Four primality testing algorithms

Summary

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or ‘very probably’ prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice.

In Section 2 we discuss the Miller–Rabin test. This is one of the most efficient probabilistic primality tests. Strictly speaking, the Miller–Rabin test is not a primality test but rather a ‘compositeness test’, since it does not prove the primality of a number. Instead, if n is not prime, the algorithm proves this in all likelihood very quickly. On the other hand, if n happens to be prime, the algorithm merely provides strong evidence for its primality. Under the assumption of the Generalized Riemann Hypothesis one can turn the Miller–Rabin algorithm into a deterministic polynomial time primality test. This idea, due to G. Miller, is also explained.

In Section 3 we describe the deterministic polynomial time primality test that was proposed by M. Agrawal, N. Kayal and N. Saxena in 2002 [Agrawal et al. 2004]. At the moment of this writing, this new test, or rather a more efficient probabilistic version of it, had not yet been widely implemented. In practice, therefore, for proving the primality of a given integer, one still relies on older tests that are either not provably polynomial time or not deterministic. In the remaining two sections we present the two most widely used such tests.