The Algerian government on Wednesday condemned the "hateful" murder of a French hostage by jihadists who abducted him while he was hiking in the mountains east of Algiers.

The beheading of Herve Gourdel in a video posted online Wednesday followed France's rejection of his kidnappers' 24-hour ultimatum to halt its air raids on the Islamic State group in Iraq.

The 55-year-old was kidnapped on Sunday by Jund al-Khilifa, or "Soldiers of the Caliphate," in a national park that was once a magnet for tourists but later became a sanctuary for Islamists.

"From the moment of the abduction of the French national, the Algerian authorities mobilised all of their energies and all of their human and material resources to free the hostage," a government statement said.

"The Algerian government presents its sincere condolences to the family of the deceased and those close to him as well as to the French government."

Algiers expressed its "determination to continue the struggle against terrorism in all its forms and guarantee the protection and security of foreign nationals present on its territory."