And they ascertained that a final April 2009 e-mail announcing a "marriage to Nadia" between the 15th and the 20th was a signal that a terrorist attack was imminent, according to British court documents.

It is unclear exactly how British intelligence linked the Pakistani e-mail address to a senior al-Qaeda operative who communicated in a kind of code to his distant allies.

But the intelligence helped stop the plot in England, and the address made its way to the US National Security Agency (NSA).

A few months later, the NSA was monitoring the Yahoo user in Pakistan when a peculiar message arrived from a man named Najibullah Zazi, an Afghan American living in Colorado. He asked about "mixing of [flavour and ghee oil] and I do not know the amount, plz right away."

Soon after, on September 9, 2009, a second message arrived that echoed the code used in the British plot: "The marriage is ready," Zazi wrote.

The e-mails led the NSA to alert the FBI, which obtained a court order to place Zazi under more extensive surveillance. Officials learned that he had visited Pakistan in 2008, the same time as one of the British plotters.

In the end, the e-mails and additional surveillance foiled a plot by Zazi and two others to conduct suicide bombings in the New York subway system just days after he sent the "marriage is ready" e-mail. In recent days, US intelligence and law enforcement officials, as well as congressional officials, have pointed to the authority that allowed them to target the Yahoo account - Section 702 of the Foreign Intelligence Surveillance Act (FISA) - as a critical tool in identifying and disrupting terrorist plots in the US and abroad.

But some critics of NSA surveillance suggested that the collection of data under a programme called Prism was not essential to Zazi's capture because the British first obtained the critical e-mail address.

Still, the case study provides a rare glimpse of how the broad surveillance practices of the United States, often in concert with allies, are deployed.

"The 702 programme has been enormously useful in a large number of terrorist cases," said a US official who has access to classified records on NSA programmes. "It's beyond dispute that it is highly effective. It operates exactly as anyone paying attention would have expected it to operate based on floor debate and plain reading of law."

Passage of Section 702 as an amendment to FISA in 2008 gave the government the authority to request information from US telecommunications companies on foreign targets located overseas without a court order for each individual case.

The broad authority is reviewed and renewed annually by the FISA court, although the law does not preclude making a specific request for surveillance.

"It appears the NSA did not need any of the expanded authorities conferred by Section 702 to monitor the communications at issue," said Elizabeth Goitein, co-director of the Brennan Centre for Justice's Liberty and National Security Programme. "The government easily could have met this standard if it certified that the targets were al-Qaeda terrorists in Pakistan."

But US officials argue that, given the flood of leads in today's interconnected world, the system would get bogged down and they could miss plots if they had to go before the court every time they got information about potential foreign suspects.

The officials said they used material from multiple sources - allies, agents, informants and other investigations - to provide rolling targeting information for the Prism program.

They also said if the Yahoo address had not been included, Zazi might not have been identified just days before the attacks were set to occur. In testimony before Congress on Tuesday, senior intelligence and law enforcement officials said that recently revealed surveillance programmes have disrupted more than 50 "potential terrorist events", including at least 10 plots with a connection in the US.

The Zazi case was one of four that officials used in recent days to defend the effectiveness of the surveillance programmes. One of the others was a planned attack on a Danish newspaper that involved a Pakistani American, David Headley.

Sean Joyce, the deputy director of the FBI, described the other two potential attacks on Tuesday in testimony before the House Intelligence Committee. In one, Joyce said, the NSA was monitoring "a known extremist in Yemen" when it learned that the individual was in contact with a man in Kansas City, Missouri.

Joyce said Khalid Ouazzani and two co-conspirators were plotting to bomb the New York Stock Exchange.

Ouazzani pleaded guilty in 2010 to supporting a terrorist organisation, bank fraud and overseas money laundering. His co-conspirators also pleaded guilty to terrorism charges.

In the other incident, phone records helped identify a San Diego man who was financing a terrorist group overseas, apparently al-Shabab in Somalia.

"Investigating terrorism is not an exact science. It's like a mosaic," Joyce said. "We try to take these disparate pieces and bring them together to form a picture. There are many different pieces of intelligence.

"We have assets. We have physical surveillance. We have electronic surveillance through a legal process, phone records through additional legal process, financial records.

"Also, these programmes that we're talking about here today, they're all valuable pieces to bring that mosaic together."

General Keith Alexander, head of the National Security Agency, said details of the two programmes disclosed by Snowden were not closely held within the secretive agency.

Alexander said after the hearing that most of the documents accessed by Snowden, a former systems analyst on contract to the NSA, were on a web forum available to NSA employees.

Others were on a site that required a special credential to access. Alexander said investigators were studying how Snowden did that. He told lawmakers Snowden's leaks had caused "irreversible and significant damage to this nation".

He also said the internet programme had helped stop 90 per cent of the 50-plus plots he cited. He said more than 10 of the plots thwarted had a link inside the US.

Still, little was offered to substantiate claims that the programmes had been successful in stopping acts of terrorism that would not have been caught with narrower surveillance.

In the New York subway bombing case, Barack Obama conceded the would-be bomber might have been caught with less sweeping surveillance.

Committee chairman Congressman Mike Rogers said the programmes were vital to the intelligence community and blasted Snowden's actions as criminal.

"It is at times like these where our enemies within become almost as damaging as our enemies on the outside," Rogers said.

Officials acknowledged that intelligence collected from US phone records under a programme authorised by the USA Patriot Act is less compelling and the case for that extensive surveillance is harder to make.

The NSA's ability to intercept "the contents of e-mail communications of bad guys overseas provides a more lucrative set of information" about terrorist activity than its access to phone records of millions of Americans, one US official said.