Average case error estimates for the strong probable prime test

Consider a procedure that chooses k-bit odd numbers independently and from the uniform distribution, subjects each number to t independent iterations of the strong probable prime test (Miller-Rabin test) with randomly chosen bases, and outputs the first number found that passes all t tests. Let {p_{k,t}} denote the probability that this procedure returns a composite number. We obtain numerical upper bounds for {p_{k,t}} for various choices of k, t and obtain clean explicit functions that bound {p_{k,t}} for certain infinite classes of k, t. For example, we show {p_{100,10}} ≤ {2^{ - 44}},{p_{300,5}} ≤ {2^{ - 60}},{p_{600,1}} ≤ {2^{ - 75}} , and {p_{k,1}} ≤ {k^2}{4^{2 - √ k }} for all k ≥ 2 . In addition, we characterize the worst-case numbers with unusually many "false witnesses" and give an upper bound on their distribution that is probably close to best possible.