Capacity of Private Linear Computation for Coded Databases

Abstract

We consider the problem of private linear computation (PLC) in a distributed storage system. In PLC, a user wishes to compute a linear combination of f messages stored in noncolluding databases while revealing no information about the coefficients of the desired linear combination to the databases. In extension of our previous work we employ linear codes to encode the information on the databases. We show that the PLC capacity, which is the ratio of the desired linear function size and the total amount of downloaded information, matches the maximum distance separable (MDS) coded capacity of private information retrieval for a large class of linear codes that includes MDS codes. In particular, the proposed converse is valid for any number of messages and linear combinations, and the capacity expression depends on the rank of the coefficient matrix obtained from all linear combinations.

Private Polynomial Computation for Noncolluding Coded Databases

“…One can verify that Λ PIR 1,2 = 1 0 1 0 0 1 0 1 is a valid PIR achievable rate matrix for C with (κ, ν) = (1,2). This is true given that, column-wise, the Hamming weight of each column in Λ PIR 1,2 is κ = 1.…”

“…Let the desired linear function index be v = 1, i.e., the user wishes to obtain X (1) . To retrieve the desired linear function evaluation X (1) , symbols are downloaded from the two databases in a total of 4 rounds as shown in Table I. Initialization (Round τ = 1): The user first downloads 1 distinct instance of x t,j from each database.…”

“…In this section, we construct a generic query generation algorithm for a PIR-like scheme, where its dependent virtual messages represent the evaluations of the µ candidate polynomial functions. The constructed generic algorithm is a generalized version of our query generation algorithm for PLC from coded DSSs, that first appeared in [1], and will act as the main building block for the PC schemes presented in this work.…”

“…The resulting queries after sign assignment are shown in Table IV. Similar to the motivating example of Section III, Lemma 3 will be used to remove the redundant queries from each database, and one can show that the desired linear function evaluation X (1) can still be recovered. This completes the recovery part.…”

“…Example 3. Consider two messages W (1) and W (2) that are stored in a noncolluding DSS using a [4,2] RS code C . Suppose that the user wishes to obtain a monomial function evaluation X (v) from the set of nonparallel monomial functions of degree at most g = 2.…”