Every time there are elections in USA, questions of the security vulnerabilities and hacking/tampering with voting machines are raised (I'm sure that it's not US-only phenomenon, either).
Obviously, preventing hardware level hacks is a whole different kettle of fish, but if we assume that the machine is physically secure - if for no other reason, because there are just so many, all over entire USA, so it's nearly impossible to tamper with meaningful enough amount even only counting swing states), would the following approach be a reasonably reliable way to prevent non-physical tampering/hacking of a voting machine?
Have a real ROM (non-writeable) storing the checking code
Checking code verifies that the voting machine software is valid (presumably, by computing checksum, and verifying the checksum against known valid values - for example, stored on a website of Federal election office).
- Use public key encryption for ensuring that the check-sums retrieved are indeed sourced correctly (using federal election office's public key).
If checksum fails, use separate hardware NOT controlled by the rest of the electronics in voting machine, to at least signal malfunction, or even lock out voting functionality.
Since the checking code is 100% in ROM, it would require physical access to the machine to mess with (and if it's sufficiently well protected, e.g. within a welded compartment, would be extra hard to screw with).
UPDATE: just to be clear, the assumption here is that any attack is external to the machine's vendor, and therefore, the software written for it by the vendor does exactly what it's supposed to do barring adequately explained by stupidity bugs. There are no clever trojans in the code, there is no attack vector in the compiler used by the vendor. The question's context is defending against an attack by a 3rd party on a perfectly valid functioning machine.
You can also assume you can trust the checksum signatures on the federal website - why that is the case is irrelevant, but heck, it's as simple as having interns from every party of interest sitting there all election day with printout of original checksum and verifying that the website still matches the printout.