Consider an asynchronous network in a shared-memory environment consisting of
n nodes. Assume that up to f of the nodes might be Byzantine (n > 12f), where
the adversary is full-information and dynamic (sometimes called adaptive). In
addition, the non-Byzantine nodes may undergo transient failures. Nodes advance
in atomic steps, which consist of reading all registers, performing some
calculation and writing to all registers.
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.