Prime Numbers

450 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
a convenient value R = B k may be used, whence the z value in Algorithm
9.2.5 can be obtained by looping k times and doing arithmetic (mod B) that
is particularly convenient for the machine. Explicit word-oriented loops that
achieve the optimal asymptotic complexity are laid out nicely in [Menezes et
al. 1997].
9.2.2 Newton methods
We have seen in Section 9.1 that the div operation may be effected via
additions, subtractions, and bit-shifts, although, as we have also seen, the
algorithm can be bested by moving away from the binary paradigm into the
domain of general base representations. Then we saw that the technique of
Montgomery mod gives us an asymptotically efficient means for powering with
respect to a fixed modulus. It is interesting, perhaps at first surprising, that
general div and mod may be effected via multiplications alone; that is, even
the small div operations attendant to optimized div methods are obviated, as
are the special precomputations of the Montgomery method.
One approach to such a general div and mod scheme is to realize that the
classical Newton method for solving equations may be applied to the problem
of reciprocation. Let us start with reciprocation in the domain of real numbers.
If one is to solve f(x) = 0, one proceeds with an (adroit) initial guess for x,
call this guess x0, and iterates
xn+1 = xn − f(xn)/f ′ (xn), (9.9)
for n =0, 1, 2 ..., whence—if the initial guess x0 is good enough—the sequence
(xn) converges to the desired solution. So to reciprocate a real number a>0,
one is trying to solve 1/x − a = 0, so that an appropriate iteration would be
xn+1 =2xn − ax 2 n. (9.10)
Assuming that this Newton iteration for reciprocals is successful (see Exercise
9.13), we see that the real number 1/a can be obtained to arbitrary accuracy
with multiplies alone. To calculate a general real division b/a, onesimply
multiplies b by the reciprocal 1/a, so that general division in real numbers
can be done in this way via multiplies alone.
But can the Newton method be applied to the problem of integer div?
Indeed it can, provided that we proceed with care in the definition of a
generalized reciprocal for integer division. We first introduce a function B(N),
defined for nonnegative integers N as the number of bits in the binary
representation of N, except that B(0) = 0. Thus, B(1) = 1,B(2) = B(3) = 2,
and so on. Next we establish a generalized reciprocal; instead of reciprocals
1/a for real a, we consider a generalized reciprocal of integer N as the integer
part of an appropriate large power of 2 divided by N.
Definition 9.2.7. The generalized reciprocal R(N) is defined for positive
integers N as ⌊4 B(N−1) /N ⌋.