Then the attacks, coming in waves, began to strike newspapers and television stations, then schools and finally banks, raising fears that what was initially a nuisance could have economic consequences.
The attacks have peaked and tapered off since then, but they have not ended, prompting officials there to declare Estonia the first.
"If you have a missile attack against, let's say, an airport, it is an act of war," a spokesman for the Estonian Defense Ministry, Madis Mikko, said Friday in a telephone interview. "If the same result is caused by computers, then how else do you describe that kind of attack?"
Officials in Estonia have accused Russia of orchestrating the attacks, officially or unofficially. They raised the issue at a meeting of NATO on Monday, with Defense Minister Jaak Aaviksoo saying that the alliance, which Estonia joined in 2004, needed to urgently debate the question--once seemingly a distant threat--of whether mass computer attacks posed a threat to national security.
"Events of this nature make a lot of people sit up," a NATO spokesman, Robert Pszczel, said in a telephone interview. "Today Estonia, tomorrow it could be somebody else."
The Kremlin has repeatedly denied government involvement in the attacks, dismissing Estonia's complaints as fabrications. Estonian officials, including Prime Minister Andrus Ansip, have said security officials traced the initial attacks to Russian computer servers, including domains registered to the government and the administration of President Vladimir V. Putin.
Since those computers could be used by another party, it is unlikely that involvement of the Russian authorities can be proved, though an Estonian government spokesman, Martin Jasko, said the timing and targets raised suspicion enough.
The attacks began on the day that the Estonian authorities removed a Soviet-era war monument that had been the source of protests and diplomatic tensions with Russia for months. Russia reacted vehemently, accusing Estonia, a former republic of the Soviet Union, of besmirching the memory of Soviet soldiers who fought against Nazi Germany.
In the days that followed, Russia suspended rail service, ostensibly for track repairs, while protesters in Moscow staged raucous demonstrations, harassing Estonia's ambassador in one instance. Senior officials have called for a boycott of Estonian goods, which at least one supermarket chain has observed.
The tensions with Estonia, along with Russian disputes with Poland and Lithuania, overshadowed a meeting in southern Russia near the city of Samara on Friday between Putin and the European Union's leaders, including Chancellor Angela Merkel of Germany, the Union's rotating president, and Jos? Manuel Barroso, the president of the European Commission, the Union's governing body.
Tensions between Russia and Estonia were discussed at the meeting, but the computer attacks were not, Taneli Lahti, a member of the European delegation, said in a telephone interview.
Estonian officials assert that the cyberattack is the largest ever against a country. NATO, which itself came under a similar though smaller attack during its war against Serbia in 1999, sent a computer expert to Estonia this week to observe the attacks firsthand.
The computer attacks have inundated Estonia's Web sites, overwhelming servers and forcing them to shut down, sometimes for a few hours, sometimes longer. Mikko, the Defense Ministry spokesman, said sites that typically received 1,000 visits a day had been buried under as many as 2,000 a second.
The attacks spiked on May 3, the day of the most boisterous protests in Moscow, and again on May 8 and 9, the days that Europe commemorates the victory over Nazi Germany in World War II. For many Estonians the end of the war ushered in nearly five decades of Soviet occupation, and the monument was a symbol of it. Another spike in attacks occurred on Tuesday, forcing one of Estonia's biggest banks to suspend access from abroad. Another bank faced an attack last week.
As the attacks have continued, they are now being traced to computers around the world, from Vietnam to the United States, according to Hillar Aarelaid, the head of the country's newly created Computer Emergency Response Team.
Aarelaid said attacks involved "botnets," networks of computers that have been compromised by an unauthorized user, who can then command and control them, surreptitiously and usually nefariously. Instructions in Russian on how to attack Estonian sites have circulated on the Internet, he added, suggesting that the world's first cyberwar would continue.
"We can't say we have seen the biggest attack yet," he said, "because each wave is bigger than the one before."