How can I decrypt a RSA2 encrypted message C given only private key d and Phi(n)?

Question

How can I decrypt a RSA encrypted message C given private key d and Phi(n)?

It is known that this is a TextBook RSA, so e is small supposedly.

I have gone through all the usual ways to decrypt RSA, however, it seems that there are little things I could do with only Phi(n).

Sorry, but that's all the information I have on hand to solve an RSA encryption. I am given Ciphertext C, private key d and phi(n). nothing more T_T — Kim lim, Sep 02 '20 at 13:41
Yes, you do have more context if this is a ctf. At a minimum you have the actual values and can add them to your question. — President James K. Polk, Sep 02 '20 at 13:44
The numbers are stored in a black box, we must provide a method to it to obtain the flag, its given that all C, d, phi(n) are all 1024-bits and the RSA is TextBook RSA — Kim lim, Sep 02 '20 at 16:14
ok, the fact that they're 1024 bits is significant as it rules out the method proposed in the answer below. — President James K. Polk, Sep 02 '20 at 16:34

guidot · Answer 1 · 2020-09-02T13:04:53.123

1

Phi(n) for an RSA-modulus (known to be of the form n = p*q, with p and q being primes), is simply:

φ(n) = (p-1) * (q-1)

Since p-1 and q-1 both are composite (no primes, since they will be even), you will need to iterate through all combinations of the factorization (assuming each factor to belong to p and subsequently q), which satisfy p and q being in the same magnitude. If the incremented product is not prime you may skip that combination.

You re-compute n from the assumed p and q and if decryption works you found it.

edited Sep 02 '20 at 13:04

answered Sep 02 '20 at 12:46

guidot

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This assumes that phi(n) is small enough or smooth enough to be factored by someone doing a ctf. – President James K. Polk Sep 02 '20 at 13:50

How can I decrypt a RSA2 encrypted message C given only private key d and Phi(n)?

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