Prime Numbers

454 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
precision modes in such cases. Another new development is an interesting
Karatsuba-like recursive divide, in [Burnikel and Ziegler 1998]. The method
has the interesting property that the complexities of finding the div or just a
mod result are not quite the same.
Newton methods apply beyond the division problem. Just one example
is the important computation of ⌊ √ N⌋. Onemayemploya(realdomain)
Newton iteration for √ a in the form
xn+1 = xn
2
to forge an algorithm for integer parts of square roots:
a
+ , (9.14)
2xn
Algorithm 9.2.11 (Integer part of square root). This algorithm returns
⌊ √ N⌋ for positive integer N.
1. [Initialize]
x =2 ⌈B(N)/2⌉ ;
2. [Perform Newton iteration]
y = ⌊(x + ⌊N/x⌋)/2⌋;
if(y ≥ x) return x;
x = y;
goto [Perform Newton iteration];
We may use Algorithm 9.2.11 to test whether a given positive integer N
is a square. After x = ⌊ √ N⌋ is computed, we do one more step and check
whether x2 = N. This equation holds if and only if N is a square. Of course,
there are other ways to rule out very quickly whether N is a perfect square,
for example to test instances of ( N
p ) for various small primes p, or the residue
of N modulo 8.
It can be argued that Algorithm 9.2.11 requires O(ln ln N) iterations
to terminate. There are many interesting complexity issues with this and
other Newton method applications. Specifically, it is often lucrative to change
dynamically the working precision as the Newton iteration progresses, or to
modify the very Newton loops (see Exercises 9.14 and 4.11).
9.2.3 Moduli of special form
Considerable efficiency in the mod operation can be achieved when the
modulus N is of special form. The Barrett method of the previous section
is fast because it exploits mod 2 q arithmetic. In this section we shall see that
if the modulus N is close to a power of 2, one can exploit the binary nature of
modern computers and carry out the arithmetic very efficiently. In particular,
forms
N =2 q + c,
where |c| is in some sense “small” (but c is allowed to be negative), admit
efficient mod N operations. These enhancements are especially important in
the studies of Mersenne primes p =2q− 1 and Fermat numbers Fn =22n+1,