all K messages) in such a way that any individual server learns no information about which message is being retrieved. The focus of this work is on a recent conjecture by Freij-Hollanti, Gnilke, Hollanti and Karpuk (FGHK conjecture, in short) in [6] which offers a capacity expression for a generalized form of PIR, called MDS-TPIR. In support of the plausible asymptotic (K → ∞) optimality of their scheme, and based on the intuition from existing capacity expressions for PIR, MDS-PIR and TPIR, Freij-Hollanti et al conjecture that if. In all these cases the problem is open on both sides, i.e., the conjectured capacity expression is neither known to be achievable, nor known to be an outer bound. The lack of any non-trivial outer bounds for MDSTPIR is recently highlighted in [8] This intriguing combination of plausibility, uncertainty and generality of the FGHK conjecture motivates our work.
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