A course in computational algebraic number theory
A course in computational algebraic number theory
The Rabin-Monier theorem for Lucas pseudoprimes
Mathematics of Computation
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Finding strong pseudoprimes to several bases
Mathematics of Computation
Mathematics of Computation
On the difficulty of finding reliable witnesses
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
Finding strong pseudoprimes to several bases. II
Mathematics of Computation
Inefficacious Conditions of the Frobenius Primality Test and Grantham's Problem
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
On the effectiveness of a generalization of Miller's primality theorem
Journal of Complexity
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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 (T2 - uT + 1), and define the base-counting functions: B(n) : #{u: 0 ≤ u is a psp(Tu)} and SB(n) : #{u: 0 ≤ u is an spsp(Tu)}. Then we give explicit formulas to compute B(n) and SB(n), and prove that, for odd composites n, B(n) and SB(n) , and point out that these are best possible. Finally, based on one-parameter quadratic-base pseudoprimes, we provide a probable prime test, called the One-Parameter Quadratic-Base Test (OPQBT), which passed by all primes ≥ 5 and passed by an odd composite n = p1r1 P2r2 ...Psrs (P1 2 s odd primes) with probability of error τ (n). We give explicit formulas to compute τ(n), and prove that τ(n) n4/3, for n nonsquare free with s = 1; 1/n2/3, for n square free with s = 2; 1/n2/7, for n square free with s = 3; 1/8s-4.166(p1+1), for n square free with s even ≥ 4; 1/16s-5.119726, for n square free with s odd ≥ 5; 1/4s Πi=1s 1/pi2(ri-1), otherwise, i.e., for n nonsquare free with s ≥ 2. The running time of the OPQBT is asymptotically 4 times that of a Rabin-Miller test for worst cases, but twice that of a Rabin-Miller test for most composites. We point out that the OPQBT has clear finite group (field) structure and nice symmetry, and is indeed a more general and strict version of the Baillie-PSW test. Comparisons with Gantham's RQFT are given.