python - extended euclidean algorithm and the concept of multiplicative inverse

Question

I have with python:

e*d == 1%etf

we know (e) and (etf) and must discover (d) using the extended euclidean algorithm and the concept of multiplicative inverse of modular arithmetic.

d = (1/e)%etf

d = (e**-1)%etf

generate a global wrong number, please help me find (d) using the rules above explained.

The solution (Modular multiplicative inverse function in Python) illustrated below gives me wrong computational result

e*d == 1 (mod etf)
d = (e**(etf-2)) % etf 
d = pow(e,etf-2,etf)

Am I making some mistake elsewhere? Is this calculation ok?

Answer 1

The trick you list with d = (e**(etf-2)) % etf will work only if etf is prime. If it's not, you have to use the EEA itself to find the modular multiplicative inverse.