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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. A composite passing this test is called a Frobenius pseudoprime.

Concept

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The test was later extended by Damgård and Frandsen to a test called extended quadratic Frobenius test (EQFT).

Algorithm

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 $B$ and $\sqrt{n}$ divides n. If yes, then stop as n is composite.

(2) Test whether $\sqrt{n}\in \mathbb{\mathbb{Z}}$. If yes, then stop as n is composite.

(3) Compute $x^{\frac{n+1}{2}}mod(n,x^2-bx-c)$. If $x^{\frac{n+1}{2}}\notin \mathbb{\mathbb{Z}}/n\mathbb{\mathbb{Z}}$ then stop as n is composite.

(4) Compute $x^{n+1}mod(n,x^2-bx-c)$. If $x^{n+1}\equiv ̸-c$ then stop as n is composite.

(5) Let $n^2-1=2^{r}s$ with s odd. If $x^s\equiv ̸1mod(n,x^2-bx-c)$, and $x^{2^{j}s}\equiv ̸-1mod(n,x^2-bx-c)$ for all $0\le j\le r-2$, then stop as n is composite.

If the QFT doesn't stop in steps (1)–(5), then n is a probable prime.

See also

• Integers modulo n
• Multiplicative group of integers modulo n

References

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