Explicit Low-Bandwidth Evaluation Schemes for Weighted Sums of Reed-Solomon-Coded Symbols

Motivated by applications in distributed storage, distributed computing, and homomorphic secret sharing, we study communication-efficient schemes for computing linear combinations of coded symbols. Specifically, we design low-bandwidth schemes that evaluate the weighted sum of ℓ coded symbols in a codeword c ∈ F n , when we are given access to d of the remaining components in c . Formally, suppose that F is a field extension of B of degree t . Let c be a codeword in a Reed-Solomon code of dimension k and our task is to compute the weighted sum of ℓ coded symbols. In this paper, for some s < t , we provide an explicit scheme that performs this task by downloading d ( t - s ) sub-symbols in B from d available nodes, whenever d ≥ ℓ|B| s -ℓ + k . In many cases, our scheme outperforms previous schemes in the literature. Furthermore, we provide a characterization of evaluation schemes for general linear codes. Then in the special case of Reed-Solomon codes, we use this characterization to derive a lower bound for the evaluation bandwidth.