Simple Gradecast Based Algorithms

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 4 n Byzantine nodes and has message complexity of O(n), where n is the number of nodes and k is the number of rounds. Our solution to the approximate agreement problem is optimal, simple and reduces the message complexity to O(k · n), while supporting up to 1 3 n Byzantine nodes. Multi consensus was first presented by Bar-Noy et al. It consists of consecutive executions of l Byzantine consensuses. Bar-Noy et al., show an optimal amortized solution to this problem, assuming that all nodes start each consensus instance at the same time, a property that cannot be guaranteed with early stopping. Our solution is simpler, preserves round complexity optimality, allows early stopping and does not require synchronized starts of the consensus instances.