Prime Numbers

9.2 Enhancements to modular arithmetic 453
where b = B(R) − 1. In particular, such splitting can render the relevant
multiplications somewhat simpler. In fact, one sees that
R x/2 b−1 /2 b+1 = ⌊x/N⌋−j (9.13)
for j =0, 1, or 2. Thus using the left-hand side for d in Algorithm 9.2.10
involves at most two passes through the while loop. And there is an apparent
savings in time, since the length of x can be about 2b, and the length of R
about b. Thus the multiplication xR in Algorithm 9.2.10 is about 2b × b
bits, while the multiplication inherent in (9.12) is only about b × b bits.
Because a certain number of the bits of xR are destined to be shifted into
oblivion (a shift completely obscures the relevant number of lower-order bits),
one can intervene into the usual grammar-school multiply loop, effectively
cutting the aforementioned rhombus into a smaller tableau of values. With
considerations like this, it can be shown that for 0 ≤ x