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And if I am on the right track, how do I reach the solution from here?

I'm doing a CS game and I've hit a wall (I'm a rank beginner when it comes to encryption) on the encryption portion. I've been given the encrypted output and what I think (from my research) is an encoded version of the key using this:

RSA Mod Equation

For me, C is what I'm looking for, d = 7, N = 98286183916580532129117891164396375520832348985166654489180704043300879122347 and P = 89511674269883781051088511817946305647658886680759682801944853433540569894936.

p and q for N are 294156383975755063828794349953398256143 and 334129018681034197306379539716603659429

C was created randomly and is known to be 16 bytes. P is known because it was the first 32 bytes of the encrypted file (that I ripped out using a hex editor and turned into base 10) and d and N were both hardcoded.

I tried reversing it manually (using modular arithmetic, since a = b mod n then b = a mod n so C = root7(PmodN) ) but I got a 37 byte number that I know must be incorrect. Can someone please tell me where I'm going wrong?

I can post the source if it's necessary.

Engie
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1 Answers1

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I think you should review how RSA encryption and decryption work.

You are given all the elements you need:

$P = C^d \mod N$

You know $d$, you know $p$, $q$, and thus $pq = N$. Usually people use $e$ to the encryption exponent and $d$ for the decryption exponent, but from what you said, $d$ is being used to encrypt.

You can recover $C$ with $P^u \mod N$, where $u$ solves:

$ud = 1 \mod (p-1)(q-1)$

Can you figure it out from there? (Hint: The Extended Euclidean algorithm will help.)

  • You keep editing the post so I'm a little bit confused. Just what is that second equation supposed to be? – Engie May 17 '17 at 23:36
  • You want to find the modular inverse of d, modulo (p-1)(q-1). The Extended Euclidean algorithm will do this. –  May 17 '17 at 23:38
  • Looking over it – Engie May 17 '17 at 23:43
  • This part in particular –  May 17 '17 at 23:43
  • Alright, I can write that algorithm no problem. So a would be d and n would be (p−1)(q−1) right? I want to make sure I'm not misinterpreting. – Engie May 17 '17 at 23:47
  • Yes, in that expression $at = 1 \mod n$, that's correct. –  May 17 '17 at 23:59
  • That gave me an integer answer which is far better than can be said for my attempts earlier, but it still gave me an answer that's too large (32 bytes) instead of the 16 byte answer I'm looking for (66700634945270197619886428952694736069398970829100240402059886548334902240273) can you take a guess at why? – Engie May 18 '17 at 00:04
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    Your implementation may be wrong--you can test it by multiplying it by 7, taking the answer mod N and seeing if you get 1. If not, then the implementation is wrong. Rosetta Code may have something you can use too. –  May 18 '17 at 00:11
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    As much as I hate to say it you were 100% correct. I searched around for an already written algorithm and used that and it gave me exactly what I was looking for. Thanks very much I feel like I've learned a lot today. – Engie May 18 '17 at 00:50