On the improbability of reaching Byzantine agreements

It is well known that for the Byzantine Generals Problem, no deterministic protocol can exist for an <italic>n</italic>-processor system if the number <italic>t</italic> of faulty processors is allowed to be as large as <italic>n</italic>/3. In this paper we investigate the maximum achievable agreement probability Β<subscrpt><italic>n,t</italic></subscrpt> in a model in which the faulty processors can be as devious and powerful as possible. We also discuss a restricted model with Β<subscrpt><italic>n,t</italic></subscrpt> denoting the corresponding maximum achievable probability. We will prove that: (i) for <italic>n</italic> = 3, <italic>t</italic> = 1 (the first nontrivial case), Β<subscrpt>3,1</subscrpt> = (√5 - 1)/2 (the reciprocal of the golden ratio); (ii) for all ε with 0 < ε < 1, if <italic>t</italic>/<italic>n</italic> > 1 - log (1 -ε)<supscrpt>1/2</supscrpt>/ log (1 - (1 -ε)<supscrpt>1/2</supscrpt>) then Β<supscrpt><italic>t</italic></supscrpt><subscrpt><italic>n,t</italic></subscrpt> < ε