Theory of Relative Squares

[AJMansfield]

What I mean, is that if all you knew was , normally you have to take its square root (a tedious, process, where the current best for ) to find , and then you can just do to get . But if you could figure out how to get from without needing to calculate , you could crack RSA using the difference-of-squares method much more quickly than existing methods.

Basically, find a way of computing a square root of a perfect square that is significantly faster than multiplying two numbers of the same lemgth, and you win. (Not that I expect this to ever happen.)