Prime Numbers

4.5 The primality test of Agrawal, Kayal, and Saxena (AKS test) 201
it is remarkable in that the test itself is quite simple. And further, two of
the authors, Kayal and Saxena, had worked on this problem for their senior
project, having just received their bachelor’s degrees three months before the
announcement. A short time later, after suggestions from various quarters,
Agrawal,Kayal,andSaxenacameoutwithanevensimplerversionofthe
test. These two versions may be found in [Agrawal et al. 2002], [Agrawal et al.
2004].
In this section we shall present the second version of the Agrawal–Kayal–
Saxena algorithm, as well as some more recent developments. As of this
writing, it remains to be seen whether the AKS test will be useful in proving
large numbers prime. The quartic time test at the end of the section stands
the best chance.
4.5.1 Primality testing with roots of unity
If n is prime, then
g(x) n ≡ g(x n )(modn),
for any polynomial g(x) ∈ Z[x]. In particular,
(x + a) n ≡ x n + a (mod n) (4.24)
for any a ∈ Z. Further, if (4.24) holds for just one value of a with gcd(a, n) =1,
then n must be prime; see Exercise 4.25. That is, (4.24) is an if-and-only-if
primality criterion. The trouble is that we know no speedy way of verifying
(4.24) even for the simple case a = 1; there are just too many terms on the
left side of the congruence.
If f(x) ∈ Z[x] is an arbitrary monic polynomial, then (4.24) implies that
(x + a) n ≡ x n + a (mod f(x),n) (4.25)
for every integer a. So,ifn is prime, then (4.25) holds for every integer a
and every integer monic polynomial f(x). Further, it should be possible to
rapidly check (4.25) if deg f(x) is not too large. As an example, take a =1
and f(x) =x − 1. Then (4.25) is equivalent to
2 n ≡ 2(modn),
the Fermat congruence to the base 2. However, as we have seen, while this
congruence is necessary for the primality of n, itisnotsufficient.So,by
introducing the modulus f(x) we gain speed, but perhaps lose our primality
criterion.
But (4.25) allows more generality; we are not required to take f(x) of
degree 1. For example, we might take f(x) = x r − 1 for some smallish
number r, and so be implicitly dealing with the r-th roots of unity. Essentially,
allthatneedstobedoneistochooser appropriately (but bounded by a
polylogarithmic expression in n), and to verify (4.25) for every a up to a
certain point (again bounded by a polylogarithmic expression in n).
The new primality test is so simple and straightforward that we cannot
resist stating it first as pseudocode, discussing the details only afterward.