So about the Extended Euclidean algorithm, the algorithm that finds, a, b, and d in the equation am+bn=d, where m and n are two positive integers, d is their GCD, and a and b are two integers (not necessarily positive)
So the book I am following has the following instructions for implementing this algorithm:
E1. [Initialize.] Set a′ ← b ← 1, a ← b′ ← 0, c ← m, d ← n.
E2. [Divide.] Let q and r be the quotient and remainder, respectively,
of c divided by d. (We have c = qd + r and 0 ≤ r < d.)
E3. [Remainder zero?] If r = 0, the algorithm terminates;
we have in this case am + bn = d as desired.
E4. [Recycle.] Set c ← d, d ← r, t ← a′, a′ ← a, a ← t − qa, t ← b′, b′ ← b, b ← t − qb,
and go back to E2.
My problem is that I don't particularly understand all the math around this algorithm, I understand the normal Euclidean algorithm well enough, so I understand half of the extended one as well. What I don't understand is the need for a' and b' as well t (I get that t is a temporary variable but I don't get the a = t - qa and b = t - qb part). That as well as the initialization of the variables a, a', b, and b' as well as the swapping of the values between them on each iteration. Can anybody help me bridge these gaps in my understanding of this algorithm?
