Parallelized Pollard's Rho algorithm for ECDLP + Jacobian coordinates My implementation of the parallelized Pollard's Rho algorithm is using Jacobian coordinates to avoid the costly inversion operation when performing point addition.

I am wondering if there are any methods to check for distinguished points while in Jacobian form, since switching back to affine coordinates requires an inversion?
The bad news is that projective coordinates do not work with Pollard's Rho like you want it to. Rho needs an unambiguous point representation to find meaningful collisions, and in projective coordinates each point can have up to $p-1$ valid distinct representations.

The good news is that, sticking to affine coordinates, you can avoid most of the cost of the inversions by performing a bunch of them in parallel, and trading $N$ inversions by $1$ inversion plus $3N-3$ multiplications. This is the so-called Montgomery trick (Section 2.5.1), and is the most common approach in ECDLP Rho implementations.