Private Repair of a Single Erasure in Reed-Solomon Codes
<|reference_start|>Private Repair of a Single Erasure in Reed-Solomon Codes: 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.<|reference_end|>
@article{kruglik2024private,
    title={Private Repair of a Single Erasure in Reed-Solomon Codes},
    author={Stanislav Kruglik, Han Mao Kiah, Son Hoang Dau, Eitan Yaakobi},
    journal={arXiv preprint arXiv:2405.06583},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.06583},
    primaryClass={cs.IT math.IT}
}