Karl Schwarzschild

The Fermat primality test is a probabilistic test to determine if a number is probable prime.

Concept

Fermat's little theorem states that if p is prime and $1\leq a<p$, then

$a^{p-1}\equiv 1{\pmod {p}}.$

If we want to test if p 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 p is composite. If the equality does hold for many values of a, then we can say that p is probable prime.

It might be in our tests that we do not pick any value for a such that the equality fails. Any a such that

$a^{n-1}\equiv 1{\pmod {n}}$

when n is composite is known as a Fermat liar. Vice versa, in this case n is called Fermat pseudoprime to base a.

If we do pick an a such that

$a^{n-1}\not \equiv 1{\pmod {n}}$

then a is known as a Fermat witness for the compositeness of n.

Example

Suppose we wish to determine if n = 221 is prime. Randomly pick 1 ≤ a < 221, say a = 38. We check the above equality and find that it holds:

$a^{n-1}=38^{220}\equiv 1{\pmod {221}}.$

Either 221 is prime, or 38 is a Fermat liar, so we take another a, say 26:

$a^{n-1}=26^{220}\equiv 169\not \equiv 1{\pmod {221}}.$

So 221 is composite and 38 was indeed a Fermat liar.

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 [1, n − 1]

If $a^{n-1}\not \equiv 1{\pmod {n}}$, then return composite

If composite is never returned: return probably prime

Using fast algorithms for modular exponentiation, the running time of this algorithm is O(k × log²n × log log n × log log log n), where k is the number of times we test a random a, and n is the value we want to test for primality.

Flaw

There are infinitely many values of $n$ (known as Carmichael numbers) for which all values of $a$ for which $gcd(a,n)=1$ are Fermat liars. While Carmichael numbers are substantially rarer than prime numbers, there are enough of them that Fermat's primality test is often not used in the above form. Instead, other more powerful extensions of the Fermat test, such as Baillie-PSW, Miller-Rabin, and Solovay-Strassen are more commonly used.

In general, if $n$ is not a Carmichael number then at least half of all

$a\in (\mathbb {Z} /n\mathbb {Z} )^{*}$

are Fermat witnesses. For proof of this, let $a$ be a Fermat witness and $a_{1}$, $a_{2}$, ..., $a_{s}$ be Fermat liars. Then

$(a\cdot a_{i})^{n-1}\equiv a^{n-1}\cdot a_{i}^{n-1}\equiv a^{n-1}\not \equiv 1{\pmod {n}}$

and so all $a\times a_{i}$ for $i=1,2,...,s$ are Fermat witnesses.

Applications

The encryption program PGP uses this primality test in its algorithms. The chance of PGP generating a Carmichael number is less than 1 in 10⁵⁰, which is more than adequate for practical purposes.

References

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