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
$[n,k]_{\mathbb{F}_{q^{\ell}}}$
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.