I think you misunderstand. Perry and I are talking about the algormithm (If it exists) being O(log_2(n)). That is, "log base 2 of n". This means that the time taken is proportional to the log to the base two of the number of keys.
Fascinating as this speculation is, I see no way to craft such an algorithm. The nature of the modular space makes "larger" and "smaller" difficult to distinguish.
I have made submission of a short text which details my thoughts relating to a mod function attack. I am under no illusion about the complexity of mounting a factor attack. I do see the mod function as the next natural hole to look at the algorithm through. I can find no work relating to periodicities in the mod function and it occurs to me that such relationships might point the way...