In this work we focus on two basic secure distributed computation tasks- Probabilistic Weak Secret Sharing (PWSS) and Probabilistic Verifiable Secret Sharing (PVSS). PVSS allows a dealer to share a secret among several players in a way that would later allow a unique reconstruction of the secret with negligible error probability. PWSS is slightly weaker version of PVSS where the dealer can choose not to disclose his secret later. Both of them are well-studied problems. While PVSS is used as a building block in every general probabilistic secure multiparty computation, PWSS can be used as a building block for PVSS protocols. Both these problems can be parameterized by the number of players (n) and the fault tolerance threshold (t) which bounds the total number of malicious (Byzantine) players having unbounded computing power. We focus on the standard secure channel model, where all players have access to secure point-topoint channels and a common broadcast 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 we show the following: (a) 1-round PWSS is possible iff n> 3t and (b) 3-round PWSS is possible if n> 2t. All our protocols are efficient. Comparing our results with the existing trade-off results for perfect (zero error probability) VSS and WSS, we find that probabilistically relaxing the conditions of VSS/WSS helps to increase fault tolerance significantly
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