Declaring that the nation's fundamental security was at stake in the presidential election, Vice President Dick Cheney said yesterday that Senator John Kerry had repeatedly "made the wrong call" on critical foreign policy challenges and failed to appreciate the severity of the threat the nation faced after the Sept. 11 attacks.

Mr. Cheney's remarks were part of a vigorous assault that he and his party mounted on Mr. Kerry's domestic and foreign policy credentials, coupled with a spirited defense of President Bush's economic stewardship, as the Republicans gathered for the third night of their nominating convention in New York.

"He talks about leading a more sensitive war on terror, as though Al Qaeda will be impressed with our softer side,'' Mr. Cheney said of Mr. Kerry, speaking in a somber tone to a crowd that interrupted him with shouts of "U.S.A." as he recalled the terrorist attacks of Sept. 11.

"He declared at the Democratic Convention that he will forcefully defend America -- after we have been attacked,'' Mr. Cheney continued. "My fellow Americans, we have already been attacked, and faced with an enemy who seeks the deadliest of weapons to use against us, we cannot wait for the next attack. We must do everything we can to prevent it -- and that includes the use of military force."

"George W. Bush will never seek a permission slip to defend the American people," he said, drawing an explosion of applause as he mocked Mr. Kerry's call for alliance building.

Mr. Cheney led a parade of Republicans - and one Democrat - in an orchestrated evening intended to undercut Mr. Kerry's commander-in-chief credentials while also trying to seize the ground on economic issues. The speakers stood in front of a banner that read "Land of Opportunity" as they attributed the economic difficulties of the past four years to a downturn that began under President Bill Clinton and was aggravated by the terrorist attacks. They asserted that the economy was on the rise again.

The party formally nominated Mr. Bush to run for a second term at 8:09 p.m.

Mr. Kerry's running-mate, Senator John Edwards of North Carolina, issued a statement shortly after the Republicans adjourned, saying: "There was a lot of hate coming from that podium tonight. What John Kerry and I offer to the American people is hope."

Republicans enlisted the Democrat who delivered the keynote address at the 1992 Democratic convention that nominated Bill Clinton, Senator Zell Miller of Georgia, to offer the keynote for the Republicans, which amounted to a memorably brutal attack on Mr. Kerry and the Democratic Party. Mr. Miller, a Southern conservative who has grown increasingly alienated from his party, said Democrats had placed partisan politics over national security during this time of war and portrayed Mr. Kerry as "faint-hearted," self-indulgent and indecisive.

"For more than 20 years, on every one of the great issues of freedom and security, John Kerry has been more wrong, more weak and more wobbly than any other national figure,'' Mr. Miller said, adding: "George Bush wants to grab terrorists by the throat and not let them go to get a better grip. From John Kerry, they get a 'yes-no-maybe' bowl of mush that can only encourage our enemies and confuse our friends."

But it was a full-throated attack on his own party that roused the Republicans who filled Madison Square Garden.

"At the same time young Americans are dying in the sands of Iraq and the mountains of Afghanistan, our nation is being torn apart and made weaker because of a Democrat's manic obsession to bring down our commander in chief,'' Mr. Miller said, staring sternly across the hall. "What has happened to the party I've spent my life working in?"Mr. Cheney and Mr. Miller paired up in a one-two punch of prime-time attractions for the Republicans on a day where, once again, New York witnessed protests directed against the convention - this time, two that made it inside Madison Square Garden. The police arrested 12 members of the group Act Up who infiltrated the convention hall in a protest of Mr. Bush's AIDS policy during a morning session of Republican youth. Another protester was dragged out after she disrupted Mr. Cheney's speech.

As much presidential campaigning was taking place away from the convention as inside Madison Square Garden. Mr. Kerry, breaking from a tradition in which opposing candidates strike a low profile during nominating conventions, flew from his vacation home in Nantucket to Nashville to tell veterans that "extremism has gained momentum" across the globe under Mr. Bush's policies.

Mr. Bush came to New York the night before he is to accept the nomination to run for a second term, to join firefighters in Queens, in a scene that was shown on monitors in the hall, and to accept the endorsement of leaders of the city's main firefighters union, the Uniformed Firefighters Association.

For the second time in three nights, the challenge to Mr. Kerry's national security credentials was the main line of attack But setting the groundwork for Mr. Bush's acceptance speech on Thursday, the speakers last night also moved to seize the issue of the economy.Republican speakers sought to portray Mr. Kerry as an advocate of an old Democratic Party philosophy of sprawling government and punishing taxes, from his days as a lieutenant governor under former Gov. Michael S. Dukakis of Massachusetts to his 20 years in the Senate. The debate over the economy was waged from the podium - where three families talked about what they said was the benefit of Mr. Bush's tax cuts - and the streets, where thousands of protesters lined up waving fliers that declared, "The next pink slip might be yours!"

Representative Rob Portman of Ohio, a close White House ally from a pivotal state, told the delegates last night: "This president inherited an economy spiraling into recession, and already losing jobs in states like Ohio. Yes, we have more work to do, but we are on track for economic growth."

And Representative Paul D. Ryan, an influential conservative Republican from Wisconsin, said that while Mr. Kerry "claims he wants to relieve the squeeze on the middle class," he had "never met a tax increase he didn't like."

Mr. Bush's economic record is considered a major vulnerability this year, with weak job growth in recent months, a decline in real wages over the past year, and a net loss of 1.8 million private sector jobs since he took office.

"When Republicans talk about a 'Land of Opportunity,' we must ask: Opportunity for whom?" Elliot Spitzer, the attorney general from New York, said at a Democratic Party briefing yesterday. "For the past four years, people like me have had to protect the public because George Bush has not."Republicans turned not only to a Democrat to go after Mr. Kerry, but also to natives of his own state, Massachusetts. "If you think that during the great national policy debate of the 1980's Ronald Reagan was wrong and Ted Kennedy was right, then by all means send in John Kerry," said Gov. Mitt Romney, a Massachusetts Republican.

But the night belonged to Mr. Cheney and Mr. Miller, speaking during prime-time, to a hall filled with delegates who booed at every mention of Mr. Kerry's name, and waved signs that said, "Let Freedom Reign." If Mr. Miller was fiery and provocative, Mr. Cheney delivered equally brutal lines in an understated fashion.

"History has shown that a strong and purposeful America is vital to preserving freedom and keeping us safe - yet time and again Senator Kerry has made the wrong call on national security,'' Mr. Cheney said. "Even in this post-9/11 period, Senator Kerry doesn't appear to understand how the world has changed,'' he said, adding: "A senator can be wrong for 20 years, without consequence to the nation. But a president - a president - always casts the deciding vote. And in this time of challenge, America needs - and America has - a president we can count on to get it right.''

A moment later, when he mocked Mr. Kerry for changing his positions on issues, the audiences responded, "Flip-flop, flip-flop."

Mr. Miller split his time attacking Mr. Kerry and his party. "They don't believe there is any real danger in the world except that which America brings upon itself through our clumsy and misguided foreign policy,'' he said of Democrats. "It is not their patriotism -- it is their judgment that has been so sorely lacking."

The Republicans in the hall were delighted with both speakers - but particularly, at the image of a Democrat of their midst offering what was arguably the toughest attack on Mr. Kerry of the week. "Vintage Zell Miller,'' said Mayor Bob Young of Augusta, Ga.: "With his barbs and his spitballs. He's a very plain speaker and a traditional populist."

And Mr. Cheney, on taking the podium before beginning his 23-minute speech, said archly: "I'm glad Zell Miller is on our side."

Mr. Kerry's aides accused the Republicans of distorting his record - and were particularly dismissive of Mr. Miller.