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120 CHAPTER 4. MODULAR METHODS
of the g.c.d. modulo 3, and modulo 3 B has degree 4, so the gc.d. must have
degree at most 4.
Can this be generalised, in particular can we update our estimate “on the
fly” as upper bounds on the degree of the g.c.d change, and is it worth it? In
view of the ‘early success’ strategies discussed later, the answer to the last part
is probably negative.
4.1.4 Computing the g.c.d.: several small primes
While algorithm 13 does give us some control on the size of the numbers being
considered, we are still often using numbers larger than those which hindsight
would show to be necessary. For example, in (4.1), (4.2) we could deduce coprimeness
using the prime 5, rather than 1021 from (4.4) or 149 from (4.5). If
instead we consider (x − 1)A and (x − 1)B, the norms change, giving 812.35 in
(4.4) (a prime of 1627) and 116.05 in (4.5) (a prime of 239). Yet primes such
as 5, 11, 13 etc. will easily show that the result is x − 1. Before we leap ahead
and use such primes, though, we should reflect that, had we taken (x − 10)A
and (x − 10)B, 5 would have suggested x as the gcd, 11 would have suggested
x + 1, 13 would have suggested x + 3 and so on.
The answer to this comes in observing that the smallest polynomial (in terms
of coefficient size) which is congruent to x modulo 5 and to x + 1 modulo 11 is
x−10 (it could be computed by algorithm 40). More generally, we can apply the
Chinese Remainder Theorem (Theorem 35) to enough primes of good reduction,
as follows. We assume that find_prime(g) returns a prime not dividing g, a
different one each time. The algorithm is given in Figure 4.3, with a diagram
in Figure 4.4.
Observation 4 The reader may think that Algorithm 14 is faulty: line (*)
in Figure 4.3 iterates until n ≥ 2M, which would be fine if we were actually
computing the g.c.d. But we have forced the leading coefficient to be g, which
may be overkill. Hence aren’t we in danger of trying to recover g times the true
g.c.d., whose coefficients may be greater than 2M?
In fact there is not a problem. The proof of Corollary 8 relies on estimating
the leading coefficient of gcd(A, B) by g, and so the bound is in fact a bound for
the coefficients after this leading coefficient has been imposed.
Having said that, we can’t “mix and match”. If we decide that Corollary
9 provides a better lower bound than Corollary 8, then we must go for “imposed
trailing coefficients” rather than “imposed leading coefficients”, or, more
pragmatically, compute the g.c.d. of  and ˆB, and reverse that.
We should note the heavy reliance on Corollary 10 to detect bad reduction. We
impose g as the leading coefficient throughout, and make the result primitive
at the end as in the large prime variant.