Output Phase. Same as protocol Output in Optimal-A-cast with t being replaced by tj, A being replaced by Æ . t Moreover, CORE contains 2tj + 1 parties and CORE = P \ CORE.
Output Phase. The segment failure may occur only in the second phase and hence only the first two phases of a segment may be repeated several times (bounded by t); once the first two phases are successful for a segment, the segment will always be successfully completed after robustly executing the third phase. So at the end of segment α, every honest party will agree on a common A bits, denoted by m∗α. Moreover if the honest parties start with common input (i.e. miα’s are equal for all honest parties), then m∗α will be same as that common input. t P P P ⊆ P
Output Phase. Here the (honest) parties in CORE propagate the common message held by them (which they have received from S) to all other (honest) parties in P \ CORE. P X X C D D P C D Informally, in Distribution Phase, S sends his message m to every party in . In Verification & Agreement on CORE Phase, party Pi on receiving a message, say mi from S, computes n hash values of mi corresponding to n distinct random hash keys, say ri1, . . . , rin chosen from F. To enable Pj to check whether his received message mj is same as Xx’s received message mi, party Pi privately sends rij and ij = κ(mi, rij) to Pj. Party Pj, on receiving these values from Pi checks whether ij = κ(mj, rij) (for honest S and Pi, it should hold). Xx Xxxxxx-A-casts a confirmation signal OK(Pj, Pi) if the above check passes. Now based on the confirmation signals, a graph with the parties in as vertex set is formed and applying Find-STAR on the graph, an (n, t)-star ( , ) is obtained. The ( , ) is then agreed among all the parties and component of it is taken as CORE. The protocols for Distribution Phase and Verification & Agreement on CORE Phase are given below. Before outlining Output Phase, we prove Lemma 1-3.
Output Phase. The segment failure may occur only in first phase and hence only the first phase of a segment may be repeated several times (bounded by t); once the first phase is successful for a segment, the segment will always be successfully completed after robustly executing second phase.
Output Phase. Here the parties in CORE help the parties in P \ CORE (not P' \ CORE) to learn the common l/t message m∗α held by the honest parties in CORE. After this phase, current segment terminates with common output m∗α and the parties proceed to the computation of next segment. The purpose and the implementation of this phase is almost identical to the Output Phase of Optimal-A-cast. Now the overall structure of Optimal-ABA is as follows: Code for Pi: Protocol Optimal-ABA(P)
Output Phase. Output 1 if and only if Verify(pp, {vk1, . . . , vkn}, m′, σ′) = 1 and m′ =ƒ m.
Output Phase. Here the parties in Pe′x help the parties in P \Pe′x (not P′ \ Pe′x) to learn the common l/t message m held by the honest parties in Pe′x. After this phase, current segment terminates with common output m∗α and the parties proceed to the computation of next segment. The implementation of this phase is very similar to the implementation of the Output Phase of [21] and the Claiming Stage of the BA protocol of [15]. Now the overall structure of Optimal-ABA is presented below. Protocol Optimal-ABA(P) Code for Pi: Every party in P executes this code.
