Send Sample Clauses

Send or “sent electronically” means to send or have sent any information or instructions by way of fax, email or any other form of electronic communication or messaging; and
Send. Bˆ, Pˆi) to obtain Y .
Send. Bˆ, Pˆi, Y, Zi0 ).l
Send. CommitAggPvss(ID, H(pvss), Σ) to all parties
Send. KeyRec(shA, shB, h) to all parties
Send. KeyShare(ID, C, A(j), B(j)) to Pj for each j ∈ [n]j
Send. Seed(ID, H(pvss), Σ, seed) to all partiesDeferred proofs for Lemma 1. Here we prove that the protocol in Alg. 5 satisfies all properties of Reliable Leaded Seeding given in Definition 3 and analyze how does it incur only quadratic messages and communications.Proof. Here prove that Alg. 5 satisfy the Seeding properties one by one:– Totality. Assume that an honest party outputs in the Seeding, it must have received 2f + 1 SeedReady messages. At least f + 1 of the messages are sent from honest parties. Therefore, all parties will eventually receive f + 1 SeedReady messages from these honest parties and then send a SeedReady messages as well (if a SeedReady message has not been sent yet). So the totality property always holds.HH −−– Correctness. In the seed aggregation phase, the leader will collect 2f + 1 valid pvssi scripts and aggregate them into a pvss whose weight has 2f + 1 positions to be 1. In the seed commitment phase, all honest parties sign for this pvss so that the leader can collect at least n f valid signatures for (pvss) to form valid Σ. In the seed recovery phase, the leader can collect at least n f valid shares which are committed to the pvss they have signed for. All honest parties can receive the same (pvss), Σ and seed that pass verifications. So they would broadcast the same SeedEchomessage and the same SeedReady messages, thus finally outputting in the Seeding instance.PHP P ƒP P– Commitment. Assume that Pi is the first party who starts to run the protocol’s revealing phase, it implies that i received a valid CommitAggPvss(ID, h, Σ) message from leader L. If another honest j received a valid CommitAggPvss(ID, hj, Σj) message from leader L, where hj = h, since a valid Σ contains 2f + 1 valid signatures for a same hash value from distinct parties, it induces that at least one honest party signed for both h and hj, which is impossible. Hence, when some honest party i starts to run the protocol’s revealing phase, the h from any valid CommitAggPvss(ID, h, Σ) message is unique. Following the commitment of the PVSS scheme, there exists a fixed value seed corresponding to the pvss, where h = (pvss).Suppose that some honest party outputs seedj from the Seeding. By the code, it receives 2f + 1SeedReady messages containing seedj. Then at least one honest party received 2f + 1 valid SeedEcho messages with the same seedj from distinct parties, which means that at least f + 1 honest parties received valid Seed(ID, h, Σ, seedj) message from the leader. ...
Send. SeedEcho(ID, seed) to all parties
Send. SeedReady(ID, seed) to all parties if SeedReady not sent yet