End-to-End Verifiable Election technology has been studied by the cryptographic research community for more than a quarter century. Unlike traditional systems, end-to-end verifiable elections use cryptographic techniques to enable individual voters and even simple observers to verify that every vote has been counted accurately. This can be done without placing any trust whatsoever in election equipment (including both hardware and software) or in election officials.
This is just a skeleton of links on the subject, but it's enough to get started. Some of my own papers on the subject can be found at my home page.