Having been through a three major surgeries in the family, kids' week off from school, a few illnesses and several impending work deadlines - I have completely neglected this poor haven of strange ideas... yup, I'm a slug...
My RSA adventures have lead to some interesting shortening of attack vectors, but nothing of true value as of yet - though I thought I had it licked at one point! Still, some great insight for me.
I've learned a LOT on approaching impossible problems - analysis saves a lot of development work, but only if you're willing to identify and gather real data. Also, you need to analyze sets, not just points on a graph. Knowing your functional, set values and computational limitations beforehand gives you an enormous edge in quickly performing tests - but you must be careful not to be boxed into your own boundaries.
For instance, here's the boundaries for all RSA factors, taken as a percentage of distance from the square root. This trend holds for all magnitudes of prime and presents a rather reduced attack space than I first anticipated. The blue line is the running minimum "center point," the green space represents all possible distances between one the real factor and the square root of the quotient, and the blue space represents all possible distances between the one real factor and the center point (p + q / 2)
In laymen's terms - this means that there's no need to go outside of this space to look for possible p factors for the quotient pq - thus reducing the attack surface. In practical terms, the distance between the square root and one of the factors is AT MOST 2.5%!!! And though the distance may increase to 10% - it is a known search space of much reduced size. Yes, that is still a large area when dealing with 300 digit numbers - but VASTLY smaller than I originally anticipated. Perhaps this is well known among mathematicians - but it was an eye opener for me!
I definitely have gained much even though I have yet to solve this beast!
