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122 CHAPTER 4. MODULAR METHODS
Figure 4.5: “Early termination” g.c.d. code
then D ′ := D
D := Algorithm 40(C, D, p, N);
if D = D ′ #We may have found the answer
then E := pp(D);
if E divides A and B
then return E;
# Otherwise this was a false alert, and we continue as normal.
4.1.5 Computing the g.c.d.: early success
While Algorithm 14 will detect a g.c.d. of 1 early, it will otherwise compute as
far as the Landau–Mignotte bound if the g.c.d. is not 1. While this may be
necessary, it would be desirable to terminate earlier if we have already found
the g.c.d. This is easily done by replacing the line
then D := Algorithm 40(C, D, p, N);
by the code in Figure 4.5. We should note that we return an E which divides
the inputs, and is derived from modular images, and therefore has to be the
greatest common divisor by Corollary 10.
4.1.6 An alternative correctness check
So far we have suggested computing the putative g.c.d. G, then checking that
it really divides both, and relying on Corollary 10 to say that G is therefore a
greatest common divisor. An alternative approach is to compute the co-factors,
i.e. A ′ such that A = A ′ G and B ′ such that B = B ′ G at the same time, and
use these as the check. So let us assume that modular gcd cofactors returns
a triple [G, A ′ , B ′ ] modulo p. The Algorithm (15) is given in Figure 4.6, and the
diagram in Figure 4.4 is still relevant.
Observation 5 It is tempting to conjecture that we do not need to make both
the multiplication checks at the end, but this is false: consider A = H, B =
H + 3p 1 · · · p k , when the algorithm will find H as the putative g.c.d., since the
Landau–Mignotte bound will ignore the large extra term in B, and only the
multiplication check for B will detect this.
Observation 6 Early termination can perfectly well be applied to this variant:
at any time gA = DA ′ and gB = DB ′ over the integers, we can finish.
Observation 7 We have presented this material as if there were a choice between
one large prime (Algorithm 13) and several small ones (Algorithms 14,
15). In practice, of course, a computer regards all numbers less than 32 bits (and