Average case error estimates for the strong probable prime test

Abstract

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 {p_{k,t}} denote the probability that this procedure returns a composite number. We obtain numerical upper bounds for p k , t {p_{k,t}} for various choices of k, t and obtain clean explicit functions that bound p k , t {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 {p_{100,10}} \leq {2^{ - 44}},{p_{300,5}} \leq {2^{ - 60}},{p_{600,1}} \leq {2^{ - 75}} , and p k , 1 ≤ k 2 4 2 − k {p_{k,1}} \leq {k^2}{4^{2 - \sqrt k }} for all k ≥ 2 k \geq 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.