Abstract
We look at two basic coding theoretic and cryptographic mechanisms developed separately and investigate relationships between them and their implications. The first mechanism is Proactive Secret Sharing (PSS), which allows randomization and repair of shares using information from other shares. PSS enables constructing secure multi-party computation protocols that can withstand mobile dynamic attacks. This self-recovery and the redundancy of uncorrupted shares allows a system to overcome recurring faults throughout its lifetime, eventually finishing the computation (or continuing forever to maintain stored data). The second mechanism is Regenerating Codes (RC) which were extensively studied and adopted in distributed storage systems. RC are error correcting (or erasure handling) codes capable of recovering a block of a distributedly held codeword from other servers’ blocks. This self-healing nature enables more robustness of a code distributed over different machines. Given that the two mechanisms have a built-in self-healing (leading to stabilizing) and that both can be based on Reed Solomon Codes, it is natural to formally investigate deeper relationships between them. We prove that a PSS scheme can be converted into an RC scheme, and that under some conditions RC can be utilized to instantiate a PSS scheme. This allows us, in turn, to leverage recent results enabling more efficient polynomial interpolation (due to Guruswami and Wooters) to improve the efficiency of a PSS scheme. We also show that if parameters are not carefully calibrated, such interpolation techniques (allowing partial word leakage) may be used to attack a PSS scheme over time. Secondly, the above relationships give rise to extended (de)coding notions. Our first example is mapping the generalized capabilities of adversaries (called generalized adversary structures) from the PSS realm into the RC one. Based on this we define a new variant of RC we call Generalized-decoding Regenerating Code (GRC) where not all network servers have a uniform sub-codeword (motivated by non-uniform probability of attacking different servers case). We finally highlight several interesting research directions due to our results, e.g., designing new improved GRC, and more adaptive RC re-coding techniques.
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- 1.
Our condition is that an RC code is MDS. This is to simplify this first treatment of the topic, we note that there is more work required to understand what conditions on the RC side imply certain types of security on the PSS side.
- 2.
- 3.
Regenerating codes are studied from the point of view of erasure recovery in coding theory literature. However, here we refer to it as a subset of error-correction because we also care about corruption of nodes along with node failures.
- 4.
For example, Corollary 9 in [8] constructs these polynomials as \(\mu _{\zeta , \alpha }(\alpha ^*) = p(\alpha )\cdot \frac{\prod _{\beta \in A\setminus \{\alpha ^*\}}(\beta - \alpha ^*)}{\prod _{\beta \in A\setminus \{\alpha \}}(\beta - \alpha )}\) where p is a polynomial dependent only on the evaluation points A.
- 5.
That is, we only consider perfect secret sharing schemes [27, Definition 11.59].
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Eldefrawy, K., Genise, N., Kshirsagar, R., Yung, M. (2021). On Regenerating Codes and Proactive Secret Sharing: Relationships and Implications. In: Johnen, C., Schiller, E.M., Schmid, S. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2021. Lecture Notes in Computer Science(), vol 13046. Springer, Cham. https://doi.org/10.1007/978-3-030-91081-5_23
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