Evaluation and comparison of two efficient probabilistic primality testing algorithms

Monier, Louis

doi:10.1016/0304-3975(80)90007-9

Cited by 76 publications

(55 citation statements)

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“…Write n − 1 = 2 k q with q odd; and p i − 1 = 2 ki q i with q i odd for 1 ≤ i ≤ s, ordering the p i s such that k 1 ≤ · · · ≤ k s . Monier [8] proved that SB(n) = 1 + 2 k1s − 1 2 s − 1 s i=1 gcd(q, q i ).…”

Section: Discussionmentioning

confidence: 99%

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Finding strong pseudoprimes to several bases. II

Zhang

Tang

2003

Math. Comp.

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Abstract. Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n < ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ 9 , ψ 10 and ψ 11 were first given by Jaeschke, and those for ψ 10 and ψ 11 were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).In this paper, we first follow the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) n < 10 24 to the first five or six prime bases, which have the form n = p q with p, q odd primes and q − 1 = k(p−1), k = 4/3, 5/2, 3/2, 6; then we tabulate all Carmichael numbers < 10 20 , to the first six prime bases up to 13, which have the form n = q 1 q 2 q 3 with each prime factor q i ≡ 3 mod 4. There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for ψ 9 , ψ 10 and ψ 11 are lowered from 20-and 22-decimal-digit numbers to a 19-decimal-digit number:We conjecture that ψ 9 = ψ 10 = ψ 11 = 3825 12305 65464 13051, and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor q 3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.

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Section: Discussionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Finding strong pseudoprimes to several bases. II

Zhang

Tang

2003

Math. Comp.

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“…Rabin [11] and Monier [9] independently showed that (M, A 2 ) has probability of failure less than 1 4 . Example 2.5.…”

Section: Definition 21 ([5]) Let a ⊆ N An Elementary Probabilisticmentioning

confidence: 99%

A generalization of Miller’s primality theorem

Berrizbeitia

Olivieri

2008

Proc. Amer. Math. Soc.

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“…However, we will allow m to assume nonintegral values. Since α(n) ≤ 1/4 for odd composite n = 9 (see [4] or [5]) and since α(9) = 1/3, we will have C m = ∅ for 0 < m ≤ ln 3/ ln 2 and C m = {9} for ln 3/ ln 2 < m ≤ 2.…”

Section: Estimatesmentioning

confidence: 99%

“…If n is prime, then S(n) = n − 1, and if n is an odd composite number, then S(n)/(n − 1) ≤ 1/4 (see Monier [4], Rabin [5]). Now if for a given n we can find an integer a ∈ [1, n − 1] such that a / ∈ S(n), then we know that n is composite.…”

Section: Introductionmentioning

confidence: 99%

Further investigations with the strong probable prime test

Burthe¹

1996

Math. Comp.

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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.

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Evaluation and comparison of two efficient probabilistic primality testing algorithms

Cited by 76 publications

References 3 publications

Finding strong pseudoprimes to several bases. II

Finding strong pseudoprimes to several bases. II

A generalization of Miller’s primality theorem

Further investigations with the strong probable prime test

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