This paper studies the problem of Byzantine consensus in a synchronous message-passing system of n processes. The first deterministic algorithm, and also the simplest in its principles, was the Exponential Information Gathering protocol (EIG) proposed by Pease, Shostak and Lamport in . The algorithm requires processes to send exponentially long messages. Many follow-up works reduced the cost of the algorithm. However, they had to either lower the maximum number of faulty processes t from the optimal range t < n/3 to some smaller range of t [4, 11, 18], or increase the maximum worst-case number of rounds needed for termination (the lower bound being t + 1) [3, 9, 20]. Garay and Moses were the first and only who solved the problem by using a polynomial number of communication bits, for the whole optimal range t < n/3 of the number of Byzantine processes and within the optimal number (t+1) of communication rounds. Their solution, though very complex and sophisticated, requires processes to send O(n9) bits in total. In this work, we present much simpler solution that also holds for the whole optimal range t < n/3 and the optimal number t + 1 of communication rounds, and at the same time lowers the number of exchanged communication bits to O(n3 log n). For achieving such an improvement, processes no more exchange relayed proposed values, but information on suspicions "who suspects who", the size of which is quadratic in n in the worst case.