negative exponent in modular exponentiation for RSA

Question

I am trying to write an RSA code in python3.6 for educational purposes.

The key generation and message encryption work fine, but I have a problem with decryption. As I understand the decryption algorithm is M = C^d mod n, where M is the message, C is the encrypted message (using the public key of the receiver) , d is the private key of the receiver. The problem is when d is negative, which in my experience is very often. I'm using right-to-left algorithm for modular exponentiation, but I don't know how to make it work with negative exponent. Here's my code for m. e.:

def mod_pow(b, e, m):
if m == 1:
    return 0
res = 1
b = b % m
while e > 0:
    if e % 2 == 1:
        res = (res * b) % m
    e = e >> 1
    b = (b * b) % m
return res

See [Computing modular inverses](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Computing_multiplicative_inverses_in_modular_structures): "if t < 0 then t := t + n". That is why d will always be positive for a correct modular inverse algorithm. — President James K. Polk, Jun 13 '17 at 12:23

score 1 · Accepted Answer · answered Jun 13 '17 at 12:43

1

The question was answered by James K Polk, I just had to add this to the extended Euclidean algorithm code:

if t < 0:
    t += n

answered Jun 13 '17 at 12:43

Ivan Gorin

303
2
12

negative exponent in modular exponentiation for RSA

1 Answers1