The three adults and four children were kidnapped in north Cameroon near the Nigerian border last week. In a video posted online, the gunmen said France had declared war on Islam with its campaign in Mali and threatened to kill the hostages unless authorities in Nigeria and Cameroon freed militants there.
"We do not negotiate on that kind of basis, with these kind of groups," French Defense Minister Jean-Yves Le Drian told RTL radio. "We will use all (other) possible means to ensure these and other hostages are freed."
Le Drian said the fighting was not close to an end and troops in Mali's remote mountain and desert north were facing stiff resistance from the "strongest and most organized" rebels, underscoring the risk of French and African forces becoming entangled in a messy guerrilla war.
The kidnapping brought to 15 the number of French citizens held in West Africa and highlighted the danger to French nationals and interests in the region since Paris sent troops to Mali last month to help oust Islamist rebels.
It was the first abduction of foreigners in the mostly Muslim north of Cameroon, a former French colony. But the region - with its porous borders - is within the operational sphere of Boko Haram and fellow Nigerian Islamist militants Ansaru.
Boko Haram, one of a number of al-Qaida linked groups in the region, has killed hundreds of people in recent years in an attempt to establish an Islamist state in Nigeria.
"The principle of terrorism is the same whether you are in Somalia with the Al Shabaab, in Mali with Ansar Dine or in Nigeria with Boko Haram or Ansaru," Le Drian said. "It's the same system, the same methods, which threaten us."
The video posted online on Monday showed the hostages, including the four boys, surrounded by three gunmen wearing turbans and camouflage gear.
"The president of France has launched a war on Islam and we are fighting it everywhere," said one of the apparent kidnappers, identifying himself as a member of Boko Haram.
Mali rebel resistance
In Mali, French and Chadian troops are encountering strong resistance from die-hard al-Qaida-linked Islamists in the mountainous north, Le Drian said.
Chadian troops launched an offensive at the weekend against fighters holed up in the Adrar des Ifoghas mountains near the Algerian border but suffered the heaviest losses since the international offensive began last month.
Chad's armed forces said some 23 of its soldiers and about 90 rebels were killed in the fighting. French fighter jets and helicopters were forced to support the Chadian offensive.
"The most fundamentalist elements are there," Le Drian said. "The strongest and most organized forces. We expected resistance and we've had some extremely violent battles."
Paris intervened in its former West African colony six weeks ago to stop a southward offensive by Islamist fighters who seized control of the north last April.
After quickly driving the rebels out of major urban areas, France and its African allies have focused on the remote northeast - an area the size of France that includes networks of caves, passes and porous borders.
Asked about the timing for pulling out the 4,000 French troops, Le Drian said it was hard to give a precise timetable.
"If things evolve normally, we could begin leaving before the end of March," Le Drian said, adding that the operation had cost about $130.73 million so far.
Rebels have staged bombings and raids mainly targeting Mali's poorly trained and equipped army in northern cities.
A spokesman for Mali's military said on Tuesday a total of 37 Malian soldiers had been killed and 138 injured since the start of the offensive. He said five Malian soldiers suspected of ethnic reprisals after the recapture of Timbuktu had been called back to Bamako by military authorities.