The Rabin-Monier theorem for Lucas pseudoprimes

Arnault, François

doi:10.1090/s0025-5718-97-00836-3

Cited by 10 publications

(6 citation statements)

References 9 publications

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“…Similar results to those for the MR primality test can be established for the Lucas test: a single Lucas test will declare a given composite number as being composite with probability at least 1 − (4/15) and as being prime with probability at most (4/15), with these bounds being tight [4]. [35] is a deterministic primality test consisting of a single Miller-Rabin test with base 2 followed by a single Lucas test.…”

Section: Lucassupporting

confidence: 54%

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A Performant, Misuse-Resistant API for Primality Testing

Massimo

Paterson

2020

Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security

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Primality testing is a basic cryptographic task. But developers today are faced with complex APIs for primality testing, along with documentation that fails to clearly state the reliability of the tests being performed. This leads to the APIs being incorrectly used in practice, with potentially disastrous consequences. In an effort to overcome this, we present a primality test having a simplest-possible API: the test accepts a number to be tested and returns a Boolean indicating whether the input was composite or probably prime. For all inputs, the output is guaranteed to be correct with probability at least 1 − 2 −128 . The test is performant: on random, odd, 1024-bit inputs, it is faster than the default test used in OpenSSL by 17%. We investigate the impact of our new test on the cost of random prime generation, a key use case for primality testing. The OpenSSL developers have adopted our suggestions in full; our new API and primality test are scheduled for release in OpenSSL 3.0.

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Section: Lucassupporting

confidence: 54%

“…Let C i denote the cost of a trial division for prime p i and let C M R denote the cost of a single MR test. 4 Then the total cost of MRAC on random prime k-bit inputs is:…”

Section: Mrac On Randommentioning

confidence: 99%

A Performant, Misuse-Resistant API for Primality Testing

Massimo

Paterson

2020

Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security

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“…There are 155 psp(T 3 )'s among 279 lpsp(3, 1)'s < 10 6 . Arnault [3] and Grantham [9,10] cited a preprint of Mo and Jones, who introduced a test via slpsp(u, 1), which has probability of error < 1/8. So far I have not been able to access the preprint.…”

Section: Theoremmentioning

confidence: 99%

“…If n is composite and relatively prime to 2QD such that condition (1.4) or (1.5) holds, then we call n a Lucas pseudoprime [4,19] or a strong Lucas pseudoprime [3,4] to parameters P and Q, or lpsp(P, Q) or slpsp(P, Q) for short.…”

Section: Introductionmentioning

confidence: 99%

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A one-parameter quadratic-base version of the Baillie-PSW probable prime test

Zhang

2002

Math. Comp.

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Abstract. The well-known Baillie-PSW probable prime test is a combination of a Rabin-Miller test and a "true" (i.e., with (D/n) = −1) Lucas test. Arnault mentioned in a recent paper that no precise result is known about its probability of error. Grantham recently provided a probable prime test (RQFT) with probability of error less than 1/7710, and pointed out that the lack of counter-examples to the Baillie-PSW test indicates that the true probability of error may be much lower.In this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter: Tu = T mod (T 2 − uT + 1), and define the base-counting functions: B(n) = #{u : 0 ≤ u < n, n is a psp(Tu)} and SB(n) = #{u : 0 ≤ u < n, n is an spsp(Tu)}.Then we give explicit formulas to compute B(n) and SB(n), and prove that, for odd composites n,

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A Probable Prime Test with High Confidence

Grantham

1998

Journal of Number Theory

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The Rabin-Monier theorem for Lucas pseudoprimes

Abstract: Abstract. We give bounds on the number of pairs (P, Q) with 0 ≤ P, Q < n such that a composite number n is a strong Lucas pseudoprime with respect to the parameters (P, Q).

Cited by 10 publications

References 9 publications

A Performant, Misuse-Resistant API for Primality Testing

A Performant, Misuse-Resistant API for Primality Testing

A one-parameter quadratic-base version of the Baillie-PSW probable prime test

A Probable Prime Test with High Confidence

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