Prime Numbers

4.5 The primality test of Agrawal, Kayal, and Saxena (AKS test) 207
cannot do better than Õ(ln6 n) for the total running time. Note too that from
Exercise 4.30, this total running time is indeed bounded by Õ(ln6 n) for almost
all primes n. (For most composite numbers n, the running time is less.)
But in our never-ending quest for the best possible algorithm, we ask
whether Õ(ln6 n) can be achieved for all numbers n. It seems as if this should
be the case; that is, it seems as if we should be able to choose r = Õ(ln2 n)
always. Such a result follows independently from strong forms of two different
conjectures. One of these is the Artin conjecture asserting that if n is neither
−1 nor a square (which is certainly an allowable hypothesis for us), then there
are infinitely many primes r with n a primitive root for r. Anysuchprimer
with r>1+lg 2 n may be used in Algorithm 4.5.1, and it seems reasonable
to assume that there is always such a prime smaller than 2 lg 2 n (for n>2).
It is interesting that in [Hooley 1976] there is a proof of the Artin conjecture
assuming the GRH (see the comments in Exercise 2.39), and it may be that
this proof can be strengthened to show that there is good value for rlg 2 n with q = Õ(ln2 n)
and r =2q + 1 not dividing n ± 1; see [Agrawal et al. 2004]. Such a value for
r is valid in Algorithm 4.5.1. Indeed, it would suffice if the order of n modulo
r is either q or 2q. But otherwise, its order is 1 or 2, and we have stipulated
that r does not divide n ± 1. These conjectures strengthen our view that the
complexity of Algorithm 4.5.1 should be Õ(ln6 n).
Using a deep theorem in [Fouvry 1985], one can show that r may be chosen
with r = O(ln 3 n); see [Agrawal et al. 2004]. Thus, the total bit complexity
for the algorithm is Õ(ln7.5 n). This is nice, but there is a drawback to using
Fouvry’s theorem. The proof is not only difficult, it is ineffective. This means
that from the proof there is no way to present a numerically explicit upper
bound for the number of bit operations. This ineffectivity is due to the use of
a theorem of Siegel; we have already seen the consequences of Siegel’s theorem
in Theorem 1.4.6, and we will see it again in our discussion of class numbers
of quadratic forms.
So using Fouvry’s result, we get close to the natural limit of Õ(ln6 n), but
not quite there, and the time estimate is ineffective. In the next subsection
we shall discuss how these defects may be removed.
4.5.3 Primality testing with Gaussian periods
In Theorem 4.5.2 we are concerned with the polynomial x r −1. In the following
result from [Lenstra and Pomerance 2005] we move toward a more general
polynomial f(x).