Prime Numbers

9.5 Large-integer multiplication 473
It is natural to ask whether there exist extended forms of such recursivegcd
algorithms, along the lines, say, of Algorithm 2.1.4 or Algorithm 9.4.3, to
effect asymptotically fast modular inversion. The answer is yes, as explained
in [Stehlé and Zimmermann 2004] and [Cesari 1998].
9.5 Large-integer multiplication
When numbers have, say, hundreds or thousands (even millions) of decimal
digits, there are modern methods for multiplication. In practice, one finds
that the classical “grammar-school” methods just cannot effect multiplication
in certain desired ranges. This is because, of course, the bit complexity of
grammar-school multiply of two size-N numbers is O ln 2 N . It turns out that
by virtue of modern transform and convolution techniques, this complexity
can be brought down to
O(ln N(ln ln N)(lnlnlnN)),
as we discuss in more detail later in this section.
The art of large-integer arithmetic has, especially in modern times,
sustained many revisions. Just as with the fast Fourier transform (FFT)
engineering literature itself, there seems to be no end to the publication of
new approaches, new optimizations, and new applications for computational
number theory. The forest is sufficiently thick that we have endeavored in
this section to render an overview rather than an encyclopedic account of this
rich and exotic field. An interesting account of multiplication methods from
a theoretical point of view is [Bernstein 1997], and modern implementations
are discussed, with historical references, in [Crandall 1994b, 1996a].
9.5.1 Karatsuba and Toom–Cook methods
The classical multiplication methods can be applied on parts of integers to
speed up large-integer multiplication, as observed by Karatsuba. His recursive
scheme assumes that numbers be represented in split form
x = x0 + x1W,
with x0,x1 ∈ [0,W − 1], which is equivalent to base-W representation, except
that here the base will be about half the size of x itself. Note that x is
therefore a “size-W 2 ” integer. For two integers x, y of this approximate size,
the Karatsuba relation is
where
xy =
t + u
2
− v + t − u
2 W + vW 2 , (9.17)
t =(x0 + x1)(y0 + y1),
u =(x0 − x1)(y0 − y1),
v = x1y1,