CAIRO (Reuters) - Egypt branded Islamist gunmen who killed 16 police near the Israeli border as "infidels" and promised on Monday to launch a crackdown following the massacre that strained Cairo's ties with both Israel and Palestinians. An Egyptian official has said "Jihadist elements" crossed from the Gaza Strip into Egypt before leading the assault on a border station. They then stole two armored vehicles and headed to nearby Israel, where they were killed by Israeli fire. Israeli Defense Minister Ehud Barak said on Monday that eight assailants died in the attack, adding that he hoped the incident would serve as a "wake-up call" to Egypt, long accused of losing its grip in the desert Sinai peninsula. The bloodshed represented an early diplomatic test for Egyptian President Mohamed Mursi, an Islamist who took office at the end of June after staunch U.S. ally Hosni Mubarak was overthrown last year in a popular uprising. Mubarak cooperated closely with Israel on security and suppressed Islamist movements such as Mursi's Muslim Brotherhood whose leaders often voiced hostility towards the Jewish state. Egypt's military, which still holds many levers of power in the most populous Arab nation, called the attackers "infidels" and said it had been patient until now in the face of the instability in Sinai. "But there is a red line and passing it is not acceptable. Egyptians will not wait for long to see a reaction to this event," it said in a statement on its Facebook page. A demilitarized Sinai is the keystone of the historic 1979 peace deal between the two countries. Mursi has promised to honor Egypt's 1979 peace treaty with Israel and has done little to suggest a major shift in ties. He has also reached out to Hamas, the Islamist rulers of the Gaza Strip that borders Egypt and Israel, and Sunday's killings put an instant strain on those closer ties. Egypt announced it was closing its border crossing into Gaza "indefinitely", cutting off the sole exit route for most Palestinians at the height of the Muslim-fast month of Ramadan. Hamas, which condemned the killings of the Egyptians, immediately sealed the warren of smuggling tunnels that connect Gaza with Egypt after Cairo said the gunmen had used these links to reach their territory. Many key goods, including oil, pass through the tunnels, and a prolonged closure could stymie life in the coastal enclave. Hamas said it was working with Egypt to try to identify those behind the bloody operation. "No Palestinian could take part in such an ugly crime and in the killing of our beloved Egyptian army men in such a horrible manner," said Taher Al-Nono, spokesman of the Hamas Gaza government.