Quadratic Frobenius test
The quadratic Frobenius test (QFT) is a probabilistic primality test to test whether a number is a probable prime. It is named after Ferdinand Georg Frobenius. The test uses the concepts of quadratic polynomials and the Frobenius automorphism. A composite passing this test is called a Frobenius pseudoprime.
Grantham's stated goal when developing the algorithm was to provide a test that primes would always pass and composites would pass with a probability of less than 1/7710.:33
Let n be a positive integer such that n is odd, (b2 + 4c | n) = −1 and (−c | n) = 1, where (x | n) denotes the Jacobi symbol. Set B = 50000. Then a QFT on n with parameters (b, c) works as follows:
- (1) Test whether one of the primes less than or equal to the lower of the two values and divides n. If yes, then stop as n is composite.
- (2) Test whether . If yes, then stop as n is composite.
- (3) Compute . If then stop as n is composite.
- (4) Compute . If then stop as n is composite.
- (5) Let with s odd. If , and for all , then stop as n is composite.
If the QFT doesn't stop in steps (1)–(5), then n is a probable prime.
- Grantham, J. (1998). "A Probable Prime Test With High Confidence". Journal of Number Theory (Academic Press) 72 (1): 32–47. doi:10.1006/jnth.1998.2247.
- Damgård, Ivan Bjerre; Frandsen, Gudmund Skovbjerg (2003). "An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates". Lecture Notes in Computer Science. Fundamentals of Computation Theory (Springer Berlin Heidelberg) 2751: 118–131. doi:10.1007/978-3-540-45077-1_12. ISBN 978-3-540-45077-1. ISSN 1611-3349.