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 ${\mathbf{c}} \in {\mathbb{F}^n}$, when we are given access to d of the remaining components in c. Formally, suppose that $\mathbb{F}$ is a field extension of $\mathbb{B}$ 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 $\mathbb{B}$ from d available nodes, whenever $d \geq \ell |\mathbb{B}{|^s} - \ell + k$. 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.