Private Repair of a Single Erasure in Reed-Solomon Codes

TL;DR

This work tackles privately repairing a single erasure in Reed-Solomon codes under limited communication by extending subspace-polynomial repair to $t$-privacy via randomization. It presents two constructions: a hidden-subspace 1-private scheme and a secret-sharing based $t$-private scheme, both achieving bandwidth $(n-1)(\ell-m)$ sub-symbols in $\mathbb{F}_q$ subject to $n-k \ge |\mathbb{B}|^{m}+t-1$, with optimality shown for $n=q^{\ell}$ under a mild assumption. The paper also connects private repair to private retrieval and establishes a lower bound on private repair bandwidth, which matches the proposed scheme in key parameter regimes. Numerical results illustrate the bandwidth performance and privacy tradeoffs, highlighting practical relevance for RS-coded data stores. Overall, the work advances privacy-aware RS repair and retrieval, revealing fundamental bandwidth-privacy tradeoffs and guiding future explorations in multi-node privacy and high sub-packetization settings.

Abstract

We investigate the problem of privately recovering a single erasure for Reed-Solomon codes with low communication bandwidths. For an $[n,k]_{q^\ell}$ code with $n-k\geq q^{m}+t-1$, we construct a repair scheme that allows a client to recover an arbitrary codeword symbol without leaking its index to any set of $t$ colluding helper nodes at a repair bandwidth of $(n-1)(\ell-m)$ sub-symbols in $\mathbb{F}_q$. When $t=1$, this reduces to the bandwidth of existing repair schemes based on subspace polynomials. We prove the optimality of the proposed scheme when $n=q^\ell$ under a reasonable assumption about the schemes being used. Our private repair scheme can also be transformed into a private retrieval scheme for data encoded by Reed-Solomon codes.

Figures (2)

• Figure 1: A toy example of a private repair scheme operating in a distributed storage system that stores two objects $\pmb{a}=(a_1,a_2)\in \mathbb{F}_4$ and $\pmb{b}=(b_1,b_2)\in \mathbb{F}_4$ with two parities using a systematic $[4,2]_{\mathbb{F}_4}$ Reed-Solomon code (see Example \ref{['ex:toy']} for more details). The client, which plays the role of a repair node, requests $a_2+b_1=(a_1+a_2+b_2)+(a_1+b_1+b_2)$ from the shaded parity node, which cannot tell if the request is for repairing $\pmb{a}$ or $\pmb{b}$, as both cases are equally probable. A similar example can also be constructed for the bottom parity node. Such a repair scheme (based on trace polynomials) is private against one curious node.
• Figure 2: Recovery bandwidth for private repair of RS code with $k=99$ over $\textrm{GF}(2^{8})$

Theorems & Definitions (23)

• Definition 1: originalRSpaper
• Definition 2
• Proposition 1: Subspace-polynomial repair scheme guruswami2016repairingguruswami2017repairingdau2017optimaldau2021repairing
• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof : Proof of Theorem \ref{['thm:hidden_subspace']}