Prime Numbers

248 Chapter 5 EXPONENTIAL FACTORING ALGORITHMS
In a computational (and theoretical) tour de force, [Watkins 2004] shows
unconditionally that h(D) > 100 for −D >2384797.
The following formula for h(D) is attractive (but admittedly not very
efficient when |D| is large) in that it replaces the infinite sum implicit in
L(1,χD) with a finite sum. The formula is due to Dirichlet, see [Narkiewicz
1986]. For D < 0, D a fundamental discriminant (this means that either
D ≡ 1(mod4)andDis squarefree or D ≡ 8 or 12 (mod 16) and D/4 is
squarefree), we have
h(D) = w
D
|D|
χD(n)n.
n=1
Though an appealing formula, such a summation with its |D| terms is suitable
for the exact computation of h(D) only for small |D|, say|D| < 10 8 .There
are various ways to accelerate such a series; for example, in [Cohen 2000]
one can find error-function summations of only O(|D| 1/2 ) summands, and
such formulae allow one easily to handle |D| ≈10 16 . Moreover, it can be
shown that directly counting the primitive reduced forms (a, b, c) of negative
discriminant D computes h(D) inO |D| 1/2+ɛ operations. And the Shanks
baby-steps, giant-steps method reduces the exponent from 1/2 to1/4. We
revisit the complexity of computing h(D) in the next section.
5.6.4 Ambiguous forms and factorization
It is not very hard to list all of the elements of the class group C(D) thatare
their own inverse. When D