Prime Numbers

4.4 Gauss and Jacobi sums 199
Note that the definition of χp,q and of l imply that
G(p, q) pw(p) up ≡ ζ l(p,q)
p
= ζ l(q)
p
= χp,q(g l(q)
q )=χp,q(l) (modr)
for every pair of primes p, q with q|F, p|q − 1. Thus, from (4.22) and Lemma
4.4.2,
χp,q(r) =χp,q(r) bp ≡ G(p, q) (rp−1 −1)bp = G(p, q) p w(p) upap
and so by Lemma 4.4.3 we have
≡ χp,q(l) ap = χp,q(l a )(modr),
χp,q(r) =χp,q(l a ).
The product of the characters χp,q for p prime, p|I and p|q − 1, is a character
χq of order
p|q−1 p = q − 1, as q − 1|I and I is squarefree. But a character
mod q of order q − 1 is one-to-one on Zq (see Exercise 4.24), so as
χq(r) =
χp,q(r) =
χp,q(l a )=χq(l a ),
p|q−1
p|q−1
we have r ≡ l a (mod q). As this holds for each prime q|F and F is squarefree, it
follows that (4.23) holds. This completes the proof of correctness of Algorithm
4.4.5.
It is clear that the running time is bounded by a fixed power of I, sothe
running time assertion follows immediately from Theorem 4.3.5. ✷
With some extra work one can extend the Gauss sums primality test to the
case where I is not assumed squarefree. This extra degree of freedom allows
for a speedier test. In addition, there are speed-ups that use randomness, thus
eschewing the deterministic aspect of the test. For a reasonably fast version
of the Gauss sums primality test, one might consult the new paper [Schoof
2004].
4.4.2 Jacobi sums test
We have just mentioned some ways that the Gauss sums test can be improved
in practice, but the principal way is to not use Gauss sums! Rather, as with
the original test of Adleman, Pomerance and Rumely, Jacobi sums are used.
The Gauss sums G(p, q) are in the ring Z[ζp,ζq]. Doing arithmetic in this ring
modulo n requires dealing with vectors with (p − 1)(q − 1) coordinates, with
each coordinate being a residue modulo n. It is likely in practice that we can
take the primes p to be very small, say less than ln n. But the primes q can
be somewhat larger, as large as (ln n) c ln ln ln n . The Jacobi sums J(p, q) that
we shall presently introduce lie in the much smaller ring Z[ζp], and so doing
arithmetic with them is much speedier.
Recall the character χp,q from Section 4.4.1, where p, q areprimeswith
p|q − 1. We shall suppose that p is an odd prime. Let b = b(p) be the least