Prime Numbers

198 Chapter 4 PRIMALITY PROVING
Remark. We may omit the condition F ≥ √ n and use Algorithm 4.2.11 for
the divisor search. The algorithm will remain fast if F ≥ n 1/3 .
Theorem 4.4.6. Algorithm 4.4.5 correctly identifies prime and composite
inputs. The running time is bounded by (ln n) c ln ln ln n for some positive
constant c.
Proof. We first note that a declaration of prime or composite in Step
[Preparation] is certainly correct. That a declaration of composite in Step
[Probable-prime computation] is correct follows from Lemma 4.4.2. If the gcd
calculation in Step [Coprime check] is not 1, it reveals a proper factor of n,
so it is correct to declare n composite. It is obvious that a declaration of
composite is correct in step [Divisor search], so what remains to be shown is
that composite numbers which have survived the prior steps must be factored
in Step [Divisor search] and so declared composite there.
Suppose n is composite with least prime factor r, and suppose n has
survived steps 1–4. We first show that
p w(p) |r p−1 − 1 for each prime p|I. (4.21)
This is clear if w(p) = 1, so assume w(p) ≥ 2. Suppose some l(p, q) = 0.Then
by Lemma 4.4.3
G(p, q) pw(p) up ≡ ζ l(p,q)
p
≡ 1(modn),
sothesameistrue(modr), using Lemma 4.4.3. Let h be the multiplicative
order of G(p, q) modulo r, sothatp w(p)+1 |h. But Lemma 4.4.2 implies that
h|p(r p−1 − 1), so that p w(p) |r p−1 − 1, as claimed. So suppose that each
l(p, q) = 0. Then from the calculation in Step [Coprime check] we have
G(p, q0) pw(p) up ≡ 1(modr), G(p, q0) pw(p)−1 up ≡ ζ j p (mod r)
for all j. Again with h the multiplicative order of G(p, q0) modulo r, wehave
p w(p) |h. Also,G(p, q0) m ≡ ζ j p (mod r) for some integers m, j implies that
ζ j p = 1. Lemma 4.4.2 then implies that G(p, q0) rp−1 −1 ≡ 1(modr) sothat
h|r p−1 − 1andp w(p) |h. This completes the proof of (4.21).
For each prime p|I, (4.21) implies there are integers ap,bp with
r p−1 − 1
p w(p) up
= ap
, bp≡1(modp). (4.22)
bp
Let a be such that a ≡ ap (mod p) for each prime p|I. We now show that
r ≡ l a (mod F ), (4.23)
from which our assertion about Step [Divisor search] follows. Indeed, since
F ≥ √ n ≥ r and F = r, wehaver equal to the least positive residue of l a
(mod F ), so that the proper factor r of n will be discovered in Step [Divisor
search].