Question

I’ve written out a couple of functions for doing expmod, that is, (x ** y) % n. These are both standard functions, I’ve checked and re-checked both but can’t find any silly errors.

Here’s the recursive one:

def expmod(x,y,m):
    if y == 0: return 1
    if y % 2 == 0:
        return square(expmod(x,y/2,m)) % m # def square(x): return x*x
    else:
        return (x * expmod(x,y-1,m)) % m

…and here’s the non-recursive one:

def non_recursive_expmod(x,y,m):
    x = x % m
    y = y % m
    result = 1
    while y > 0:
        if(y%2 == 1):
            result = (result * x) % m
        x = (x*x) % m
        y = y/2
    return result

They agree for small values:

>>> expmod(123,456,789) - non_recursive_expmod(123,456,789)
0

…but don’t for larger ones:

>>> expmod(24354321,5735275,654) - non_recursive_expmod(24354321,5735275,654)
-396L

What’s going on?

Answer 1

Your function non_recursive_expmod has some suspicious steps in it: Remove the %m for x and y at the beginning. Both are not needed.

Additionally make sure that the division of y is an integer division by using y = y // 2.

In total the function should look like this:

def non_recursive_expmod(x, y, m):
    result = 1
    while y > 0:
        if y % 2 == 1:
            result = (result * x) % m
        x = (x * x) % m
        y = y // 2
    return result