Euclidean Algorithm - Algorithmic Efficiency - Efficiency of Alternative Methods

Efficiency of Alternative Methods

Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. For comparison, the efficiency of alternatives to Euclid's algorithm may be determined.

One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor. The common divisors can be found by dividing both numbers by successive integers from 2 to the smaller number b. The number of steps of this approach grows linearly with b, or exponentially in the number of digits. Another inefficient approach is to find the prime factors of one or both numbers. As noted above, the GCD equals the product of the prime factors shared by the two numbers a and b. Present methods for prime factorization are also inefficient; many modern cryptography systems even rely on that inefficiency.

The binary GCD algorithm is an efficient alternative that substitutes division with faster operations by exploiting the binary representation used by computers. However, this alternative also scales like O(h²). It is generally faster than the Euclidean algorithm on real computers, even though it scales in the same way. Additional efficiency can be gleaned by examining only the leading digits of the two numbers a and b. The binary algorithm can be extended to other bases (k-ary algorithms), with up to fivefold increases in speed.

A recursive approach for very large integers (with more than 25,000 digits) leads to subquadratic integer GCD algorithms, such as those of Schönhage, and Stehlé and Zimmermann. These algorithms exploit the 2×2 matrix form of the Euclidean algorithm given above. These subquadratic methods generally scale as O(h (log h)2 (log log h)).