Quadratic Frobenius test

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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. It should not be confused with the more general Frobenius test using a quadratic polynomial -- the QFT restricts the polynomials allowed based on the input, and also has other conditions that must be met. A composite passing this test is a Frobenius pseudoprime, but the converse is not necessarily true.

Concept[edit]

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.[1]:33

The test was later extended by Damgård and Frandsen to a test called Extended Quadratic Frobenius Test (EQFT).[2]

Algorithm[edit]

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.

See also[edit]

References[edit]

  1. ^ 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. 
  2. ^ Damgård, Ivan Bjerre; Frandsen, Gudmund Skovbjerg (2003). "An Extended Quadratic Frobenius Primality Test with Average and Worst Case Error Estimates" (PDF). 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.