Further investigations with the strong probable prime test

Abstract

Abstract. Recently, Damgård, Landrock and Pomerance described a procedure in which a k-bit odd number is chosen at random and subjected to t random strong probable prime tests. If the chosen number passes all t tests, then the procedure will return that number; otherwise, another k-bit odd integer is selected and then tested. The procedure ends when a number that passes all t tests is found. Let p k,t denote the probability that such a number is composite. The authors above have shown that p k,t ≤ 4 −t when k ≥ 51 and t ≥ 1. In this paper we will show that this is in fact valid for all k ≥ 2 and t ≥ 1.

On Assessing the Performance of Randomized Algorithms

“…2− √ k k ≥ 2 and methods for finding sharper numerical bounds to p k t for k = 100 150 600 and t = 1 10 (see their Burthe [4] improves the results in [6] and provides tables of upper bounds to p k t t = 1 2 and 25 ≤ k ≤ 50, again for the strong probable test. In his computing he no longer takes P f but uses P f / P f + 1 as in our (3.6) (see his Table IV, p. 381).…”

“…The most important result of the paper is that, for all k ≥ 2 and for all t ≥ 1 the probability p k t ≤ 4 −t , which is the naive upper bound for this problem (recall that at each trial the error is at most 4 −1 ). Burthe [4] …”

“…We can now come back to the question of comparing fixed base tests with random base ones. For small values of n this comparison can be done since Burthe's [4] Table III contains exact values p k 1 while lists of pseudoprimes (or strong pseudoprimes) are available. In Table 2 we make this comparison for a few selected intervals.…”

An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates

“…Note that [5] contains sharper numeric estimates for the MR test than what the above type of analysis implies, and also more work has been done in this direction after [5], for instance [2]. However, such methods for better estimates on the MR test could also be applied to our test so that the relative strengths of the tests is likely to remain the same.…”

“…We stated above that Granthams test (our test) takes time approximately equivalent to 3 (2) Miller-Rabin tests. What we mean by this more precisely is that the running time of Miller-Rabin, resp.…”