Contents - Student subdomain for University of Bath

4.1. GCD IN ONE VARIABLE 123
Figure 4.6: Algorithm 15
Algorithm 15 (Modular GCD (Alternative small prime version))
Input: A, B polynomials in Z[x].
Output: G := gcd(A, B), A ′ , B ′ with A = A ′ G, B = B ′ G.
M :=Landau_Mignotte_bound(A, B);
g := gcd(lc(A), lc(B));
p := find_prime(g);
[D, A ′ , B ′ ] := modular_gcd_cofactors(A, B, p);
if deg(D) = 0 then return [1,A,B]
D := gD # g is our guess at the leading coefficient of the g.c.d.
N := p; # N is the modulus we will be constructing
while N < 2M repeat
p := find_prime(g);
[C, A 1 , B 1 ] := modular_gcd_cofactors(A, B, p);
if deg(C) = deg(D)
then D := Algorithm 40(C, D, p, N);
A ′ := Algorithm 40(A 1 , A ′ , p, N);
B ′ := Algorithm 40(B 1 , B ′ , p, N);
N := pN;
else if deg(C) < deg(D)
# C proves that D is based on primes of bad reduction
if deg(C) = 0 then return [1,A,B]
D := C; A ′ = A 1 ; B ′ = B 1 ;
N := p;
else #D proves that p is of bad reduction, so we ignore it
if gA = DA ′ and gB = DB ′
then G := pp(D) gcd(cont(A), cont(B)); # Theorem 5
A ′ := A ′ /lc(G); B ′ := B ′ /lc(G); # Fix leading coefficients
return [G, A ′ , B ′ ]
else all primes must have been bad, and we start again