Average case error estimates for the strong probable prime test

Abstract

Abstract. 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~™ 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.

Further investigations with the strong probable prime test

“…In this paper we will improve the results in [3] and show that p k,t ≤ 4 −t for all k with k ≥ 2 and t ≥ 1. This will be done by extending some of the ideas in [3] and sharpening the upper bounds found there as well. Some improvements are due to simple observations of the properties of certain numbers and easily lead to a lower upper bound.…”

“…Taking l = 2 in the above inequality, we can see that to show that p k,t ≤ 4 −t for all t ≥ 1 it will also suffice to show that p k,1 ≤ 1/4 and p k,2 ≤ 1/17. In [3], this is shown to be true for all k with 51 ≤ k ≤ 54. In this paper we will improve the results in [3] and show that p k,t ≤ 4 −t for all k with k ≥ 2 and t ≥ 1.…”

“…In [3], this is shown to be true for all k with 51 ≤ k ≤ 54. In this paper we will improve the results in [3] and show that p k,t ≤ 4 −t for all k with k ≥ 2 and t ≥ 1. This will be done by extending some of the ideas in [3] and sharpening the upper bounds found there as well.…”

“…In [3], it is shown that this is true for k ≥ 55. Taking l = 2 in the above inequality, we can see that to show that p k,t ≤ 4 −t for all t ≥ 1 it will also suffice to show that p k,1 ≤ 1/4 and p k,2 ≤ 1/17.…”

Finding strong pseudoprimes to several bases. II

“…Monier [8] and Rabin [13] proved that if n is an odd composite positive integer, then SB(n) ≤ (n−1)/4. In fact, as pointed by Damgård, Landrock and Pomerance [4], if n = 9 is odd and composite, then SB(n) ≤ ϕ(n)/4, i.e., P R (n) ≤ 1/4. These facts lead to the Rabin-Miller test: given a positive integer n, pick k different positive integers less than n and perform the Miller test on n for each of these bases; if n is composite, the probability that n passes all k tests is less than 1/4 k .…”