Resilient-optimal interactive consistency in constant time

For a complete network of n processors within which communication lines are private, we show how to achieve concurrently many Byzantine Agreements within constant expected time both on synchronous and asynchronous networks. As an immediate consequence, this provides a solution to the Interactive Consistency problem. Our algorithms tolerate up to (n - 1)/3 faulty processors in both the synchronous and asynchronous cases and are therefore resilient-optimal. In terms of time complexity, our results improve a time bound of O(log n) (for n concurrent agreements) which is immediately implied by the constant expected time Byzantine Agreement of Feldman and Micali (synchronous systems) and of Canetti and Rabin (asynchronous systems). In terms of resiliency, our results improve the resiliency bound of the constant time, O(4√n)-resilient algorithm of Ben-Or. An immediate application of our protocols is a constant expected time simulation of simultaneous broadcast channels over a network with private lines.