Jihadists linked to the Islamic State group have beheaded a Frenchman abducted in Algeria, in a video posted online, but President Francois Hollande has vowed France won't give in to blackmail.
Hollande said the "cruel and cowardly" murder of Herve Gourdel would only strengthen France's resolve to pursue its air war against the Islamic State group in Iraq.
The 55-year-old was kidnapped on Sunday by Jund al-Khilifa, or "Soldiers of the Caliphate", while hiking in a national park that was once a magnet for tourists but later became a sanctuary for Islamists.
His beheading followed France's rejection of the group's 24-hour ultimatum to halt its air raids in Iraq - part of a US-led campaign against IS.
Hollande, whose government was the first to join Washington in carrying out air strikes against the jihadists, pledged that France would stay the course.
"France is going through an ordeal through the murder of one of its citizens, but France will never give in to blackmail," he told the UN General Assembly.
"The fight against terrorism must continue and be stepped up."
US President Barack Obama voiced solidarity with France over the killing, which came after two American journalists and a British air worker were beheaded by IS in similar videotaped executions.
Chairing a special UN Security Council meeting, Obama looked at his French counterpart and said: "We stand with you and the French people as you grieve this terrible loss and as you stand up against terror in defence of liberty."
The video, entitled "A Message with Blood to the French Government", employs a similar style to those used in previous videos by IS showing the beheadings of three Western hostages.
It begins with a clip of Hollande at a press conference where he announced strikes against IS in Iraq.
The video then shows Gourdel on his knees with his hands behind his back in front of four armed militants, their faces covered.
The Frenchman briefly expresses his love for his family before one of the jihadists reads a speech in which he denounces the actions of the "French criminal crusaders" against Muslims in Algeria, Mali and Iraq.
The jihadist says the beheading is to "avenge the victims in Algeria... and support the caliphate" proclaimed by IS in Iraq and Syria.
The United States has built a coalition of more than 50 nations to fight IS, after the jihadists seized large parts of Syria and Iraq and committed widespread atrocities, including beheadings and crucifixions.
France has ruled out joining military operations in Syria, where a US-led coalition began strikes against IS on Tuesday.
Prime Minister Manuel Valls, in a speech on Wednesday that kicked off a parliamentary session about France's engagement in Iraq, said it was not the cause of Gourdel's abduction.
"It's not our intervention that is exposing us to terrorism. The threat has been there for a long time. And that's why we are acting," Valls said.
The Algerian government called it "hateful" and expressed its "determination to continue the struggle against terrorism in all its forms and guarantee the protection and security of foreign nationals".
Algiers also pledged to "continue to mobilise all forces possible to find the killers so that they can be punished, and to find Herve Gourdel's body", Hollande told journalists after speaking to Algerian Prime Minister Abdelmalek Sellal.
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