Common use of Asynchronous Agreement Clause in Contracts

Asynchronous Agreement. Now we give our main protocol A-Agree which uses AR- Cast as a subroutine and allows the correct nodes in an asynchronous network to agree on a reference frame. In Epoch 0 of protocol 3 A-Agree all the nodes proposes a direction which represents their local frame. They broadcast this direction using AR-Cast. All the correct nodes waits for at least (3t + 1) such broadcasts to be complete. Then they enter Epoch 1. Since, there are (3t + 1) correct nodes they will eventually arrive at Epoch 1. In this step all the correct nodes creates a bit string of length n where j’th bit represents if the j’th AR-Cast have been completed successfully in Epoch 0. Then all the nodes sends this bit string to all by playing Asynchronous- IC. After this they enters Epoch 2. In this Epoch every node has the same set of bit strings. They now look for the lowest inter k such that at least (t + 1) bit strings has a 1 in that position. If they have completed that k’th AR-Cast they output their direction received from that broad cast. If they k’th AR-Cast is not complete for him they wait until it completes and then output. The election of k ensures that at least one correct node have completed the k’th AR-Cast so by Consistency of asynchronous reference frame broadcast all the honest nodes will eventually complete the k’th AR-Cast. This ensure that the A-Agree eventually completes. As, there is no loop in this protocol and all the subroutines run in constant expected time. The A-Agree is also a constant expected time protocol.

Asynchronous Agreement. Now we give our Our main protocol A-Agree which uses AR- AR-Cast as a subroutine and allows the correct nodes in an asynchronous network to agree on a reference frame. This protocol also have several epochs (or states). In Epoch 0 of protocol 3 A-Agree all the nodes proposes a direction which represents their local frame. They broadcast broad- cast this direction using AR-Cast. All the correct nodes waits for at least (3t + 1) such broadcasts to be complete. Then they enter Epoch 1. Since, there are (3t + 1) correct nodes they will eventually arrive at Epoch Epoch 1. In this step all the correct nodes creates a bit string of length n where j’th bit represents if the j’th AR-Cast have been completed successfully in Epoch 0. Then all the nodes sends this bit string to all by playing Asynchronous- Asynchronous-IC. The Asynchronous-IC is a classical protocol given by Ben- Or and El-Yaniv in [26]. After this they enters Epoch 2. In this Epoch every node has the same set of bit strings. They now look for the lowest inter k such that at least (t + 1) bit strings has a 1 in that position. If they have completed that k’th AR-Cast they output their direction received from that broad cast. If they k’th AR-Cast is not complete for him they wait until it completes and then output. The election of k ensures that at least one correct node have completed the k’th AR-Cast so by Consistency of asynchronous reference frame broadcast all the honest nodes will eventually complete the k’th AR-Cast. This ensure that the A-Agree eventually completes. As, there is no loop in this protocol and all the subroutines run in constant expected timetime [28]. The A-Agree is also a constant expected time protocol.