Islamic State extremists are urging militants in Egypt’s Sinai Peninsula to continue with beheadings and attacks against security forces.
There are concerns about links between the terror group and insurgents in a region popular with tourists, including the beach resort of Sharm el Sheikh.
Egyptian officials, including the foreign minister, have acknowledged co-ordination between the two groups but insist there are no IS fighters in the country.
The terror group controls swathes of territory in Iraq and Syria, where it has declared an Islamic state, or caliphate.
In the latest violence, its fighters have been locked in fierce clashes with Kurdish forces to the east of a Syrian city near the border with Turkey.
The IS offensive on Kobani, also known as Ayn al Arab, has led tens of thousands of Syrian Kurds to flee, many across the Turkish frontier.
Egypt has faced an Islamist insurgency since the army ousted president Mohamed Morsi, of the Muslim Brotherhood, last year.
IS spokesman Abu Muhammad al Adnani said in a statement released online: “Rig the roads with explosives for them. Attack their bases.
“Raid their homes. Cut off their heads. Do not let them feel secure.”
He praised Egyptian militants for carrying out “blessed operations against the guards of the Jews, the soldiers of Sisi, the new Pharaoh of Egypt”.
Current president Abdel Fattah al Sisi, who as the then head of the army led the overthrow of Mr Morsi in 2013 following mass protests, has raised concerns about the threat posed by Islamist militants in Egypt and other parts of the Middle East.
A militant from the Sinai-based Ansar Bayt al Maqdis, which has killed hundreds of Egyptian security forces in the past year, has said IS has provided instructions on how to operate more effectively.
A bomb attack outside the Egyptian foreign ministry in Cairo on Sunday, claimed by a militant group, killed three policemen, including a key witness in a trial of Mr Morsi.
Meanwhile, authorities in Belgium said several arrests had been made to prevent attacks by jihadist extremists, although the European Commission said it had not been told of any specific threat against it after reports suggested its Brussels headquarters was a possible target.
And in Australia, sweeping new security powers are being sought by the government to tackle what it says is the growing threat from militant extremists.
Measures include making it a crime for Australian citizens to travel to overseas areas declared off limits.
The move follows the largest anti-terror operation the country has ever seen to thwart what the government said was a plot by IS supporters to behead a random member of the public.