Gradecast is a simple three-round algorithm presented by Feldman and Micali. The current work presents a very simple synchronous algorithm that utilized Gradecast to achieve Byzantine agreement. Two small variations of the presented algorithm lead to improved algorithms for solving the Approximate agreement problem and the Multi-consensus problem. An optimal approximate agreement algorithm was presented by Fekete, which supports up to 1 n Byzantine nodes and has message complexity of O(n k ), where n is the number of nodes and k is the number of rounds. Our solution to the approximate agreement problem is optimal,
This paper introduces some algorithms to solve crash-failure, failure-by-omission and Byzantine failure versions of the Byzantine Generals or consensus problem, where non-faulty processors need only arrive at values that are close together rather than identical. For each failure model and each value ofS, we give at-resilient algorithm usingS rounds of communication. IfS=t+1, exact agreement is obtained. In the algorithms for the failure-by-omission and Byzantine failure models, each processor attempts to identify the faulty processors and corrects values transmited by them to reduce the amount of disagreement. We also prove lower bounds for each model, to show that each of our algorithms has a convergence rate that is asymptotic to the best possible in that model as the number of processors increases.