Symmetric Private Polynomial Computation From Lagrange Encoding

Abstract

The problem of X-secure T -colluding symmetric Private Polynomial Computation (PPC) from coded storage system with B Byzantine and U unresponsive servers is studied in this paper. Specifically, a dataset consisting of M files are stored across N distributed servers according to (N, K + X) Maximum Distance Separable (MDS) codes such that any group of up to X colluding servers can not learn anything about the data files. A user wishes to privately evaluate one out of a set of candidate polynomial functions over the M files from the system, while guaranteeing that any T colluding servers can not learn anything about the identity of the desired function and the user can not learn anything about the M data files more than the desired polynomial function, in the presence of B Byzantine servers that can send arbitrary responses maliciously to confuse the user and U unresponsive servers that will not respond any information at all. Two novel symmetric PPC schemes using Lagrange encoding are proposed. Both the two schemes achieve the same PPC rate 1 − G(K+X−1)+T +2B N−U, secrecy rate G(K+X−1)+T N−(G(K+X−1)+T +2B+U ) , finite field size and decoding complexity, where G is the maximum degree over all the candidate polynomial functions. Particularly, the first scheme focuses on the general case that the candidate functions are consisted of arbitrary polynomials, and the second scheme restricts the candidate functions to be a finite-dimensional vector space (or sub-space) of polynomials over Fp but requires less upload cost, query complexity and server computation complexity. Remarkably, the PPC setup studied in this paper generalizes all the previous MDS coded PPC setups and the two degraded schemes strictly outperform the best known schemes in terms of (asymptotical) PPC rate, which is the main concern of the PPC schemes.

A Systematic Approach towards Efficient Private Matrix Multiplication

“…Coded computing has recently emerged as a technique of utilizing information/coding theoretical tools to inject redundant data and computations into distributed computing systems, to mitigate communication and straggler bottlenecks, and provide security and privacy for various computation tasks (see, e.g., [5], [10], [11], [15], [16], [19], [20], [22]- [30]). Privately retrieving a message from a distributed storage system without revealing the index of the message has been studied extensively in the problem of Private Information Retrieval (PIR) [31]- [37] in recent years.…”

“…Consider the same parameters as Example 1, i.e., V = m = p = n = T = 2. The partitions of matrices A, B (1) , B (2) and the desired computation C (1) = AB (1) are shown in (29) and (30), respectively. Firstly, we use Strassen's construction [43] with bilinear complexity R = 7 to encode the sub-matrices of A and B (1) , B (2) as…”