By Maggie Fick and Yusri Mohamed

CAIRO/ISMAILIA, Egypt, (Reuters) - Within hours of Egypt's elected president being overthrown this month, militant fellow Islamists in the Sinai peninsula were talking of making war on Cairo's security forces.

Scarcely had a video surfaced on YouTube of hundreds of men chanting "No to peace!", than police and troops were attacked in El Arish and other North Sinai towns. Ten have now been killed across the province since Mohamed Mursi was toppled on July 3.

The desert peninsula has long been a security headache for Egypt and its neighbours. Large and empty, it borders Israel and the Gaza Strip and flanks the Suez Canal linking Asia to Europe. It is also home to nomad clans disaffected with rule from Cairo.

By adding to anger and seeming to confirm low expectations of democracy among Islamist militants who viewed Mursi and his Muslim Brotherhood as too moderate, the president's removal by the army has brought new violence to Sinai. It may presage more, if the fiery rhetoric of various hardline groups is any guide.

Targets this month, in addition to Egyptian security posts near the Suez Canal and the Gaza frontier, have included: a Christian priest, shot dead in the Mediterranean port of El Arish; a gas pipeline to Jordan; and the Israeli Red Sea resort of Eilat, where remains of a rocket were found.

Egypt's armed forces are on high alert, though military sources play down talk of a major offensive. Such an operation might require Israeli approval - due to their 1979 peace treaty. And some experts say the Egyptian army is less than ideally equipped and trained for a counter-insurgency drive.

Despite banner headlines in a state-run newspaper this weekend declaring a new assault on Sinai militants in the coming days, army sources are playing down the possibility of a major operation in the near term. Resources are already stretched.

The sources, speaking on condition of anonymity because they are not authorised to brief journalists on the sensitive issue, said troops in Sinai were already on heightened alert.

If the army were to want to be more assertive, it might need to re-equip. The Abrams tanks and F-16 fighter jets it buys with $1.3 billion in annual U.S. military aid are not ideal for fighting small groups of international jihadist militants and their local Bedouin allies in remote, rugged terrain.

"We've long been urging them to change their procurement policies to give them the flexibility they need to tackle counter terror in Sinai," said Robert Springborg, who studies the Egyptian military at the U.S. Naval Postgraduate School.

ISRAELI APPROVAL

Under a U.S.-brokered treaty that ended 15 years of Israeli occupation of Sinai in 1982, Israel has a say in whether Egypt can increase its forces in the largely demilitarised peninsula.

Since an increase in militant activity after the fall of Hosni Mubarak in 2011 - including trouble on Israel's border - the Jewish state has already given a green light.

"Egypt could bring in more forces now, with Israel's blessing," said Amos Yadlin, former chief of military intelligence and now head of the Institute for National Security Studies in Tel Aviv.

"That will be a sensible thing to do. And if they were to leave 30 tanks there, we could live with that."

Army sources estimate there are around 1,000 armed militants in Sinai, divided into different groups with varying ideologies. They are spread over a region twice the size of Belgium but with only half a million residents, concentrated in coastal resorts.

The attacks since Mursi's ouster may have risen in number, but they have been relatively small in scale so far, and may not, in themselves, unduly worry General Abdel Fattah al-Sisi, the head of the armed forces who removed Mursi from office.

But various factors are keeping his eyes on the peninsula, the army sources said: weapons are flowing in - some from Libya, some possibly from Gaza, long an importer of arms from Sinai's many smuggling networks; the civil war in Syria is swelling the ranks of international Islamist fighters; and the rhetoric of militant and Islamist groups is growing increasingly aggressive.

A major militant assault in Sinai like one last August which killed 16 Egyptian guards on the Israeli border could trigger a sterner military response, the sources said.

In one of the worst attacks in Egypt's history, more than 80 people were killed and 200 injured in 2005 when suspected car bombs rocked Sharm el-Sheikh, a Red Sea resort on Sinai's southern coast popular with international tourists.

Analysts warn of violence hitting "mainland Egypt", as the African bulk of the country west of the Suez Canal is sometimes known, with the anger being voiced by militants in Sinai shared more broadly among Islamists across the country.

"They will justify their attacks by casting the ousting of Mursi as an attack on Islam," said Khalil al-Anani, a senior fellow at Washington's Middle East Institute.

"It's not all about Sinai. The situation would also encourage many young angry Islamists to commit attacks in Cairo and Alexandria. There's the potential for this. Islamists now are very angry and disenchanted with what is happening." (Additional reporting by Yasmine Saleh and Mike Collett-White in Cairo, Dan Williams in Jerusalem and Nidal al-Mughrabi in Gaza; Writing by Mike Collett-White; Editing by Alastair Macdonald)