Average Case Error Estimates for the Strong Probable Prime Test

Consider a procedure that chooses fe-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 pfc , denote the probability that this procedure returns a composite number. We obtain numerical upper bounds for pk , for various choices of k, t and obtain clean explicit functions that bound p^ , for certain infinite classes of k, t. For example, we show Pxsyo, 10 < 2-44 , Pjoo, 5 < 2-60 , P600,1 < 2~75 , and Pk,i < k242~TM 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.