A Probable Prime Test with High Confidence
نویسندگان
چکیده
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 automor-phism. The test, along with a xed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the polynomials x 2 ? bx ? c with b 2 +4c n = ?1 and ? ?c n = 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 ? 1. If a s 1 mod n or a 2 j s ?1 mod n for some 0 j r ?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 + o(1)) log 2 n multiplications mod n, where log 2 n denotes the base 2 logarithm. The catch is that a number which passes the test is not necessarily prime. Monier 9] and Rabin 13], however, showed that a composite n passes for at most 1 4 of the possible bases a. Thus, if the bases a are chosen at random, composite n will pass k iterations of the Strong Probable Prime Test with probability at most 1 4 k. Recently, Arnault 2] has shown that any composite n passes the Strong Lucas Probable Prime Test for at most 4 15 of the bases (b; c), unless n is the product of twin primes having certain properties (these composites are easy to detect). Also, Jones and Mo 8] have introduced an Extra Strong Lucas Probable Prime Test. Composite n will pass this test with probability at most 1 8 for a random choice of the parameter b. These authors did not concern themselves with the issue of running time. The methods of 10], however, show each test can be performed in twice the time it takes to perform the Strong Probable Prime Test. By contrast, two iterations of the Strong Probable Prime Test …
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