In school we learn that the RSA encryption algorithm consists of one public key and one private key. I've asked several professors if there could be a second private key that decrypts all messages in the same way as the intended private key and I've always received a negative answer.

And yet I stumbled upon two private keys in one particular example. I delved into the Mathematics behind the RSA algorithm and I found out that in fact there are at least two keys in every single example. I not only managed to prove the existence of a second private key but I was also able to find two different formulas for obtaining the second key.

One of the formulas requires knowledge of phi(n)/2 where phi(n) is Euler's Totient Function.

If d

_{1}is the intended private key and (n, e) is the public key then the second key d

_{2}can be found using the following formula:

d

_{2}*e = phi(n)/2Here's the full paper: https://marinaibrishimova.net/docs/otherkeys.pdf

Currently, I'm looking into ways of obtaining phi(n)/2 without having to factor n.