I want to determine if a value is a quadratic residue mod $2^{32}$. I've developed a very fast pre-screening method based on a Bloom Filter that identifies quadratic residues for mod $2^7=128$ in just a couple multiplies, but it can't extend to much higher powers efficiently.

The general method of testing for quadratic residues will work, using Hensel lifting to find a modular square root if it exists, but I wonder if there are any shortcuts given the specific modulus of $2^{32}$.

This is for a computer search, so efficiency using 32 or 64 bit math helps.