Primality test
A primality test is an algorithm for determining whether an input number is prime.
Naive methods
The simplest primality test is as follows: Given an input number n,
we see if each integer k from 1 to n divides n. If n is divisible by any k then n is composite, otherwise it is prime.
A slightly better method is to see if n is divisible by any integer k from 1 to <math>\sqrt{n}<math>, inclusive. If n is composite then it can be factored into two values, at least one of which is less than <math>\sqrt{n}<math>
A good way to speed up this method (and all the others mentioned below) is to pre-compute and store a list of all primes up to a certain bound, say all primes up to 200; such a list can be computed with the Sieve of Eratosthenes. Then, before testing n for primality with a serious method, one first checks whether n is divisible by any prime from the list.
Probabilistic tests
Most popular primality tests are probabilistic tests. They do not determine with certainty whether a number is prime or not, but are acceptable for practical applications such as cryptography that often critically depend on large prime numbers. The basic idea is as follows:
- Randomly pick a number a.
- Check some equality involving a and the given number n. If the equality fails to hold true, then n is a composite number,a is known as a witness for the compositeness, and the test stops.
- Repeat step 1 until the required certainty is achieved.
After several iterations, if n is not found to be a composite number, then it can be declared probably prime.
These tests are fast but often not perfect. For a given test, there may be some composite numbers that will be declared "probably prime" no matter what witness is chosen. Such numbers are called pseudoprimes for that test.
The simplest probabilistic primality test is the
Fermat primality test. It is sometimes used if a rapid screening
of numbers is needed, for instance in the key generation phase of the
RSA public key cryptographical algorithm. The
Miller-Rabin primality test and Solovay-Strassen primality test are more
sophisticated variants which detect all composites; they are often the methods
of choice.
Fast deterministic tests
A deterministic primality test is the cyclotomy test; its runtime can be proven to be O(nclog(log(n))), where n is the number of digits of N and c is a constant independent of n. This is slower than polynomial time.
The elliptic curve primality test can be proven to run in O(n6), but only if some still unproven (but widely assumed to be true) statements of analytic number theory are used. In practice, this test is slower than the cyclotomy test for numbers with up to 10,000 digits or so.
The implementation of these two methods is rather difficult, and their error probabilities in practice may therefore be even higher than those of the probabilistic methods mentioned above.
If the generalized Riemann hypothesis is assumed, the Miller-Rabin test can be turned into a deterministic version with runtime O(n4). In practice, this algorithm is slower than the other two for sizes of numbers that can be dealt with at all.
In 2002, Manindra Agarwal, Nitin Saxena and Neeraj Kayal described a new deterministic primality test which runs in O(n12), and this bound can be rigorously proven. In addition, given a certain unproven, but widely believed to be true, conjecture, it runs in O(n6). As such, this provided the first deterministic primality test with provably polynomial run-time. In practice, this algorithm is slower than the other methods.
Number-theoretic methods
Certain number-theoretic methods exist, such as the Lucas-Lehmer primality test for testing whether a number is prime.
The Lucas-Lehmer test relies on the fact that the if the multiplicative order of some number a modulo n is n-1 for a prime n when a is primitive. If we can show a is primitive for n, we can show n is prime.
