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The problem of symmetric private information retrieval (SPIR) from a codeddatabase which is distributively stored among colluding servers is studied.Specifically, the database comprises $K$ files, which are stored among $N$servers using an $(N,M)$-MDS storage code. A user wants to retrieve one filefrom the database by communicating with the $N$ servers, without revealing theidentity of the desired file to any server. Furthermore, the user shall learnnothing about the other $K-1$ files in the database. In the $T$-colluding SPIRproblem (hence called TSPIR), any $T$ out of $N$ servers may collude, that is,they may communicate their interactions with the user to guess the identity ofthe requested file. We show that for linear schemes, the information-theoreticcapacity of the MDS-TSPIR problem, defined as the maximum number of informationbits of the desired file retrieved per downloaded bit, equals$1-\frac{M+T-1}{N}$, if the servers share common randomness (unavailable at theuser) with amount at least $\frac{M+T-1}{N-M-T+1}$ times the file size.Otherwise, the capacity equals zero. We conjecture that our capacity holds alsofor general MDS-TSPIR schemes.

Linear Symmetric Private Information Retrieval for MDS Coded Distributed Storage with Colluding Servers