Probabilistic Verifiable Secret Sharing Tolerating Adaptive Adversary

In this work, we focus on two basic secure distributed computation tasksProbabilistic WeakSecret Sharing (PWSS) and Probabilistic Verifiable Secret Sharing (PVSS). PVSS allows a dealerto share a secret among several players in a way that would later allow a unique reconstruction ofthe secret with negligible error probability. PWSS is slightly weaker version of PVSS where thedealer can choose not to disclose his secret later. Both of them are well-studied problem. WhilePVSS is used as building block in every general probabilistic secure multiparty computation, PWSScan be used as building block for PVSS protocols. Both these problems can be parameterized bythe number of players (n) and the fault tolerance threshold (t) which bounds the total number ofmalicious (Byzantine) players having unbounded computing power. We focus on the standard securechannel model, where all players have access to secure point-to-point channels and a common broad-cast medium. We show the following for PVSS: (a) 1-round PVSS is possible iff t = 1 and n > 3(b) 2-round PVSS is possible if n > 3t (c) 4-round PVSS is possible if n > 2t. For the PWSS weshow the following: (a) 1-round PWSS is possible iff n > 3t and (b) 3-round PWSS is possible ifn > 2t. All our protocols are efficient. Comparing our results with the existing trade-off results forperfect (zero error probability) VSS and WSS, we find that probabilistically relaxing the conditionsof VSS/WSS helps to increase fault tolerance significantly.