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
else:
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
else:
gcd = r2
break
return gcd, s2, t2
Results
To find gcd of 17 and 667:
