Prime Numbers

9.3 Exponentiation 459
But there is a kind of complementary way to effect the powering. This
alternative is exemplified in the relation
x 13 = x ∗ (x 2 ) 2 ∗ (x 4 ) 2 ,
where there are again 2 multiplications and 3 squarings (because x 4 was
actually obtained as the middle term (x 2 ) 2 ). In fact, in this example we see
more directly the binary expansion of the exponent. The general formula
would be
x y j
yj2
= x = x y0 (x 2 ) y1 (x 4 ) y2 ···, (9.16)
where the yj are the bits of y. The corresponding algorithm is a “right-left”
ladder in which we keep track of successive squarings of x:
Algorithm 9.3.2 (Binary ladder exponentiation (right-left 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; a =1;
2. [Loop over bits of y, starting with lowest]
for(0 ≤ j