Prime Numbers

9.4 Enhancements for gcd and inverse 465
}
2. [Halve t3]
goto [Check even];
if(t1,t2 both even) (t1,t2,t3) =(t1,t2,t3)/2;
else (t1,t2,t3) =(t1 + y, t2 − x, t3)/2;
3. [Check even]
if(t3 even) goto [Halve t3];
4. [Reset max]
if(t3 > 0) (a, b, h) =(t1,t2,t3);
else (v1,v2,v3) =(y − t1, −x − t2, −t3);
5. [Subtract]
(t1,t2,t3) =(a, b, h) − (v1,v2,v3);
if(t1 < 0) (t1,t2) =(t1 + y, t2 − x)
if(t3 = 0) goto [Halve t3];
return (a, b, 2 β h);
Like the basic binary gcd algorithm, this one tends to be efficient in actual
machine implementations. When something is known as to the character of
either operand (for example, say y is prime) this and related algorithms can
be enhanced (see Exercises).
9.4.2 Special inversion algorithms
Variants on the inverse-finding, extended gcd algorithms have appeared over
the years, in some cases depending on the character of the operands x, y. One
example is the inversion scheme in [Thomas et al. 1986] for x −1 mod p, for
primes p. Actually, the algorithm works for unrestricted moduli (returning
either a proper inverse or zero if the inverse does not exist), but the authors
were concentrating on moduli p for which a key quantity ⌊p/z⌋ within the
algorithm can be easily computed.
Algorithm 9.4.4 (Modular inversion). For modulus p (not necessarily
prime) and x ≡ 0(modp), this algorithm returns x −1 mod p.
1. [Initialize]
z = x mod p;
a =1;
2. [Loop]
while(z = 1) {
q = −⌊p/z⌋; // Algorithm is best when this is fast.
z = p + qz;
a =(qa) modp;
}
return a; // a = x −1 mod p.
This algorithm is conveniently simple to implement, and furthermore (for
some ranges of primes), is claimed to be somewhat faster than the extended