[python]Extended Euclidean Algorithm

I took my exam last night, and I guessed I would fail as I did not know how to calculate extended Euclidean Algorithm required for RSA. I came across this video, which explained eGCD really well, better than the slides I had and the tutor’s explanation, the substitution method explained by my tutor was confusing.

The table to find the GCD, s2 and t2 by hand looks like below:

Find GCD with Euclidean Algorithm
In summary gcd is to find the denominator that makes the remainder of two numbers 0. Hence gcd(a, b) = gcd(b, a (mod b)), there will be left shift to test the division until the remainder is 0. So the python equivalent to find the gcd of a and b will be:

def gcd(a, b):
    while b != 0:
        remainder = a % b
        # left shift
        # gcd(a, b) = gcd(b, a (mod b))
        a = b
        b = remainder
    return a

Find s and t with extended Euclidean Algorithm
Thanks to the video, I know how this egcd is done, and hence I wrote my own python logic for this:

def egcd(a, b):
    # thanks to this video: https://www.youtube.com/watch?v=-uFc7-wOplM
    # Extended Euclidean Algorithm assumptions
    # s1 = 1, s2 = 0
    # t1 = 0, t2 = 1
    s1 = 1
    s2 = 0
    t1 = 0
    t2 = 1
    if a < b:
        r1 = b
        r2 = a
        r1 = a
        r2 = b
    while True:
        quotient = r1 // r2
        remainder = r1 % r2
        # stop when remainder is 0.
        # if there is still remainder continue the process of shifting.
        if remainder != 0:
            s = s1 - s2 * quotient
            t = t1 - t2 * quotient
            # left shift
            r1 = r2
            r2 = remainder
            s1 = s2
            s2 = s
            t1 = t2
            t2 = t
            gcd = r2
    return gcd, s2, t2

To find gcd of 17 and 667:

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