Hours after terror struck New York City, President Trump tweeted that he was ordering the Department of Homeland Security to "step up" the country's "already Extreme Vetting Program."

Yet it remained unclear Wednesday what exactly the president was calling for and what changes may come.

"You'll have to contact the White House regarding the President's tweet," said Homeland Security acting press secretary Tyler Houlton.

The federal government has broad powers to limit or suspend immigration in the name of national security, as it did following the Sept. 11, 2001, terrorist attacks. Working with Congress, President George W. Bush drastically cut the number of foreigners allowed to enter the U.S. in the years following the attacks.

Republican leaders in Congress said the government should employ similar tactics to ensure that Tuesday's truck attack in New York City is not repeated.

During appearances on Fox News, Rep. Peter King, R-N.Y., called for a "full court press" to increase vetting procedures used against foreigners, and Sen. Rand Paul, R-Ky., said it was important to learn more about incoming immigrants.

"We should extensively question people: 'Do you believe in freedom of religion? Will you pledge allegiance to our flag and to our country if you come here?'" Paul said. "If you can't answer all of those questions well ... then I don't think we should have an open border to the world."

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Immigration experts cautioned against an overreaction, pointing out that the suspect in the NYC attack — Sayfullo Saipov — appears to have been inspired to commit the attack only after entering the U.S.

Alex Nowrasteh, an immigration policy analyst at the conservative Cato Institute who has studied terrorist acts committed by immigrants, said the Uzbekistan native who legally entered the country using a diversity visa seems to fit the profile of most foreigners who commit acts of terrorism in the U.S.

"Saipov entered over seven years ago. This attack did not take seven years or more to plan," Nowrasteh said. "By all indications so far, he became radicalized after he arrived to the United States."

That's why Nowrasteh believes little can be done to identify foreigners who may go on to become radicalized in the U.S. years after they arrive. In the 16 years following Sept. 11, he said most domestic attacks committed by foreigners were committed by people who entered the country legally and had no red flags on their record.

The brothers who set off two bombs at the Boston Marathon in 2013 came from Kyrgzystan and were children when their family entered the U.S. And Omar Mateen, who killed 49 people in the Pulse nightclub in Orlando in 2016, was born in New York to Afghan parents.

Of the over 2.6 billion foreigners admitted into the U.S. from 2002 to 2016, there have been fewer than 10 terrorists who entered as a result of a vetting failure, according to an upcoming report written by Cato immigration policy analyst David Bier.

One of those was Tashfeen Malik, who helped her U.S.-born husband kill 14 people in the San Bernardino, Calif., attack in 2015.

"The current immigration screening works very well at excluding immigrants who plan terrorism or who are radicalized before they get to the United States," Nowrasteh said. "Perfect screening, unless there is a time machine involved, would not have been able to stop this fellow from coming here."

Even if Trump does try to further enhance screening procedures, he may run into some legal trouble.

His recent attempts to institute a travel ban against majority-Muslim countries have repeatedly been blocked or limited by federal judges, showing that the White House does not have absolute power to restrict foreigners entering the U.S.

The latest version of the ban, which would bar most people from eight countries from entering the U.S., was blocked by two federal judges. The Department of Justice is appealing those decisions.

On Wednesday, New York City Mayor Bill de Blasio said Trump was making a mistake by using the latest attack to enact sweeping restrictions against entire groups of people.

"We support vetting of individuals," de Blasio said during a press conference. "There should be very, very careful vetting of anyone where there's an indication of a concern, but not because of their religion, not because of their country of origin."