# Factoring large integers with the Pollard p-1 method

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]];
];

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• Per Wikipedia: In:= n = 140016480344628383; b = 2000; mbig = Product[Prime[j]^Floor[Log[Prime[j], b]], {j, PrimePi[b]}]; g = PowerMod[2, mbig, n]; GCD[g - 1, n] Out= 373607131 – Daniel Lichtblau May 8 '15 at 3:08

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

{{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.

• Wow, thank you SO much! You have saved me hours of staring at my computer. Thank you. – Nora May 8 '15 at 1:53
• @Nora see my edit, what is b^k != Mod[y*(y - 1), n]; supposed to do? – Chip Hurst May 8 '15 at 1:54
• Here is my now updated code: e = 2345671; n = 140016480344628383; b = 2; y = 0; z = 0; p = 0; Monitor[ For[k = 0, k <= 15000, k++, y = PowerMod[b, k!, n]; b^k = Mod[y*(y - 1), n]; z = y - 1; p = GCD[z, n]; If[GCD[z, n] > 1, Return[p]];], k] – Nora May 8 '15 at 1:59
• @Nora I'm still confused. You can't assign b^k a value. Can you say in words what that line is supposed to do? – Chip Hurst May 8 '15 at 2:01
• The b^k != Mod[y*(y - 1), n]; is my attempt at computing b^(k+1)! without overflow. where b is 2 and k is the iterations. That line is now not even needed in the code. It now works correctly and I was able to verify it with the FactorInteger command! – Nora May 8 '15 at 2:01