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A Probable Prime Test With High Confidence
"... . Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensure ..."
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. Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automorphism. The test, along with a fixed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the polynomials x 2 \Gamma bx \Gamma c with i b 2 +4c n j = \Gamma1 and \Gamma \Gammac n \Delta = 1. The running time of the test is asymptotically 3 times that of the Strong Probable Prime Test. x1 Background Perhaps the most common method for determining whether or not a number is prime is the Strong Probable Prime Test. Given an odd integer n, let n = 2 r s + 1 with s odd. Choose a random integer a with 1 a n \Gamma 1. If a s j 1 mod n or a 2 j s j \Gamma1 mod n for some 0 j r \Gamma 1, then n passes the test. An odd prime will pass the test for all a. The test is very fast; it requires no more than (1 +...
A oneparameter quadraticbase version of the Baillie–PSW probable prime test
 Math. Comp
"... Abstract. The wellknown BailliePSW probable prime test is a combination of a RabinMiller 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 ..."
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Abstract. The wellknown BailliePSW probable prime test is a combination of a RabinMiller 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 counterexamples to the BailliePSW 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 basecounting functions: B(n) =#{u:0 ≤ u<n, nis a psp(Tu)} and SB(n) =#{u:0 ≤ u<n, nis an spsp(Tu)}. Then we give explicit formulas to compute B(n) and SB(n), and prove that, for odd composites n, B(n) <n/2 and SB(n) <n/8, and point out that these are best possible. Finally, based on oneparameter quadraticbase pseudoprimes, we provide a probable prime test, called the OneParameter QuadraticBase Test (OPQBT), which passed by all primes ≥ 5 andpassedbyanoddcompositen = p r1 1 pr2 2 ···prs s (p1 <p2 < ·· · <ps odd primes) with probability of error τ(n). We give explicit formulas to compute τ(n), and prove that
ABSOLUTE QUADRATIC PSEUDOPRIMES
"... Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1. ..."
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Abstract. We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests. 1.