Chapter 28. The Extended Euclidean Algorithm
The Extended Euclidean Algorithm is a staple of number theory and is used to solve equations of the form ax mod n = b. This chapter reviews this algorithm and its applications. We begin with the classical algorithm and then extend it to solve simple equations.
The Euclidean Algorithm
Euclid's algorithm determines the greatest common divisor of two integers. The algorithm is based on the observation that, if x divides both a and b, then x divides their difference a – b. The trick is to find the largest such x.
Assume (without loss of generality) that a > b. If x divides a – b, then it also divides a – qb, where q is an integer. Let r = a – qb. If n ≠ 0, and x divides a – qb, then x divides r. We have now ...