Of the 860,000 Nebraskans who cast ballots in last fall’s election, only two are suspected of casting fraudulent votes. But while the actual number of illegal voters may be minuscule, State Senator John Murante says, there is an even better reason for Nebraska’s Legislature to crack down on fraud at the ballot box.

“First and foremost, we know the confidence that Americans and Nebraskans have in the election system is at an all-time low,” said Mr. Murante, a Republican who is backing a constitutional amendment that would require all voters to display photo IDs. “The perception exists that voter fraud is a serious issue, and that perception itself has to be addressed.”

Nationwide, Republican state legislators are again sponsoring a sheaf of bills tightening requirements to register and to vote. And while they have traditionally argued that such laws are needed to police rampant voter fraud — a claim most experts call unfounded — some are now saying the perception of fraud, real or otherwise, is an equally serious problem, if not worse.

Given Republicans’ history of raising undocumented claims of fraud, Democrats and voting rights advocates say that citing perceptions of tainted ballots as a reason for voting restrictions is disingenuous at best.