Prime Numbers

9.5 Large-integer multiplication 475
Evaluate symbolically z(j) =x(j)y(j) for each j ∈ [1 − D, D − 1], sothat
each z(j) is cast in terms of the original coefficients of the x and y
polynomials;
3. [Reconstruction]
Solve symbolically for the coefficients zj in the following linear system of
(2D − 1) equations:
z(t) = 2D−2
k=0 zkt k , t ∈ [1 − D, D − 1];
4. [Report scheme]
Report a list of the (2D − 1) relations, each relation casting zj in terms of
the original x, y coefficients;
The output of this algorithm will be a set of formulae that give the coefficients
of the polynomial product z(t) = x(t)y(t) in terms of the coefficients of
the original polynomials. But this is precisely what is meant by integer
multiplication, if each polynomial corresponds to a D-digit representation in
a fixed base B.
To underscore the Toom–Cook idea, we note that all of the Toom–Cook
multiplies occur in the [Evaluation] step of Algorithm 9.5.1. We give next
a specific multiplication algorithm that requires five such multiplies. The
previous, symbolic, algorithm was used to generate the actual relations of
this next algorithm:
Algorithm 9.5.2 (Explicit D = 3 Toom–Cook integer multiplication).
For integers x, y given in base B as
x = x0 + x1B + x2B 2 ,
y = y0 + y1B + y2B 2 ,
this algorithm returns the base-B digits of the product z = xy, using the
theoretical minimum of 2D − 1=5multiplications for acyclic convolution of
length-3 sequences.
1. [Initialize]
r0 = x0 − 2x1 +4x2;
r1 = x0 − x1 + x2;
r2 = x0;
r3 = x0 + x1 + x2;
r4 = x0 +2x1 +4x2;
s0 = y0 − 2y1 +4y2;
s1 = y0 − y1 + y2;
s2 = y0;
s3 = y0 + y1 + y2;
s4 = y0 +2y1 +4y2;
2. [Toom–Cook multiplies]
for(0 ≤ j