I'd not use a computer for this. I'd double the number of people elected and divide the seat between the first and second place person in each election in accordance with their percentage of the vote. So someone with 80% of the vote gets 80% of the seat and represents 80% of the people. Since the first two places generally represent 96-99% of the voters, virtually all voters get represented. Fairly. Unlike the present system, when it's quite possible that 45% get represented.
I'd probably also switch to a system where you have ten points and divide those between the candidates, no more than 7 points on any one candidate.
2. Location of voting.
You vote in the election in the State you are registered in, no matter what voting station you are at. That increases accessibility. I'd also allow for mobile voting stations, so people out in the middle of nowhere can vote.
The price for that is that I'd make voting mandatory for anyone in the continental US, optional for any eligible voter elsewhere. Overseas voters would not be using forms, they'd use the same voting system as everyone else and their votes would be counted at the same time. That way, there are no "accidents".
Anyone who is alive and was born in, was naturalized in, or is living in, the US should have the right to vote. The incompetence of State authorities, the destruction of records, etc, are not my concern. If States cannot be trusted to know who is eligible, then make everyone eligible. It is better that ten guilty go free than to have one innocent person suffer.
This makes computer voting essential because the only way to get secure voting stations everywhere is to not rely on heavy physical security.
4. Proof that one person got one vote
Use voting registration cards to hold a 4096-bit "public" key. A central voting computer carries the corresponding private key. A SHA-3 hash is used to identify which vote goes with which decryption key. A second SHA-3 hash of the vote, with a digital signature for the combined whole, will prove the vote has not been tampered with.
Votes would be transmitted in encrypted form from the voting station to a proxy server at the polling station. It cannot be decrypted there, as there's no decryption key present. It is then reliably multicast to the central computer and to independent observers. The reason for the proxy is to break any timing attack that could be used to identify who cast which encrypted vote. Reliable multicast, such as NORM, guarantees all observers and the central machine received the vote.
The voting computers, proxy and central computer would need to be open source software and open source hardware, with the software proven correct in both source and binary form. All would need to be Trusted Computers (A1+) and tamper-proof. The central computer should also be physically inaccessible.
The central computer would generate the cards, retaining the private keys and issuing only the public keys with hash. These would be fed directly into a second computer with the voter details. This would simply print the details onto the card and seal it. This way, the central computer doesn't know who the private key is associated with and the secondary computer doesn't know the private key.
(The decryption key should be printed onto parchment paper using indelible ink, together with the hash. Printouts should be fed directly into a storage bin capable of holding around 500 million pages. This should be in a distinct room that cleared individuals can enter in pairs, for maintenance. Ideally, the server room should not be entered at all, ever, once the machine is running.)
The second computer would have the name and address of every person over the age of 16 (which I'd make the new voting age) either resident in the US or born in the US regardless of where they were resident.
This sealed card would be handed off to the voting officials to mail off. They would need to certify whose card had been sent, so that discrepancies could be flagged immediately.
The voter puts the card in the machine and the vote is recorded. A physical copy should be printed out on indelible ink on parchment paper (paper with a mean time between failures of around 1,000 years) together with the hash. The physical copy should be in ASCII armoured form.
Since the voting machine is tamper-proof, uses strong encryption, prevents access to the memory being used by the software and prevents access to any privileged function by any introduced software, you can pack a bunch in an old van and take them to absolutely anywhere. It wouldn't impact the security of the system in the slightest.
So you massively increase the accessibility of the voting system, massively increase the relevance of a vote and massively expand the voices you hear.