State and/or municipal authorities could be tasked with generating identifiers and signing them cryptographically, so that any voting station can verify whether a given identifier is genuine. The federal government doesn't need to keep a central registry of all identifiers, only a list of public keys used by states and municipalities.
Obviously I'm not aware of any country doing things in such a convoluted way. But if Americans really don't want there to be a central registry of everyone's identity, modern computer science and cryptography could help.
