Proactive Secret Sharing with a Dishonest Majority

In standard Secret Sharing SS a dealer shares a secret s among n parties such that an adversary corrupting no more than t parties does not learn s, while any $$t+1$$ parties can efficiently recover s. Over a long period of time all parties may be corrupted and the threshold t may be violated, which is accounted for in Proactive Secret Sharing PSS. PSS retains confidentiality even when a mobile adversary corrupts all parties over the lifetime of the secret, but no more than a threshold t during a certain window of time, called the refresh period. Existing PSS schemes only guarantee secrecy in the presence of an honest majority with at most $$n/2-1$$ total corruptions during such a refresh period; an adversary that corrupts a single additional party beyond the $$n/2-1$$ threshold, even if only passively and only temporarily, obtains the secret. We develop the first PSS scheme secure in the presence of a dishonest majority. Our PSS scheme is robust and secure against $$t<n-2$$ passive adversaries when there are no active corruptions, and secure but non-robust but with identifiable aborts against $$t<n/2-1$$ active adversaries when there are no additional passive corruptions. The scheme is also secure with identifiable aborts against mixed adversaries controlling a combination of passively and actively corrupted parties such that if there are k active corruptions there are less than $$n-k-2$$ total corruptions. Our scheme achieves these high thresholds with $$On^4$$ communication when sharing a single secret. We also observe that communication may be reduced to $$On^3$$ when sharing On secrets in batches. Our work is the first result demonstrating that PSS tolerating such high thresholds and mixed adversaries is possible.