Broadcast for t< n Sample Clauses

Broadcast for t< n. 2 Since weak 2-cast secure against one corrupted player implies efficient global broadcast for t < n/2 (Proposition 4), it is sufficient to demonstrate the existence of an efficient protocol for weak 2-cast that tolerates one corrupted player. We now describe such a protocol for a sender s and two recipients r0 and r1.4 { } ∈ { } Let xs 0, 1 be the input of sender s, and y0 and y1 be the values the players r0 and r1 finally decide on (whereas s always implicitly decides on ys = xs). The primitive Q-Flip is invoked some m times and each player receives a sequence of m elements in 0, 1, 2 , i.e., s receives Qs = (Qs[1],..., Qs[m]), r0 receives Q0 = (Q0[1],..., Q0[m]), and r1 receives Q1 = (Q1[1],..., Q1[m]), where the triplet (Qs[i], Q0[i], Q1[i]) represents the outcome of the i-th invocation of Q-Flip. The protocol now proceeds as follows: ∈ { } First, s sends to r0 and r1 his input bit xs and the set σs of all indices i 1,..., m such that s received the complement of xs for the i-th invocation of Q-Flip (see Figure 1–A): σs = {i ∈ {1,..., m} : Qs[i] = 1 − xs} . (1) { ∈ − } ∅ ∈ { } − If this is done correctly then σs is of large size (i.e., approximately m/3, which is important for good statistics on the corresponding Q-Flips) and both recipients rk (k 0, 1 ) never received the value 1 xs for any Q-Flip invocation with respect to σs: i σs : Qk[i] = 1 xs = . − ∈ { } Let xk and σk (k 0, 1 ) be the information that (potentially faulty) s actually sent to recipient rk. The recipients now decide on yk = xk if and only if σk is of large size and the value 1 xk was never received with respect to σk: . yk := xk , if (σk large) ∧ ({i ∈ σk : Qk[i] = 1 − xk} = ∅) ⊥ , else .