Mike Duvos says:
If I have an algorithm that will take any arbitrary RSA key and produce the private key by a mechanism such as the one you propose, you are (almost certainly) proposing an algorithm that will factor arbitrary numbers that are a product of two primes.
This is likely true. However, it does not necessarily follow that such an algorithm will be any faster than current methods of factoring and might very well be a good deal slower.
Ahem. He was proposing a mechanism that will work in log(n) time. All current known methods are subexponential. As you SHOULD know, a log function will eventually be smaller than a subexponential one if you only let N grow large enough. This is baby complexity theory. I find it astonishing that I should even have to mention it.
What you seem to be overlooking is that the function Jim proposes, which tells the numerical order of two keys from an examination of the results of using them, is probably an exponential time algorithm itself as a function of keysize.
Thats not what he was proposing. Obviously one can build such an algorithm given a factoring algorithm, and we know of exponential factoring algorithms. That wasn't the idea. His notion was that there might be a CHEAP algorithm to do this. Perry