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Ted Cruz’s long-anticipated Iowa surge came a step closer on Tuesday with a new poll showing him just behind Donald J. Trump and leaping ahead of Ben Carson, as terrorism and foreign policy now drive the 2016 nominating race.

Mr. Cruz, the Republican Texas senator, was the choice of 23 percent of likely Republican caucusgoers in the new poll, from Quinnipiac University, following Mr. Trump at 25 percent and ahead of Mr. Carson at 18 percent. Senator Marco Rubio of Florida was in fourth with 13 percent. The margin of error was plus or minus four percentage points.

Just a month ago, Mr. Carson seemed to be running away with the Iowa nominating contest with support from the Christian right, polling at 28 percent in a Quinnipiac poll in late October, followed by Mr. Trump at 20 percent.

But Mr. Cruz, who in the wake of the terrorist attacks in Paris has aggressively criticized President Obama’s handling of the rise of the Islamic State in the Middle East and who has rejected letting Syrian refugees into the country, has more than doubled his Iowa support since last month. Four out of five Iowa Republicans said Syrian refugees should not be allowed to settle in the state.

The survey was taken Nov. 16-22, entirely after the Paris attacks. Terrorism and foreign policy together were the two most important issues for voters, cited by 30 percent of Republicans in deciding whom to support. Jobs and the economy followed.

Voters who named foreign policy and terrorism the most crucial issues chose Mr. Cruz as their top preference.

He was also named the most capable of handling foreign policy by voters over all, winning 24 percent of support, compared with 18 percent for Mr. Trump, 15 percent for Mr. Rubio and 6 percent for Mr. Carson, who was openly questioned last week by his own advisers in a New York Times article.

Only 49 percent in the poll said Mr. Carson had the right experience to be president, compared with 81 percent for Mr. Cruz, 68 percent for Mr. Rubio and 62 percent for Mr. Trump.

Mr. Trump, who created a stir with comments about a federal database for Muslims, was cited as the best able to handle terrorism. He was also seen as the most capable of dealing with economic issues. Mr. Carson was deemed best on social issues like abortion, and he also scored high for caring about “the needs and problems” of voters.

The poll suggested that 2016 caucus turnout would be driven by very conservative voters and those without a college degree. Majorities of both groups said they were more enthusiastic to vote than in past presidential years. Only 41 percent of voters with college degrees were more excited.

Mr. Cruz, who was endorsed last week by Representative Steve King of Iowa, one of the most hard-right members of the House, was by far the top choice of voters who called themselves “very conservative.” Mr. Trump has been shown in many polls to draw more strength from those without a college degree.

Four years ago, an establishment Republican, Mitt Romney, was in the top tier of candidates two months before the caucuses and went on to finish in a near-tie for first. Jeb Bush was supposed to own that lane this year, but he has never gained traction in the state. Mr. Bush was the choice of just 4 percent of Republicans, down from 5 percent in October. Senator Rand Paul of Kentucky was at 5 percent, and Carly Fiorina was at 3 percent. No other candidate was above 2 percent.