In what France's cyber defense chief calls an unprecedented surge, about 19,000 French websites have suffered cyber attacks since a rampage by Islamic extremists left 20 dead in Paris last week.
Admiral Arnaud Coustilliere told reporters on Thursday that many of the cyber attacks were carried out by "more or less structured" groups, including some well-known Islamic hacker groups.
"That's never been seen before. It's the first time that a country has been faced with such a large wave," he said.
The attacks, which appear to have involved mostly relatively minor denial-of-service attacks, have hit sites as varied as military regiments to pizza shops, he said.
French President Francois Hollande, meanwhile, insisted that any anti-Muslim or anti-Semitic acts must be "severely punished," as he sought to calm rising religious tensions after his country's bloodiest terrorist attacks in decades.
The country is tense since 20 people, including three gunmen, were killed in last week's rampage that began at the offices of the satirical newspaper Charlie Hebdo.
Two of the Paris terror attackers claimed allegiances to Al Qaeda in Yemen, and another - who targeted a kosher supermarket - to the Islamic State group.
The attacks occurred in an atmosphere of rising anti-Semitism in France, and have prompted scattered attacks on Muslim sites around France in an apparent backlash.
They have also put many French Muslims on the defensive.
Hollande said France's millions of Muslims should be protected and respected, "just as they themselves should respect the nation" and its strictly secular values.
"Anti-Muslim acts, like anti-Semitism, should not just be denounced but severely punished," Hollande said on Thursday at the Institute of the Arab World in Paris.
Noting that Muslims are the main victims of Islamic extremist violence, he said, "In the face of terrorism, we are all united."
With 120,000 security forces deployed to prevent future attacks, nerves jumped overnight when a car rammed into a policewoman guarding the president's palace.
The incident at the Elysee Palace had no apparent links to last week's shootings and might have been an accident, prosecutors and police said.
The car carrying four people took a one-way street in the wrong direction then drove off when the police officer tried to stop them. The officer sustained slight leg injuries, police said. Two people were later arrested, and two others in the car fled.
US and French intelligence officials are leaning toward an assessment that the Paris terror attacks were inspired by Al Qaeda but not directly supervised by the group, a view that would put the violence in a category of homegrown incidents that are extremely difficult to detect and thwart.
French justice officials have been cracking down by arresting dozens of people who glorified terrorism or made racist or anti-Semitic remarks.
Customers lined up again on Thursday to try to get copies of Charlie Hebdo's first edition since the attacks, which again had Prophet Muhammad on the cover.
Even though it has a special increased print run of 5 million copies, it sold out before dawn on Thursday in Paris kiosks for a second day straight.
Muslims believe their faith forbids depictions of the prophet, and some reacted with dismay - and occasional anger - to the new cover.
Some who had supported Charlie Hebdo after the attacks felt betrayed and others feared the cartoon would trigger yet more violence.
A leader of Yemen's Al-Qaida branch officially claimed responsibility for the attacks by the two gunmen at Charlie Hebdo, saying in a video posted online that the slayings were in "vengeance for the prophet."