Repairing Reed-Solomon Codes

Guruswami, Venkatesan; Wootters, Mary

doi:10.1109/tit.2017.2702660

Cited by 83 publications

(27 citation statements)

References 25 publications

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“…We argue that this property of SSS is directly related to the existence of linear exact repairing structures for Reed-Solomon codes [GW16,YB16]. The latter can indeed be viewed as polynomial interpolation formulae that optimize the amount of information which needs to be extracted from a reconstruction tuple to recover the shared value.…”

Section: Contributionsmentioning

confidence: 95%

Linear Repairing Codes and Side-Channel Attacks

Chabanne

Maghrebi

Prouff³

2018

TCHES

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To strengthen the resistance of countermeasures based on secret sharing,several works have suggested to use the scheme introduced by Shamir in 1978, which proposes to use the evaluation of a random d-degree polynomial into n ≥ d + 1 public points to share the sensitive data. Applying the same principles used against the classical Boolean sharing, all these works have assumed that the most efficient attack strategy was to exploit the minimum number of shares required to rebuild the sensitive value; which is d + 1 if the reconstruction is made with Lagrange’s interpolation. In this paper, we highlight first an important difference between Boolean and Shamir’s sharings which implies that, for some signal-to-noise ratio, it is more advantageous for the adversary to observe strictly more than d + 1 shares. We argue that this difference is related to the existence of so-called linear exact repairing codes, which themselves come with reconstruction formulae that need (much) less information (counted in bits) than Lagrange’s interpolation. In particular, this result implies that the choice of the public points in Shamir’s sharing has an impact on the countermeasure strength, which confirms previous observations made by Wang et al. at CARDIS 2016 for the so-called inner product sharing which is a generalization of Shamir’s scheme. As another contribution, we exhibit a positive impact of the existence of linear exact repairing schemes; we indeed propose to use them to improve the state-of-the-art multiplication algorithms dedicated to Shamir’s sharing. We argue that the improvement can be effective when the multiplication operation in the sub-fields is at least two times smaller than that of the base field.

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Section: Contributionsmentioning

confidence: 95%

Linear Repairing Codes and Side-Channel Attacks

Chabanne

Maghrebi

Prouff³

2018

TCHES

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“…Though schemes with optimal decoding bandwidth exist, improving the decoding bandwidth of Shamir's scheme remains an important problem as it is extensively used due to its simplicity. Below we describe a new family of Shamir's scheme with asymptotically optimal decoding bandwidth by extending the ideas recently developed in [5], [14] on repairing Reed-Solomon codes.…”

Section: Shamir's Scheme With Asymptotically Optimal Decoding Banmentioning

confidence: 99%

“…To reduce the decoding bandwidth, we follow the framework proposed in [5] of interpolating polynomials by querying partial polynomial evaluation. Specifically, let F be the extension field of degree l of a subfield K. During decoding, each of the n nodes applies K-linear transforms to the share over F that it holds to obtain a set of symbols over K. The decoder collects these sets of symbols and performs K-linear transforms to them to assemble the secret message over F .…”

Section: Shamir's Scheme With Asymptotically Optimal Decoding Banmentioning

confidence: 99%

“…Specifically, let F be the extension field of degree l of a subfield K. During decoding, each of the n nodes applies K-linear transforms to the share over F that it holds to obtain a set of symbols over K. The decoder collects these sets of symbols and performs K-linear transforms to them to assemble the secret message over F . Formally, viewing F as a vector space of dimension l over K, it is shown in [5] that Lemma 4. For a finite field K and its degree-l extension field F , let f be a polynomial over F of degree < t, and f (λ 1 ), · · · , f (λ n ) be n evaluations.…”

Section: Shamir's Scheme With Asymptotically Optimal Decoding Banmentioning

confidence: 99%

“…Specifically, as opposed to the original n − r symbols, the decoding DRAFT bandwidth reduces to n/(1 + r) symbols as the field size increases. The decoding algorithm follows the framework proposed in [5] of interpolating polynomials by querying partial polynomial evaluation. Our scheme is inspired by the family of Reed-Solomon (RS) codes constructed in [14] which has asymptotically optimal repair bandwidth.…”

Section: Introductionmentioning

confidence: 99%

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Secret sharing with optimal decoding and repair bandwidth

Huang

Bruck

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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This paper studies the communication efficiency of threshold secret sharing schemes. We construct a family of Shamir's schemes with asymptotically optimal decoding bandwidth for arbitrary parameters.We also construct a family of secret sharing schemes with both optimal decoding bandwidth and optimal repair bandwidth for arbitrary parameters. The construction also leads to a family of regenerating codes allowing centralized repair of multiple node failures with small sub-packetization.

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