Case 3 Sample Clauses

Case 3. 𝑣𝑎𝑙𝑢𝑒𝑠𝑟 𝑗 𝑣𝑎𝑙𝑢𝑒𝑠𝑟+2 ≠ {𝑣 }

Case 3. Any outsider or valid malicious user wants to imitate the server from sending the messages (RSA, VSA, RSB, VSB) to achieve the purpose of the at- tack. By our Definition 2, it is still computationally in- feasible since it is hard for O or C to calculate the points (rA ∗ US, rB ∗ US) by using the points (RA1, RB1, US).

Case 3. By contradiction, consider that, at the end of round t/k + 1, there exist k + 1 distinct processors q0, . . . , qk with different estimates, and which decide on their estimates. By Claim 2, there exist k + 1 processors with distinct estimates at the end of round r − 1. By Claim 4 and because r = |t/k∫ + 1, |crashedr−1| > k(r − 1) = k|t/k∫ > t − k. By Claim 3, there exist k processors that crash in round |t/k∫ + 1. Thus |crashedr| ≥ k + |crashedr−1| = k + k|t/k∫ > t. A contradiction. Q The next proposition asserts the efficiency of the algorithm.