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136 CHAPTER 4. MODULAR METHODS
Algorithm 18 (Sparse g.c.d.)
Input: A, B polynomials in R[y][x].
Output: gcd(A, B)
Figure 4.12: Algorithm 18: Sparse g.c.d.
A c := cont x (A); A := A/A c ; # Using Algorithm 16, which
B c := cont x (B); B := B/B c ; # calls this algorithm recursively
g :=Algorithm 18(lc x (A), lc x (B));
# one fewer variable
G :=failed
while G =failed # Expect only one iteration
G :=Algorithm 19(g, gA, gB);
return pp x (G)× Algorithm 18(A c , B c ); # one fewer variable
Figure 4.13: Algorithm 19: Inner sparse g.c.d.
Algorithm 19 (Inner sparse g.c.d.)
Input: g leading coefficient; A, B polynomials in R[y][x].
Output: gcd(A, B), but with leading coefficient g
or failed if we don’t have the Zippel assumption
d n := min(deg yn
(A), deg yn
(B)) # (over)estimate for deg yn
(gcd(A, B))
v 0 :=random(lc(A), lc(B)); #assumed good and satisfying Zippel for gcd(A, B);
P 0 :=Algorithm 19(g| yn=v 0
, A| yn=v 0
, B| yn=v 0
)
if P 0 =failed
return failed # Don’t try again
d x := deg x (P 0 )
i := 0
while i < d
v :=random(lc(A), lc(B)); #assumed good
P i+1 :=Algorithm 20(Sk(P 0 ), g| yn=v, A| yn=v, B| yn=v)
if deg x (P i+1 ) > d x # Either v j in Algorithm 20 or
continue round the loop # | yn=v was not a good evaluation
if deg x (P i+1 ) < d x
return failed # | yn=v 0
was not a good evaluation
i := i + 1; v i := v # store v and the corresponding P
C :=Algorithm 42({P i }, {v i }) # Reconstruct
if C divides both A and B
return C
else return failed
random(lc(A), lc(B)) chooses a value not annihilating both leading coefficients