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Average Case Error Estimates for the Strong Probable Prime Test

Ivan Damgård, Peter Landrock and Carl Pomerance
Mathematics of Computation
Vol. 61, No. 203, Special Issue Dedicated to Derrick Henry Lehmer (Jul., 1993), pp. 177-194
DOI: 10.2307/2152945
Stable URL: http://www.jstor.org/stable/2152945
Page Count: 18
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Average Case Error Estimates for the Strong Probable Prime Test
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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 pk, t denote the probability that this procedure returns a composite number. We obtain numerical upper bounds for pk, t for various choices of k, t and obtain clean explicit functions that bound pk, t for certain infinite classes of k, t. For example, we show p100, 10 ≤ 2-44, p300, 5 ≤ 2-60, p600, 1 ≤ 2-75, and $p_{k, 1} \leq k^2 4^{2 - \sqrt k}$ for all $k \qeq 2$. 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.

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