Prime Numbers
Prime Numbers Prime Numbers
5.6 Binary quadratic forms 247 In the case that D 1000 for −D >1.9 · 1011 . Probably even this greatly lowered bound is about 100 times too high. It may well be possible to establish this remaining factor of 100 or so conditionally on the ERH.
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
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5.6 Binary quadratic forms 247<br />
In the case that D 1000 for −D >1.9 · 1011 . Probably even this greatly lowered<br />
bound is about 100 times too high. It may well be possible to establish this<br />
remaining factor of 100 or so conditionally on the ERH.
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