Getting different results for the same equation in a function and the shell

Question

I've implemented Pollard's Rho for logarithms using Sage, as the following program stored in pollardrho.py.

def pollardrho(g, h, G):
    k, m = 1, 0
    t = g**k * h**m
    i, j = 1, 0
    r = g**i * h**j

    def step(t, k, m):
        if lift(t) % 3 == 0:
            return (t * g, k+1, m)
        if lift(t) % 3 == 1:
            return (t * h, k, m+1)
        if lift(t) % 3 == 2:
            return (t ** 2, 2*k, 2*m)

    while True:
        t, k, m = step(t, k, m)
        r, i, j = step(*step(r, i, j))
        if t == r:
            print("Found a cycle")
            print("g^%s h^%s == g^%s h^%s" % (k, m, i, j))
            print("g^(%s - %s) == h^(%s - %s)" % (i, k, m, j))
            l = g.multiplicative_order()
            print("(%s - %s) / (%s - %s) %% %s" % (i, k, m, j, l))
            return (i - k) / (m - j) % l  # this is where everything goes wrong.

Running this with G = GF(1013), g = G(3), h = G(245) gives the following output:

sage: pollardrho(g, h, G)
Found a cycle
g^262 h^14 == g^16870 h^1006
g^(16870 - 262) == h^(14 - 1006)
(16870 - 262) / (14 - 1006) % 1012
995

However:

sage: (16870 - 262) / (14 - 1006) % 1012
375

Note that this is a completely different result!

If I check the types of i, j, k, m, they are all of type int...

Answer 1

It turns out that typing an integer in the sage shell gives a different result than doing the same inside a python program that uses Sage libraries:

sage: type(1234)
<type 'sage.rings.integer.Integer'>

This isn't the same as the <type 'int'> I got inside of my own program!

Using k, m = Integer(1), Integer(0) solved my problem and I now get the correct discrete log.