Verifiable Secret Sharing Based on Hyperplane Geometry with Its Applications to Optimal Resilient Proactive Cryptosystems

Abstract

Secret sharing, first introduced by Shamir and Blakley independently, is an important technique to ensure secrecy and availability of sensitive information. It is also an indispensable building block in various cryptographic protocols. In the literature, most of these existing protocols are employing Shamir’s secret sharing, while Blakley’s one has attracted very little attention. In this paper, we revisit Blakley’s secret sharing that is based on hyperplane geometry, and illustrate that some of its potentials are yet to be employed. In particular, it has an appealing property that compared with Shamir’s secret sharing, it not only handles (t, n) secret sharing with similar computational costs, but also handles (n, n) secret sharing with better efficiency. We further apply this property to design a provably secure and optimal resilient proactive secret sharing scheme. Our proposed protocol is versatile to support proactive cryptosystems based on various assumptions, and it employs only one type of verifiable secret sharing as the building block. By contrast, the existing proactive secret sharing schemes with similar properties all employ two different types of verifiable secret sharing. Finally, we briefly discuss some possible extensions of our proposed protocol as well as how to explore more potentials of Blakley’s secret sharing.