Prime Numbers

9.4 Enhancements for gcd and inverse 471
266 = (−3/4)392 + 560;
392 = (−1/2)560 + 672;
560 = (−1/2)672 + 896;
672 = (3/4)896 + 0.
Now gcd(525, 266) is seen to be the odd part of 896, namely 7. At each step
we choose the fractional “quotient” so that the 2-power in the remainder
increases. Thus the algorithm below is entirely 2-adic, and is especially suited
for machinery with fast binary operations, such as vector-shift and so on. Note
that the divbin procedure in Algorithm 9.4.7 is merely a single iteration of the
above type, and that one always arranges to apply it when the first integer is
not divisible by as high a power of 2 as the second integer.
Following Stehlé–Zimmermann, we employ a signed modular reduction
x cmod m defined as the unique residue of x modulo m that lies in [−⌊m/2⌋+
1, ⌊m/2⌋]. The function v2, returning the number of trailing zero bits, is as
in Algorithm 9.4.2. As with previous algorithms, B(n) denotes the number of
bits in the binary representation of a nonnegative integer n.
Algorithm 9.4.7 (Stehlé–Zimmermann binary-recursive gcd). For nonnegative
integers x, y this algorithm returns gcd(x, y). The top-level function
SZgcd() calls a recursive, half-binary function hbingcd(), with a classical binary
gcd invoked when operands have sufficiently decreased.
1. [Initialize]
thresh = 10000; // Tunable breakover threshold for binary gcd.
2. [Set up top-level function that returns the gcd]
SZgcd(x, y) { // Variables u, v, k, q, r, G are local.
(u, v) =(x, y);
if(v2(v)