Tools Used Sample Clauses

Tools Used. Asynchronous Verifiable Secret Sharing (AVSS): An AVSS scheme consisting of a pair of pro- tocols (AVSS-Share, AVSS-Rec), can be viewed as a distributed commitment mechanism where a (▇▇▇▇▇- ▇▇▇ corrupted) special party in P, called dealer (denoted as D), commits a secret s ∈ F in AVSS-Share, where commitment information is distributed among the parties in P. Later in AVSS-Rec, the com- mitment s can be uniquely and privately reconstructed by any specific party, say Pr ∈ P (we may call it as Pr-Private-Reconstruction) even in the presence of At, except with probability 2−Ω(κ). Moreover, if D and Pr are honest, then secrecy of s from At is maintained throughout. Any AVSS satisfies the following termination property: If D is honest, then every honest party will eventually terminate AVSS-Share and AVSS-Rec, except with probability 2−Ω(n). Moreover, if some honest party has ter- minated AVSS-Share and participated in AVSS-Rec then every honest party will eventually terminate AVSS-Share and participate (and eventually terminate) AVSS-Rec, except with probability 2−Ω(n). The best known communication efficient AVSS with n = 3t+1 is due to [25]. The protocol requires a private communication O(n6n) bits (after simulating the A-cast with the protocol of [8]) to commit a secret from F. Here, we use the following syntax to invoke AVSS of [25]: AVSS-Share(D, , s) and AVSS-Rec(D, , s, Pr). Though in [25], AVSS-Rec was designed to allow all parties in to publicly reconstruct s, we can easily modify it so as to allow for Pr-Private-Reconstruction of s. Agreement on a Common Subset (ACS): In our (δ, ϵ)-ABA protocol, we come across the follow- ing situation: There exists a set of parties with t + 1, such that each party in is asked to A-cast (AVSS-Share) some value(s). While the honest parties in will eventually do the A-cast (AVSS-Share), the corrupted parties in may or may not do the same. So the (honest) parties in want to decide on a common set , with 1 t, such that A-cast (AVSS-Share) instance of each party in will be eventually terminated by the (honest) parties in . For this, the parties use the ACS primitive, presented in [7]. For the sake of completeness, the ACS protocol, along with its properties is given in APPENDIX B. The communication complexity of ACS protocol is same as that of |R| = Θ(n) executions of ABA protocol. If we use the best known communication efficient (δ, 0)-ABA of [25], then we get a (δ, 0)-ACS protocol with a private communication of O(n7n)