Disincentivize Collusion in Verifiable Secret Sharing

In verifiable secret sharing (VSS), a dealer shares a secret input among several parties, ensuring each share is verifiable. Motivated by its applications in the blockchain space, we focus on a VSS where parties holding shares are not allowed to reconstruct the dealer's secret (even partially) on their own terms, which we address as privacy-targeted collusion if attempted. In this context, our work investigates mechanisms deterring such collusion in VSS among rational and malicious parties. For this problem, we make both algorithmic and combinatorial contributions:

1. We provide two collusion-deterrent mechanisms to discourage parties from colluding and recovering the dealer's secret. Notably, when it is desired to achieve fairness-where non-colluding parties are not at a loss-while allowing for the best achievable malicious fault tolerance, we define "trackable access structures" (TAS) and design a deterrence mechanism tailored for VSS on these structures. 2. We estimate the size of the optimal TAS, construct them from Steiner systems, provide highly robust TAS using partial Steiner systems, and present efficient secret sharing schemes for the latter close-to-optimal TAS for various parameter regimes. 3. We demonstrate that trackability in access structures is connected to combinatorial objects like (partial) Steiner systems, uniform subsets with restricted intersections, and appropriate binary codes. The robustness of access structures is equivalent to the minimum vertex cover of hypergraphs. We believe these connections between cryptography, game theory, and discrete mathematics will be of broader interest.