Secret Sharing with Binary Shares
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Secret sharing is a fundamental cryptographic primitive. One of the main goals of secret sharing is to share a long secret using small shares. In this paper we consider a family of statistical secret sharing schemes indexed by N, the number of players. The family is associated with a pair of relative thresholds τ and κ, that for a given N, specify a secret sharing scheme with privacy and reconstruction thresholds, Nτ and Nκ, respectively. These are non-perfect schemes with gap N(κ-τ) and statistical schemes with errors ϵ(N) and δ(N) for privacy and reconstruction, respectively. We give two constructions of secret sharing families as defined above, with security against (i) an adaptive, and (ii) a non-adaptive adversary, respectively. Both constructions are modular and use two components, an invertible extractor and a stochastic code, and surprisingly in both cases, for any κ>τ, give explicit families for sharing a secret that is a constant fraction (in bits) of N, using binary shares. We show that the construction for non-adaptive adversary is optimal in the sense that it asymptotically achieves the upper bound N(κ-τ) on the secret length. We relate our results to known works and discuss open questions.
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