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Fermat Primality TestThe Fermat primality test is a probabalistic test to determine if a number is composite or probably prime. Concept Fermat's little theorem states that if p is prime and , then - .
If we want to test if n is prime, then we can pick random a's in the interval and see if the equality holds. If the equality does not hold for a value of a, then n is composite. If the equality does hold for many values of a, then we can say that n is probably prime, or a pseudoprime. It may be in our tests that we do not pick any value for a such that the equality fails. Any a such that -
when n is composite is known as a Fermat liar. If we do pick an a such that -
then a is known as a Fermat witness for the compositeness of n. Algorithm and running time The algorithm can be written as follows: - Inputs: n: a value to test for primality; k: a parameter that determines the number of times to test for primality
- Output: composite if n is composite, otherwise probably prime
- repeat k times:
-
- pick a randomly in the range n − 1
-
- if an − 1 mod n ≠ 1 then return composite
- return probably prime
Using fast algorithms for modular exponentiation, the running time of this algorithm is O(k × log3n), where k is the number of times we test a random a, and n is the value we want to test for primality. Flaws There are certain values of n known as Carmichael numbers for which all values of a for which gcd(a,n)=1 are Fermat liars. Although Carmichael numbers are rare, there are enough of them that Fermat's primality test is often not used in favor of other primality tests such as Miller-Rabin and Solovay-Strassen. In general, if n is not a Carmichael number then at least half of all -
are Fermat witnesses. For proof of this let a be a Fermat witness and a1, a2, ..., as be Fermat liars. Then -
and so all a × ai for i = 1, 2, ..., s are Fermat witnesses. Usage The encryption program PGP uses this primality test in its algorithms.
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