How to find a square root of a perfect square but negative modulo a semiprime?

I might have found a factoring algorithm based on it. Usually if you want the square root of -49 or -121 you need the factorisation as -7 and -11 are never the answer. But what about the case of finding any root of a negative perfect square with just needing 1? Can it be made easier than factoring a large semiprime?