Gradecast is a simple three-round algorithm presented by Feldman and Micali. The current work presents a very simple 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^k), 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^3), 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.
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