Factoring large integers with the Pollard p-1 method

Question

I am trying to use the Pollard $p-1$ method to find the factors of a large integer. Here is the problem:

An RSA-type cipher is based on the integer $n = 140016480344628383$ and exponent $2345671$. Factor n into a product of two primes, $p$ and $q$, using the Pollard $p-1$ method with base 2.

Once you have found $p$ and $q$, find the decryption index $d$ satisfying $de \equiv1\, ({\rm mod}\,(p-1)(q-1))$

None of my code is working. I can't seem to get it setup without having an overflow. Any tips and pointers would be so appreciated!

Here was one of my code attempts:

n = 140016480344628383;
b = 2;
y = 0;
z = 0;
ls = {};
p = 0;
For[k = 0, k <= 1500, k++,
  y = Mod[b^k!, n];
  b^k != Mod[y*(y - 1), n];
  z = y - 1;
  p = GCD[z, n];
  If[GCD[z, n] > 1, ls = Append[ls, p]];
];

Answer 1

Essentially all you need to get your code to terminate is

• Convert Mod[b^k!, n] to PowerMod[b, k!, n] (b^k! caused the overflow).
• Break out of your loop once p has been found.

Here's your code with these slight modifications. (I also added Monitor to see the progress.)

n = 140016480344628383;
b = 2;
y = 0;
z = 0;
p = 0;

Monitor[
  For[k = 0, k <= 15000, k++,
    y = PowerMod[b, k!, n];
    z = y - 1;
    p = GCD[z, n];
    If[p > 1, Return[p]];
  ], k
]

 373607131

Finally we can verify to make sure everything went smoothly.

FactorInteger[140016480344628383]

{{373607131, 1}, {374769293, 1}}

Edit: I think there still might be a problem in your code. What is the line

b^k != Mod[y*(y - 1), n];

supposed to do? The way you have it, whether it's True or False, it doesn't effect evaluation.

Edit 2: I have removed that line from the code after getting clarification in the comments section.