“We only had one,” said Laura Strimple, Nebraska’s assistant secretary of state. “It hasn’t been confirmed.”

“We haven’t received any complaints to our office or any word of suspicious activity, and we would definitely hear it,” said Matt Roberts, the spokesman for Arizona’s secretary of state.

Some state officials qualified their estimates, saying they had not yet reviewed all questionable ballots, or that voter fraud was a local matter that was usually — but not always — reported to them. Ohio officials declined to offer totals, saying they were still assessing complaints; Pennsylvania and Mississippi officials said they did not track fraud cases.

Many Republicans insist significant problems persist, and that much fraud goes undetected. The conservative Heritage Foundation has published online what it calls an incomplete list of voter fraud and other election-law violations dating to 1982, roughly 450 cases involving both voters and public officials. Properly written, laws requiring voters to display IDs “could increase the fairness of the election process for everyone, regardless of party,” Hans von Spakovsky, the manager of the foundation’s Election Law Reform Initiative, said.

Voting-rights advocates note that the current system caught those violations — and that the numbers, less than one per state per year — constitute a tiny sliver of the millions of votes cast in any election cycle.

No one doubts that election fraud has occurred and needs to be monitored. Election outcomes have been changed by officials who altered vote tallies, and in theory hackers could pick winners by playing havoc with voter rolls, voting machines or electronic reporting networks. But voter fraud, in which someone deliberately casts an invalid ballot or a ballot under someone else’s name, is exceedingly rare.

Its prevalence is at the heart of the debate on restrictions like voter ID. Critics say that cracking down on abuses that barely exist can cost hundreds of thousands of people or more — often the poor and minorities — their ability to vote.