Donald Trump. Joe Raedle/Getty Images Donald Trump has accused Sen. Ted Cruz (R-Texas) of "illegally" stealing his Monday-night Iowa caucus victory and suggested that Cruz's vote total should be "nullified."

Cruz won the Republican Iowa caucuses on Monday in something of an upset over Trump, who had been leading polls for weeks before the voting started.

In the first in a string of attacks against Cruz, Trump tweeted Wednesday: "Ted Cruz didn't win Iowa, he stole it. That is why all of the polls were so wrong and why he got far more votes than anticipated. Bad!"

An earlier version of Trump's tweet had said Cruz "illegally stole" the caucus win.

Laying out a series of Cruz actions that he suggested were "fraud," Trump later summarized in a tweet: "Based on the fraud committed by Senator Ted Cruz during the Iowa Caucus, either a new election should take place or Cruz results nullified."

In other tweets, Trump explained that he was referring to a controversy for which Cruz apologized to retired neurosurgeon Ben Carson on Tuesday.

Carson's campaign accused Cruz of playing "dirty tricks" on the night of the caucuses.

At the time, Carson reportedly signaled that he was speaking early Monday and flying home to Florida, which caused some Cruz supporters to publicly speculate that Carson was withdrawing from the race. Rep. Steve King (R-Iowa), a prominent Cruz surrogate, wrote on Twitter, for example: "Carson looks like he is out. Iowans need to know before they vote. Most will go to Cruz, I hope."

Carson, who placed fourth in the voting, blasted what he called the "deceit and dirty tricks and lies" during a speech for his supporters in Iowa. Cruz apologized to Carson on Tuesday and said his team should have promoted Carson's statement denying the rumors.

"During primetime of the Iowa Caucus, Cruz put out a release that @RealBenCarson was quitting the race, and to caucus (or vote) for Cruz," Trump tweeted.

Republican US presidential candidates Donald Trump, Ben Carson, and Ted Cruz. REUTERS/Jim Young

Trump also hit Cruz over a provocative mailer the senator's campaign had sent some voters in Iowa.

"Many people voted for Cruz over Carson because of this Cruz fraud," he said. "Also, Cruz sent out a VOTER VIOLATION certificate to thousands of voters. ... The Voter Violation certificate gave poor marks to the unsuspecting voter(grade of F) and told them to clear it up by voting for Cruz. Fraud."

The real-estate mogul was humble in his concession speech Monday night, but he has since launched attacks on Cruz and retweeted supporters who have called the Iowa caucuses "meaningless" and given Trump other encouragements heading into New Hampshire, the next voting state.

At a rally Tuesday night, Trump repeated his attacks about the legality of Cruz's candidacy for president, saying Cruz was "born in Canada" and warning that if Cruz locked down the Republican nomination, Democrats were "going to sue his a-- off."

Cruz was born in Canada to an American mother, but most legal experts believe he meets the Constitution's requirement that presidents be "natural born" citizens.

The Cruz campaign didn't immediately respond to a request for comment Wednesday morning.

Here are Trump's tweets on the issue:

Ted Cruz didn't win Iowa, he stole it. That is why all of the polls were so wrong and why he got far more votes than anticipated. Bad! — Donald J. Trump (@realDonaldTrump) February 3, 2016

During primetime of the Iowa Caucus, Cruz put out a release that @RealBenCarson was quitting the race, and to caucus (or vote) for Cruz. — Donald J. Trump (@realDonaldTrump) February 3, 2016

Many people voted for Cruz over Carson because of this Cruz fraud. Also, Cruz sent out a VOTER VIOLATION certificate to thousands of voters. — Donald J. Trump (@realDonaldTrump) February 3, 2016

The Voter Violation certificate gave poor marks to the unsuspecting voter(grade of F) and told them to clear it up by voting for Cruz. Fraud — Donald J. Trump (@realDonaldTrump) February 3, 2016

And finally, Cruz strongly told thousands of caucusgoers (voters) that Trump was strongly in favor of ObamaCare and "choice" - a total lie! — Donald J. Trump (@realDonaldTrump) February 3, 2016

Based on the fraud committed by Senator Ted Cruz during the Iowa Caucus, either a new election should take place or Cruz results nullified. — Donald J. Trump (@realDonaldTrump) February 3, 2016

Trump later tweeted a photo of what appears to be a Cruz campaign communication citing the Carson news and instructing people to caucus for Cruz:

This was sent out from Ted Cruz- as Iowans arrived at their caucus sites to vote. #CruzFraud pic.twitter.com/tRM7KUCrSU — Donald J. Trump (@realDonaldTrump) February 3, 2016

The State of Iowa should disqualify Ted Cruz from the most recent election on the basis that he cheated- a total fraud! — Donald J. Trump (@realDonaldTrump) February 3, 2016

Colin Campbell contributed to this report.