"It is a war against terrorism, against jihadism, against radical Islam, against everything that is aimed at breaking fraternity, freedom, solidarity," Valls said during a speech in Evry, south of Paris.
The authorities started the day hunting for the companion of one of the killers, only to learn later that she appeared to have fled to Turkey and then probably to Syria days before the first assault in Paris on Wednesday. The police had suspected that the woman - Hayat Boumeddiene, 26, the girlfriend of Amedy Coulibaly, believed to be one of the gunmen - might have played a role in one or more of the attacks.
"We are 99 percent sure that she traveled to Syria from Urfa," said a Turkish intelligence official, referring to a city in southern Turkey.
"There is no evidence that suggests she was involved in the terrorist attacks in France this week."
France remained on edge a day after security forces killed Coulibaly, who the police said was responsible for the deaths of four hostages at a kosher supermarket near the Porte de Vincennes in eastern Paris on Friday, and Said and Cherif Kouachi, the brothers who fatally shot 12 people on Wednesday in and around the offices of Charlie Hebdo, a satirical newspaper.
The French government said it would put 500 additional troops on the streets over the weekend amid preparations for a giant unity rally in Paris on Sunday. A number of European officials said they would attend, including Prime Minister Ahmet Davutoglu of Turkey, the most prominent Muslim leader scheduled to be there, and Prime Minister Benjamin Netanyahu of Israel.
On Saturday, hundreds of thousands of people marched in Paris, Toulouse, Nice and other cities in a show of solidarity, and rallies were held in places as far away as Madagascar and Bangui, Central African Republic.
Top ministers in the French government held an emergency session to discuss measures to prevent a repeat of the attacks, which shocked the country and raised questions about why law enforcement agencies had failed to thwart terrorism suspects well known to the police and intelligence services.
Some of the surviving hostages shared chilling accounts of their ordeals at the hands of heavily armed captors, who they said had seemed prepared to die as police forces amassed outside the kosher supermarket and a printing plant northeast of Paris that the Kouachi brothers had seized early on Friday.
Coulibaly, in an interview with a French television outlet not long before he was killed, claimed to be affiliated with the Islamic State, which has its headquarters in northern Syria. Officials identified him as the gunman in the fatal shooting of a female police officer in a Paris suburb on Thursday.
The crisis and its aftermath presented a major challenge to President Francois Hollande and his government, which are facing deep religious and cultural rifts in a nation with a rapidly growing Muslim population while simultaneously coping with the security threats stemming from Islamic extremists. Large numbers of French citizens have been traveling to Syria and Iraq to fight with the Islamic State.
Hollande, appealing for unity, has warned against seeing Muslims as the enemy, and Valls called again on Saturday for citizens to join the rally planned for Sunday.
"There needs to be a firm message about the values of the republic and of secularism," Valls said in Evry. "Tomorrow, France and the French can be proud. Everyone must come tomorrow."
European leaders, including Chancellor Angela Merkel of Germany and Prime Minister David Cameron of Britain, were expected to come to Paris for the rally.
But the event was already proving polarizing in some quarters.
Marine Le Pen, the leader of the far-right National Front, which was not invited, urged her followers to stay away from the rally, saying it had been "taken over by parties which represent what the French hate: partisan spirit, electioneering and indecent polemic."
Valls' statement that France was at war with jihadis and radicals also reflected the depth of the concerns over security in France and across much of Western Europe.
U.S. officials said Attorney General Eric H. Holder Jr. would attend a meeting in Paris on Sunday convened by the French interior minister.
Terrorist threats, foreign fighters and violent extremism were expected to be at the top of the agenda.
Simone Rodan Benzaquen, the director of the American Jewish Committee in Paris, said there was deep anxiety among French Jews, who had already been grappling with a spate of anti-Semitism in recent months, prompting some to declare their intention to leave France.
"We wonder whether it is the beginning of a series of further attacks or if it is the end, or a wake-up call that will help the country understand what Islamic radicalism is," Rodan Benzaquen said.
The Defense Ministry said it had added 250 soldiers to the 850-person force already deployed in the greater Paris area, and would add 250 more on Sunday. The police said security forces would reinforce vulnerable targets.
Much remained unknown or unclear about the attacks, the suspects and the efforts to hunt them down. After initially suggesting that the Kouachi brothers had another accomplice, the police have said little since an 18-year-old with the name of the person they were seeking turned himself in late on Wednesday.
The precise nature of the relationships among the Kouachi brothers, Coulibaly, the Islamic State and another terrorist group, al-Qaida in the Arabian Peninsula, remained murky. With security services still on high alert, there was little public indication of whether the authorities had specific information about possible further attacks or simply remained worried about the potential for other terrorists to take cues from the French gunmen and from the jihadis and militant groups that had quickly applauded them as heroes.
Le Monde, citing counter-terrorism officials, said that the links between Coulibaly and Cherif Kouachi appeared to stretch back to 2010, and that telephone wiretaps showed that they had frequently visited Djamel Beghal, a French-Algerian champion of jihad who had once been imprisoned for planning an attack on the U.S. Embassy in Paris in 2001. The police said Coulibaly and Kouachi had been followers of Beghal.
Coulibaly was an associate of Cherif, the younger Kouachi brother, a sometime pizza delivery man and fishmonger.
The brothers were being tracked by intelligence officials in the United States, France and Yemen, but large gaps remain in the chronology of their actions and in accounts of their affiliations. A senior U.S. official said on Saturday that Said Kouachi had traveled to Yemen with another person, probably an accomplice, in the summer of 2011. The authorities are trying to determine whether that person was Cherif, who claimed to have gone to Yemen in 2011.
The official said that the Kouachi brothers had known Coulibaly, and possibly Boumeddiene, also a French citizen, since childhood.
Reports said that Coulibaly and Boumeddiene had visited Beghal in 2010 in Murat, a village in France's Cantal department, after Beghal was released from prison and placed under house arrest. Le Monde showed photographs of Coulibaly with Beghal.
Photographs of Boumeddiene published on Saturday showed her transition from a bikini-wearing young woman, posing on a beach with Coulibaly on a vacation, to a Muslim woman covered head to toe with only her eyes showing.
According to the Paris prosecutor, Francois Molins, phone records showed that Boumeddiene and Cherif Kouachi's wife had exchanged more than 500 phone calls in 2014.
The police said Boumeddiene had married Coulibaly in a Muslim ceremony. There were reports that she did not attend her own wedding, where she was represented by her father.
In 2010, Coulibaly was arrested and detained in connection with an attempt to free Smain Ait Ali Belkacem, who was in prison while facing charges of helping engineer attacks in Paris in 1995 in which eight people were killed.
As the investigation unfolded, family members and witnesses began to speak out about what they had endured.
The brother of Ahmed Merabet, a police officer killed during the Charlie Hebdo attack, who was Muslim, called for tolerance.
"We must not confuse extremists with Muslims," said the brother, Malek Merabet.
New details began to emergeon Saturday about the standoffs, including the role played by a hostage at the printing plant in Dammartin-en-Goele, northeast of Paris.
Prosecutors said that an employee at the plant, seized by the Kouachi brothers earlyon Friday, had hidden under a sink in the cafeteria on the second floor, undetected and terrified. The chief executive of the company, Michel Catalano, told Sky News that he had told the employee to hide in the back of the building when he saw the gunmen invade the premises. He said the man, Lilian Lepere, had secretly texted the police before the Kouachi brothers clashed with counter-terrorism forces.
Catalano told Sky News he had been petrified that the brothers would discover Lepere.
"I could see from a window that there was a man with a rocket launcher and a Kalashnikov," the broadcaster quoted him as saying.
"I could immediately see there was a situation of danger. I told my employee to hide. I knew two of us couldn't hide."
Catalano added that he had helped bandage the wound of one of the Kouachi brothers.
"When I thought one of them was tense, I said, 'I can look after you,'" he recalled. Eventually, they let him leave, he told Sky News.
At the same time, Coulibaly, armed with AK-47s and explosives, took hostages, including women and children, at the kosher supermarket in eastern Paris.
The police said Coulibaly had killed four people after he forced his way into the supermarket.
As negotiations began, said Mohamed Douhane, a senior police official, the police were aware of Coulibaly's identity and his links to the Kouachi brothers. He said the raids at the printing factory and the supermarket had been simultaneous because of concerns that Coulibaly would kill his hostages if he learned of an attack against the Kouachi brothers.
Negotiations with Coulibaly proved fruitless, and he declared that he wanted to die as a martyr, Douhane said.
When the police lifted the shutters of the supermarket, Coulibaly emerged from the back of the store carrying his weapons, Douhane said, adding, "He understood at a given moment that he was not going to get out of this alive, that he would have a life in jail."
The French radio network RTL reported that it had recorded, via an open phone line in the store, fragments of conversations between Coulibaly and some of his hostages, including an argument over whether, by paying their taxes, French citizens were guilty of supporting attacks by France and its allies on Islamic fighters in Syria and other countries.
"You pay taxes and stuff, and so you agree with that," Coulibaly said.
One of the hostages replied, "Well, we are required to."
Coulibaly shot back: "Huh? You don't have to. I also have to pay taxes, but I don't."
A young employee at the supermarket, Lassana Bathily, of Mali, helped hide people in a cold storage room in the basement.
One witness, Mickael, who declined to give his last name, told Le Point magazine that he had gone to the market with his 3-year-old son to buy bread and chicken for the Sabbath, and heard a loud blast.
"I thought it was a firecracker," he said. "But when I turned my head, I saw a black man armed with two Kalashnikovs. I understood what was happening. I ran with my son by his collar to the back of the store." They then hid in the basement room.
"We were terrorized," Mickael said.
"Five minutes later, a grocery employee sent by the attacker came down and told us to come upstairs, or the attacker said there would be a bloodbath," he continued. Upstairs, they found a dead man in a pool of blood and a calm Coulibaly, who introduced himself as a Muslim from Mali who belonged to the Islamic State. He said Coulibaly had defended the attack by invoking Palestine and his "brothers in Syria."
Coulibaly ordered the hostages to put their phones down, but Mickael said he had discreetly recovered his and contacted the police. He said Coulibaly had appeared ready to die, with guns in both hands and a box of bullets nearby.
When the shutters were raised, the hostages realized that a police assault was beginning. "The noise was deafening," Mickael said. "He was dead. It was over."
The Representative Council of Jewish Institutions, France's most prominent Jewish association, identified the four hostages killed in the attack on the supermarket as Yoav Hattab, Philippe Braham, Yohan Cohen and Francois-Michel Saada.
On Friday night, in his address to the nation, Hollande called the attack a horrific anti-Semitic crime. And the council called on the Jewish community to participate in the rally on Sunday "against the barbarism that had been committed against Charlie Hebdo's journalists, against policemen and against democracy, and freedom of speech."