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