Prime Numbers

200 Chapter 4 PRIMALITY PROVING
positive integer with (b +1) p ≡ b p +1(modp 2 ). (As shown in [Crandall et
al. 1997] we may take b = 2 for every prime p up to 10 12 except p = 1093 and
p = 3511, for which we may take b = 3. It is probably true that b(p) =2or3
for every prime p. Wecertainlyhaveb(p) < ln 2 p; see Exercise 3.19.)
We now define a Jacobi sum J(p, q). This is
J(p, q) =
q−2
m=1
b
χp,q m (m − 1) .
The connection to the supposed primality of n is made with the following
more general result. Suppose n is an odd prime not divisible by p. Letf be
the multiplicative order of n in Z ∗ p. Then the ideal (n) inZ[ζp] factors into
(p−1)/f prime ideals N1, N2,...,N (p−1)/f each with norm n f .Ifα is in Z[ζp]
but not in Nj, then there is some integer aj with α (nf −1)/p ≡ ζ aj
p (mod Nj).
The Jacobi sums test tries this congruence with α = J(p, q) for the same
pairs p, q (with p>2) that appear in the Gauss sums test. To implement this,
one also needs to find the ideals Nj. This is accomplished by factoring the
polynomial x p−1 + x p−2 + ···+ 1 modulo n into h1(x)h2(x) ···h (p−1)/f (x),
where each hj(x) is irreducible of degree f. Then we can take for Nj the
ideal generated by n and hj(ζp). These calculations can be attempted even
if we don’t know that n is prime, and if they should fail, then n is declared
composite.
For a complete description of the test, the reader is referred to [Adleman
et al. 1983]. For a practical version and other improvements see [Bosma and
van der Hulst 1990].
4.5 The primality test of Agrawal, Kayal, and Saxena
(AKS test)
In August 2002, M. Agrawal, N. Kayal, and N. Saxena announced a
spectacular new development, a deterministic, polynomial-time primality test.
This is now known as the AKS test. We have seen in Algorithm 3.5.13 that
such a test exists on the assumption of the extended Riemann hypothesis.
Further, in Algorithm 3.5.6 (the “Miller–Rabin test”), we have a random
algorithm that expects to prove that composite inputs are composite in
polynomial time. We had known a random algorithm that expects to prove
that prime inputs are prime in polynomial time; this is the Adleman–Huang
test, which will be briefly described in Section 7.6. Finally, as we just saw in
Theorem 4.4.6, Algorithm 4.4.5 is a fully proved, deterministic primality test
that runs within the “almost polynomial” time bound (ln n) c ln ln ln n .Wesay
“almost polynomial” because the exponent ln ln ln n growssoveryslowlythat
for practical purposes it might be considered bounded. (A humorous way of
putting this: Though we have proved that ln ln ln n tends to infinity with n,
we have never observed it doing so!)
The new test is not just sensational because it finally settles the theoretical
issue of primality testing after researchers were so close in so many ways,