Abstract
The round complexity of verifiable secret sharing (VSS) schemes has been studied extensively for threshold adversaries. In particular, Fitzi et al. showed an efficient 3-round VSS for n ≥ 3t + 1 [4], where an infinitely powerful adversary can corrupt t (or less) parties out of n parties. This paper shows that for non-threshold adversaries:
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1
Two round perfectly secure VSS is possible if and only if the underlying adversary structure satisfies the \({\cal Q}^4\) condition;
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2
Three round perfectly secure VSS is possible if and only if the underlying adversary structure satisfies the \({\cal Q}^3\) condition.
Further as a special case of our three round protocol, we can obtain a more efficient 3-round VSS than the VSS of Fitzi et al. for n = 3t + 1. More precisely, the communication complexity of the reconstruction phase is reduced from \({\cal O}(n^3)\) to \({\cal O}(n^2)\). We finally point out a flaw in the reconstruction phase of the VSS of Fitzi et al., and show how to fix it.
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Choudhury, A., Kurosawa, K., Patra, A. (2011). The Round Complexity of Perfectly Secure General VSS. In: Fehr, S. (eds) Information Theoretic Security. ICITS 2011. Lecture Notes in Computer Science, vol 6673. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20728-0_14
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