Prime Numbers

474 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
and we obtain xy, which is originally a size-W 2 multiply, for the price of only
three size-W multiplies (and some final carry adjustments, to achieve base-W
representation of the final result). This is in principle an advantage, because
if grammar-school multiply is invoked throughout, a size-W 2 multiply should
be four, not three times as expensive as a size-W one. It can be shown that if
one applies the Karatsuba relation to t, u, v themselves, and so on recursively,
the asymptotic complexity for a size-N multiply is
ln 3/ ln
O (ln N)
2
bit operations, a theoretical improvement over grammar-school methods.
We say “theoretical improvement” because computer implementations will
harbor so-called overhead, and the time to arrange memory and recombine
subproducts and so on might rule out the Karatsuba method as a viable
alternative. Still, it is often the case in practice that the Karatsuba approach
does, in fact, outperform the grammar-school approach over a machine- and
implementation-dependent range of operands.
But a related method, the Toom–Cook method, reaches the theoretical
boundary of O ln 1+ɛ N bit operations for the multiplicative part of size-N
multiplication—that is, ignoring all the additions inherent in the method.
However, there are several reasons why the method is not the final word
in the art of large-integer multiply. First, for large N the number of
additions is considerable. Second, the complexity estimate presupposes that
multiplications by constants (such as 1/2, which is a binary shift, and so on)
are inexpensive. Certainly multiplications by small constants are so, but the
Toom–Cook coefficients grow radically as N increases. Still, the method is
of theoretical interest and does have its practical applications, such as fast
multiplication on machines whose fundamental word multiply is especially
sluggish with respect to addition. The Toom–Cook method hinges on the idea
that given two polynomials
x(t) =x0 + x1t + ...+ xD−1t D−1 , (9.18)
y(t) =y0 + y1t + ...+ yD−1t D−1 , (9.19)
the polynomial product z(t) =x(t)y(t) is completely determined by its values
at 2D −1 separate t values, for example by the sequence of evaluations (z(j)),
j ∈ [1 − D, D − 1]:
Algorithm 9.5.1 (Symbolic Toom–Cook multiplication). Given D, thisalgorithm
generates the (symbolic)Toom–Cook scheme for multiplication of (Ddigit)-by-(D-digit)
integers.
1. [Initialize]
Form two symbolic polynomials x(t),y(t) each of degree (D − 1), asin
equation (9.18);
2. [Evaluation]