Prime Numbers

458 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
the exponent, and so on. Let us first summarize the categories of powering
ladders:
(1) Recursive powering ladder (Algorithm 2.1.5).
(2) Left-right and right-left “unrolled” binary ladders.
(3) Windowing ladders, to take advantage of certain bit patterns or of
alternative base expansions, a simple example of which being what is
essentially a ternary method in Algorithm 7.2.7, step [Loop over bits ...],
although one can generally do somewhat better [Müller 1997], [De Win et
al. 1998], [Crandall 1999b].
(4) Fixed-x ladders, to compute x y for various y but fixed x.
(5) Addition chains and Lucas ladders, as in Algorithm 3.6.7, interesting
references being such as [Montgomery 1992b], [Müller 1998].
(6) Modern methods based on actual compression of exponent bit-streams, as
in [Yacobi 1999].
The current section starts with basic binary ladders (and even for these,
various options exist); then we turn to the windowing, alternative-base, and
fixed-x ladders.
9.3.1 Basic binary ladders
We next give two forms of explicit binary ladders. The first, a “left-right”
form (equivalent to Algorithm 2.1.5), is comparable in complexity (except
when arguments are constrained in certain ways) to a second, “right-left”
form.
Algorithm 9.3.1 (Binary ladder exponentiation (left-right form)).
This algorithm computes x y . We assume the binary expansion (y0,...,yD−1)
of y>0, where yD−1 =1is the high bit.
1. [Initialize]
z = x;
2. [Loop over bits of y, starting with next-to-highest]
for(D − 2 ≥ j ≥ 0) {
z = z 2 ; // For modular arithmetic, do modN here.
if(yj == 1) z = zx; // For modular arithmetic, do modN here.
}
return z;
This algorithm constructs the power x y by running through the bits of the
exponent y. Indeed, the number of squarings is (D − 1), and the number of
operations z = z ∗ x is clearly one less than the number of 1 bits in the
exponent y. Note that the operations turn out to be those of Algorithm 2.1.5.
A mnemonic for remembering which of the left-right or right-left ladder forms
is equivalent to the recursive form is to note that both Algorithms 9.3.1 and
2.1.5 involve multiplications exclusively by the steady multiplier x.