Game-Based Security Proofs for Secret Sharing Schemes

Secret sharing schemes allow the secret to be shared among a group of parties, so that a quorum of these parties can work together to recover the secret, but less number of parties cannot learn any information of the secret. In the literature, secret sharing schemes are normally analysed using heuristic arguments rather than strict security proofs. However, such a method may overlook some security flaws, especially when it is used to analyse the secrecy property. In this paper, we illustrate this issue using some concrete examples. We show that in two existing secret sharing schemes, the secrecy property was originally conjectured to be satisfied, but the adversary still can employ some security flaws to violate this property. We then introduce a game-based model that can be used to formally analyse the secrecy property in secret sharing schemes. We prove that our model captures the definition of the secrecy property. And as an example, we show how our method can be used to analyse Shamir secret sharing scheme. Note that our method might find applications in other secret sharing schemes as well.