On the Complexity of the Extended Euclidean Algorithm (extended abstract)

Euclid’s algorithm for computing the greatest common divisor of 2 numbers is considered to be the oldest proper algorithm known ([10]). This algorithm can be amplified naturally in various ways. The GCD problem for more than two numbers is interesting in its own right. Thus, we can use Euclid’s algorithm recursively to compute the GCD of more than two numbers. Also, we can do a constructive computation, the so-called extended GCD, which expresses the GCD as a linear combination of the input numbers. Extended GCD computation is of particular interest in number theory (see [1, chapters 2 and 3]) and in computational linear algebra ([3, 4, 9]), in both of which it takes a basic role in fundamental algorithms. An overview of some of the earlier history of the extended GCD is given in [1], showing that it dates back to at least Euler. Motivated by many efforts to find good algorithms for extended GCD computation, Majewski and Havas ([12]) showed that this is genuinely difficult. There are a number of problems for which efficient solutions are not readily available.