Prime Numbers

2.1 Modular arithmetic 85
The first part of this theorem stems from an interesting connection
between Euclid’s algorithm and the theory of simple continued fractions (see
Exercise 2.4). The second part involves the measure theory of continued
fractions.
If x, y are each of order of magnitude N, and we employ the Euclid
algorithm together with, say, a classical mod operation, it can be shown that
the overall complexity of the gcd operation will then be
O ln 2 N
bit operations, essentially the square of the number of digits in an operand
(see Exercise 2.6). This complexity can be genuinely bested via modern
approaches, and not merely by using a faster mod operation, as we discuss in
our final book chapter.
The Euclid algorithm can be extended to the problem of inversion. In fact,
the appropriate extension of the Euclid algorithm will provide a complete
solution to the relation (2.1):
Algorithm 2.1.4 (Euclid’s algorithm extended, for gcd and inverse). For
integers x, y with x ≥ y ≥ 0 and x > 0, this algorithm returns an integer
triple (a, b, g) such that ax + by = g =gcd(x, y). (Thus when g =1and y>0,
the residues b (mod x), a (mod y) are the inverses of y (mod x),x (mod y),
respectively.)
1. [Initialize]
(a, b, g, u, v, w) =(1, 0,x,0, 1,y);
2. [Extended Euclid loop]
while(w >0) {
q = ⌊g/w⌋;
(a, b, g, u, v, w) =(u, v, w, a − qu, b − qv, g − qw);
}
return (a, b, g);
Because the algorithm simultaneously returns the relevant gcd and both
inverses (when the input integers are coprime and positive), it is widely
used as an integral part of practical computational packages. Interesting
computational details of this and related algorithms are given in [Cohen
2000], [Knuth 1981]. Modern enhancements are covered in Chapter 8.8
including asymptotically faster gcd algorithms, faster inverse, inverses for
special moduli, and so on. Finally, note that in Section 2.1.2 we give an “easy
inverse” method (relation (2.3)) that might be considered as a candidate in
computer implementations.
2.1.2 Powers
It is a celebrated theorem of Euler that
a ϕ(m) ≡ 1(modm) (2.2)