Hi. Skim read your other replies. Am confident that you and I have enough common cause, overlap that we're pulling in the same direction. So this is a sincere response of my current worldview. My position has changed A LOT over time and I reserve the right to change more in the future. :)
I was an election integrity activist. I had some minor local victories. Like delaying the rollout of new gear (until 2009, a much less risky election) and prohibiting unique bar codes on our ballots (which were linked to voter id).
I've studied crypto voting proposals and other privacy preserving stuff like translucent databases and differential privacy. I'm just a simple bear, so I don't claim to be authoritative.
The core idea of the crypto systems I studied is to emulate the secure one-way hash of physical shuffle of the ballot box (Australian ballot) by inducing hash collisions. So there is no algorithm way to reverse the transformation.
Great in theory. Hard to implement in practice.
Over simplifying: you need simple (enough) ballots and large (enough) political jurisdictions (precincts) to ensure that each voter's ballot gets lost in the herd of ballots.
Otherwise straight forward brute force process of elimination (big data) allows trivial reidentification of each ballot's voter.
This is already a problem with our paper-based systems. Especially postal ballots. So us election integrity activists advocate very specific procedures to preserve voter privacy. Which are expensive and onerous, so rarely done.
For future, I will remain open minded about crypto voting systems. But it'll always remain a tough sell.
How this relates back to voter registration and eligibility? I don't know. Identity and authentication is very hard. My very best guess, per the translucent database strategies, is that once we issue everyone unique identifiers, such as Real ID, then we'll be able to technically unlock a lot of privacy protections.
Until then, because we have to link identifiers across heterogenous systems, we have to store PII as cleartext. Which defeats the purpose.