Ever since Google disclosed in January that Internet intruders had stolen information from its computers, the exact nature and extent of the theft has been a closely guarded company secret. But a person with direct knowledge of the investigation now says that the losses included one of Google’s crown jewels, a password system that controls access by millions of users worldwide to almost all of the company’s Web services, including e-mail and business applications.

The program, code named Gaia for the Greek goddess of the earth, was attacked in a lightning raid taking less than two days last December, the person said. Described publicly only once at a technical conference four years ago, the software is intended to enable users and employees to sign in with their password just once to operate a range of services.

The intruders do not appear to have stolen passwords of Gmail users, and the company quickly started making significant changes to the security of its networks after the intrusions. But the theft leaves open the possibility, however faint, that the intruders may find weaknesses that Google might not even be aware of, independent computer experts said.

The new details seem likely to increase the debate about the security and privacy of vast computing systems such as Google’s that now centralize the personal information of millions of individuals and businesses. Because vast amounts of digital information are stored in a cluster of computers, popularly referred to as “cloud” computing, a single breach can lead to disastrous losses.

The theft began with an instant message sent to a Google employee in China who was using Microsoft’s Messenger program, according to the person with knowledge of the internal inquiry, who spoke on the condition that he not be identified.

By clicking on a link and connecting to a “poisoned” Web site, the employee inadvertently permitted the intruders to gain access to his (or her) personal computer and then to the computers of a critical group of software developers at Google’s headquarters in Mountain View, Calif. Ultimately, the intruders were able to gain control of a software repository used by the development team.

The details surrounding the theft of the software have been a closely guarded secret by the company. Google first publicly disclosed the theft in a Jan. 12 posting on the company’s Web site, which stated that the company was changing its policy toward China in the wake of the theft of unidentified “intellectual property” and the apparent compromise of the e-mail accounts of two human rights advocates in China.

The accusations became a significant source of tension between the United States and China, leading Secretary of State Hillary Rodham Clinton to urge China to conduct a “transparent” inquiry into the attack. In March, after difficult discussions with the Chinese government, Google said it would move its mainland Chinese-language Web site and begin rerouting search queries to its Hong Kong-based site.

Company executives on Monday declined to comment about the new details of the case, saying they had dealt with the security issues raised by the theft of the company’s intellectual property in their initial statement in January.

Google executives have also said privately that the company had been far more transparent about the intrusions than any of the more than two dozen other companies that were compromised, the vast majority of which have not acknowledged the attacks.

Google continues to use the Gaia system, now known as Single Sign-On. Hours after announcing the intrusions, Google said it would activate a new layer of encryption for Gmail service. The company also tightened the security of its data centers and further secured the communications links between its services and the computers of its users.

Several technical experts said that because Google had quickly learned of the theft of the software, it was unclear what the consequences of the theft had been. One of the most alarming possibilities is that the attackers might have intended to insert a Trojan horse — a secret back door — into the Gaia program and install it in dozens of Google’s global data centers to establish clandestine entry points. But the independent security specialists emphasized that such an undertaking would have been remarkably difficult, particularly because Google’s security specialists had been alerted to the theft of the program.

However, having access to the original programmer’s instructions, or source code, could also provide technically skilled hackers with knowledge about subtle security vulnerabilities in the Gaia code that may have eluded Google’s engineers.

“If you can get to the software repository where the bugs are housed before they are patched, that’s the pot of gold at the end of the rainbow,” said George Kurtz, chief technology officer for McAfee Inc., a software security company that was one of the companies that analyzed the illicit software used in the intrusions at Google and at other companies last year.

Rodney Joffe, a vice president at Neustar, a developer of Internet infrastructure services, said, “It’s obviously a real issue if you can understand how the system works.” Understanding the algorithms on which the software is based might be of great value to an attacker looking for weak points in the system, he said.

When Google first announced the thefts, the company said it had evidence that the intrusions had come from China. The attacks have been traced to computers at two campuses in China, but investigators acknowledge that the true origin may have been concealed, a quintessential problem of cyberattacks.

Several people involved in the investigation of break-ins at more than two dozen other technology firms said that while there were similarities between the attacks on the companies, there were also significant differences, like the use of different types of software in intrusions. At one high-profile Silicon Valley company, investigators found evidence of intrusions going back more than two years, according to the person involved in Google’s inquiry.

In Google’s case, the intruders seemed to have precise intelligence about the names of the Gaia software developers, and they first tried to access their work computers and then used a set of sophisticated techniques to gain access to the repositories where the source code for the program was stored.

They then transferred the stolen software to computers owned by Rackspace, a Texas company that offers Web-hosting services, which had no knowledge of the transaction. It is not known where the software was sent from there. The intruders had access to an internal Google corporate directory known as Moma, which holds information about the work activities of each Google employee, and they may have used it to find specific employees.

Rage and love, those consuming emotions felt with a particularly acute pang in youth, all but burn up the stage in “American Idiot,” the thrillingly raucous and gorgeously wrought Broadway musical adapted from the blockbuster pop-punk album by Green Day.

Pop on Broadway, sure. But punk? Yes, indeed, and served straight up, with each sneering lyric and snarling riff in place. A stately old pile steps from the tourist-clogged Times Square might seem a strange place for the music of Green Day, and for theater this blunt, bold and aggressive in its attitude. Not to mention loud. But from the moment the curtain rises on a panorama of baleful youngsters at the venerable St. James Theater, where the show opened on Tuesday night, it’s clear that these kids are going to make themselves at home, even if it means tearing up the place in the process.

Which they do, figuratively speaking. “American Idiot,” directed by Michael Mayer and performed with galvanizing intensity by a terrific cast, detonates a fierce aesthetic charge in this ho-hum Broadway season. A pulsating portrait of wasted youth that invokes all the standard genre conventions — bring on the sex, drugs and rock ’n’ roll, please! — only to transcend them through the power of its music and the artistry of its execution, the show is as invigorating and ultimately as moving as anything I’ve seen on Broadway this season. Or maybe for a few seasons past.

Burning with rage and love, and knowing how and when to express them, are two different things, of course. The young men we meet in the first minutes of “American Idiot” are too callow and sullen and restless — too young, basically — to channel their emotions constructively. The show opens with a glorious 20-minute temper tantrum kicked off by the title song.

“Don’t want to be an American idiot!” shouts one of the gang. The song’s signature electric guitar riff slashes through the air, echoing the testy challenge of the cry. A sharp eight-piece band, led by the conductor Carmel Dean, is arrayed around the stage, providing a sonic frame for the action. The simple but spectacular set, designed by Christine Jones, suggests an epically scaled dive club, its looming walls papered in punk posters and pimpled by television screens, on which frenzied video collages flicker throughout the show. (They’re the witty work of Darrel Maloney.)

Who’s the American idiot being referred to? Well, as that curtain slowly rose, we heard the familiar voice of George W. Bush break through a haze of television chatter: “Either you are with us, or with the terrorists.” That kind of talk could bring out the heedless rebel in any kid, particularly one who is already feeling itchy at the lack of prospects in his dreary suburban burg.

But while “American Idiot” is nominally a portrait of youthful malaise of a particular era — the album dates from 2004, the midpoint of the Bush years, and the show is set in “the recent past” — its depiction of the crisis of post-adolescence is essentially timeless. Teenagers eager for their lives to begin, desperate to slough off their old selves and escape boredom through pure sensation, will probably always be making the same kinds of mistakes, taking the same wrong turns on the road to self-discovery.

“American Idiot” is a true rock opera, almost exclusively using the music of Green Day and the lyrics of its kohl-eyed frontman, Billie Joe Armstrong, to tell its story. (The score comprises the whole of the title album as well as several songs from the band’s most recent release, “21st Century Breakdown.”) The book, by Mr. Armstrong and Mr. Mayer, consists only of a series of brief, snarky dispatches sent home by the central character, Johnny, played with squirmy intensity by the immensely gifted John Gallagher Jr. (“Spring Awakening,” “Rabbit Hole”).

“I held up my local convenience store to get a bus ticket,” Johnny says with a smirk as he and a pal head out of town.

“Actually I stole the money from my mom’s dresser.”

Beat.

“Actually she lent me the cash.”

Such is the sheepish fate of a would-be rebel today. But at least Johnny and his buddy Tunny (Stark Sands) do manage to escape deadly suburbia for the lively city, bringing along just their guitars and the anomie and apathy that are the bread and butter of teenage attitudinizing the world over. (“I don’t care if you don’t care,” a telling lyric, could be their motto.)

The friend they meant to bring along, Will (Michael Esper), was forced to stay home when he discovered that his girlfriend (Mary Faber) was pregnant. Lost and lonely, and far from ready for the responsibilities of fatherhood, he sinks into the couch, beer in one hand and bong in the other, as his friends set off for adventure.

Beneath the swagger of indifference, of course, are anxiety, fear and insecurity, which Mr. Gallagher, Mr. Esper and Mr. Sands transmit with aching clarity in the show’s more reflective songs, like the hit “Boulevard of Broken Dreams” or the lilting anthem “Are We the Waiting.” The city turns out to be just a bigger version of the place Johnny and Tunny left behind, a “land of make believe that don’t believe in me.” The boys discover that while a fractious 21st-century America may not offer any easy paths to fulfillment, the deeper problem is that they don’t know how to believe in themselves.

Johnny strolls the lonely streets with his guitar, vaguely yearning for love and achievement. He eventually hooks up with a girl (a vivid Rebecca Naomi Jones) but falls more powerfully under the spell of an androgynous goth drug pusher, St. Jimmy, played with mesmerizing vitality and piercing vocalism by Tony Vincent. Tunny mostly stays in bed, clicker affixed to his right hand, dangerously susceptible to a pageant of propaganda about military heroism on the tube, set to the song “Favorite Son.” By the time the song’s over, he’s enlisted and off to Iraq.

In both plotting and its emotional palette, “American Idiot” is drawn in brash, primary-colored strokes, maybe too crudely for those looking for specifics of character rather than cultural archetypes. But operas — rock or classical — often trade in archetypes, and the actors flesh out their characters’ journeys through their heartfelt interpretations of the songs, with the help of Mr. Mayer’s poetic direction and the restless, convulsive choreography of Steven Hoggett (“Black Watch”), which exults in both the grace and the awkwardness of energy-generating young metabolisms.

Line by line, a skeptic could fault Mr. Armstrong’s lyrics for their occasional glibness or grandiosity. That’s to be expected, too: rock music exploits heightened emotion and truisms that can fit neatly into a memorable chorus. The songs are precisely as articulate — and inarticulate — as the characters are, reflecting the moment in youth when many of us feel that pop music has more to say about us than we have to say for ourselves. (And, really, have you ever worked your way through a canonical Italian opera libretto, line by line?)

In any case the music is thrilling: charged with urgency, rich in memorable melody and propulsive rhythms that sometimes evolve midsong. The orchestrations by Tom Kitt (the composer of “Next to Normal”) move from lean and mean to lush, befitting the tone of each number. Even if you are unfamiliar with Green Day’s music, you are more likely to emerge from this show humming one of the guitar riffs than you are to find a tune from “The Addams Family” tickling your memory.

But the emotion charge that the show generates is as memorable as the music. “American Idiot” jolts you right back to the dizzying roller coaster of young adulthood, that turbulent time when ecstasy and misery almost seem interchangeable states, flip sides of the coin of exaltation. It captures with a piercing intensity that moment in life when everything seems possible, and nothing seems worth doing, or maybe it’s the other way around.

AMERICAN IDIOT

Music by Green Day; lyrics by Billie Joe Armstrong; book by Mr. Armstrong and Michael Mayer; directed by Mr. Mayer; choreography by Steven Hoggett; musical supervision, arrangements and orchestrations by Tom Kitt; sets by Christine Jones; costumes by Andrea Lauer; lighting by Kevin Adams; sound by Brian Ronan; video and projections by Darrel Maloney; technical supervision by Hudson Theatrical Associates; music coordinator, Michael Keller; music director, Carmel Dean; associate choreographer, Lorin Latarro; associate director, Johanna McKean. Presented by Tom Hulce and Ira Pittelman, Ruth and Stephen Hendel, Vivek J. Tiwary and Gary Kaplan, Aged in Wood and Burnt Umber, Scott M. Delman, Latitude Link, HOP Theatricals and Jeffrey Finn, Larry Welk, Bensinger Filerman and Maellenberg Taylor, Allan S. Gordon and Élan V. McAllister and Berkeley Repertory Theater, in association with Awaken Entertainment and John Pinckard and John Domo. At the St. James Theater, 246 West 44th Street, Manhattan; (212) 239-6200. Running time: 1 hour 30 minutes.

WITH: John Gallagher Jr. (Johnny), Stark Sands (Tunny), Michael Esper (Will), Rebecca Naomi Jones (Whatshername), Christina Sajous (the Extraordinary Girl), Mary Faber (Heather) and Tony Vincent (St. Jimmy).

Britain’s sudden lifting of a six-day ban on air traffic allowed stranded travelers to begin making their way home on Wednesday, as airlines around the world began to confront a huge backlog of passengers seeking to travel, a situation that could take weeks to resolve.

With nearly 100,000 flights having been canceled due to the vast cloud of volcanic ash over Europe, carriers in Asia began resuming some European service on Wednesday. But things were far from back to normal even though most European airports were operating.

Qantas, the Australian carrier, said it would take “approximately two to three weeks to clear the current backlog.” Cathay Pacific, based in Hong Kong, said it was not taking any new bookings on flights to Europe until after May 10.

“We have to be realistic,” Cathay said in a statement. “When services resume, all airlines around the world will be competing for landing slots at airports, and airspace and airports are going to be horribly congested.”

The chaos has already lasted twice as long as the three-day closing of American airspace after the attacks of Sept. 11, 2001, which devastated many airlines financially. The opening of British airports late Tuesday night — including Heathrow near London, Europe’s busiest — came as the broader European crisis had begun to unwind with the gradual resumption of flights in some of Europe’s busiest flight paths above France and Germany.

British Airways, Air France and Lufthansa said they would run a full schedule of long-haul flights on Wednesday, while most Asian and American carriers planned limited service to and from Europe.

Eurocontrol, the agency that coordinates regional air-traffic management, said that roughly 75 percent of the airspace over Europe was open — northern Europe was still being affected — and that all airspace was open above 20,000 feet, enabling intercontinental overflight traffic to resume.

Britain had seemed the last major holdout, with aviation authorities citing an unexpected new cloud of ash closing on its airspace.

Then, in late evening, the whine of jet engines was heard for the first time in almost a week as trans-Atlantic flights began to land at Heathrow. The first one to land was a British Airways flight from Vancouver.

The British air traffic control agency, or NATS, said late Tuesday only a small area over northwest Scotland remained closed and “continues to be affected by a dense concentration of volcanic ash.”

The reopening of European airspace was certain to be accompanied by a potentially acrimonious debate about the indecision of governments in handling the crisis.

The Civil Aviation Authority said the “major barrier to resuming flights had been understanding tolerance levels of aircraft to ash,” suggesting that the authorities had been especially cautious in assessing the danger from the volcanic ash’s ability to clog jet engines, forcing planes to stall in midflight with potentially catastrophic consequences.

But, with every day of closings, the cost and disruption mounted and airlines pressured governments to reconsider their assessment of the risks. Eurocontrol said that by the end of the day Tuesday, 95,000 flights would have been canceled as a result of the ash cloud.

The British transport secretary, Andrew Adonis, said safety had been the “paramount concern” but after talking to airplane manufacturers, airlines and scientific specialists, the authorities had reached a better understanding of “how different concentrations of ash affect aircraft engines.”

Willie Walsh, the chief executive of British Airways, said the airline was “delighted the authorities have paid heed to the arguments we and the industry have put forward.” But, like others, he said he did not believe it had been necessary to impose a “blanket ban.”

With hundreds of airplanes idle on airport aprons, aviation authorities and airlines warned that the restoration of full schedules would take time. The Civil Aviation Authority said there would be a “phased reintroduction” of access to airspace, but restricted areas would be “very much smaller” than in recent days.

The cancellations have cost the United States economy some $650 million, the United States Travel Association reported Tuesday, as businesses lost out on an estimated $450,000 spent by every flight of international travelers arriving in the United States.

The closings also caused major financial strains for Asian airlines. Andrew Herdman, director general of the Association of Asia Pacific Airlines in Kuala Lumpur, Malaysia, said Tuesday in a statement that flights to and from Europe accounted for about 15 percent of total passenger revenues for the region’s main carriers, worth some $40 million a day.