I made a mistake with my description of finding the decryption key. You actually need to find loop exponents for both the cipher text and a reencryption of it (the second search is scaled to be faster and sometimes the first exponent works for both). This is to make sure that the exponent picked is of a period which fits the orbits both numbers lie on.
Once that exponent is found, it can be used in a manner similar to the totient and the decryption exponent will be the modular multiplicative inverse of the encryption exponent in that space. Which is nifty. I've coded up a proof of concept showing an actual attack on RSA type keys and decrypting data given a semiprime and encryption exponent. This approach appears to be novel and investigation to date leads me to conclude that some real world keys might be vulnerable to it. Though the proof of the pudding is in the eating so I'm looking for a way to test this on a computer fast enough to do something interesting (let me know if you have any thoughts on this). I hope this helps. Gabriel
