How come we can't make secure voting machines?

I just saw this article and I'm wondering: why can't we make secure voting machines. It is (at least currently) a mathematical fact that a public/private key system (such as AES) cannot be broken, in any practical scenario. Why can we not leverage something like this to create an electronic voting system?

I understand that this is probably not one of the major concerns, compared to things like "how do we know the machine wasn't tampered with?" To that I'd reply, the same way you know the ballots weren't tampered with.