A New Efficient and Secure Secret Reconstruction Scheme (SSRS) with Verifiable Shares Based on a Symmetric Bivariate Polynomial

Secret sharing (SS) schemes have been widely used in secure computer communications systems. Recently, a new type of SS scheme, called the secure secret reconstruction scheme (SSRS), was proposed, which ensures that the secret can only be recovered by participants who present valid shares. In other words, if any outside adversary participated in the secret reconstruction without knowing any valid share, the secret cannot be recovered by anyone including the adversary. However, the proposed SSRS can only prevent an active attacker from obtaining the recovered secret, but cannot prevent a passive attacker from obtaining the secret since exchange information among participants is unprotected. In this paper, based on bivariate polynomials, we propose a novel design for the SSRS that can prevent both active and passive attackers. Furthermore, we propose a verification scheme which can verify all shares at once, i.e., it allows all shareholders to efficiently verify that their shares obtained from the dealer are generated consistently without revealing their shares and the secret. The proposed scheme is really attractive for efficient and secure secret reconstruction in communications systems.