In a statement read out on state TV, the military said it has started a joint military-police ground operation in Sinai, backed by warplanes, to "restore stability and regain control" of the Sinai. It provided little detail.
"The Armed Forces and the Interior Ministry, backed by warplanes, started on Tuesday night implementing a plan to restore stability and security control and to pursue and target the terrorist and armed elements in Sinai," it said. It said initial operations had been successful and that the campaign was continuing.
"We call on the tribes and residents of Sinai to cooperate to regain security control" of Sinai, the statement said.
The Sunday ambush was one of the bloodiest attacks in Sinai in years and the deadliest against Egyptian troops, underlining the growing lawlessness of the territory, where security forces repeatedly have been targeted by militants, some loosely linked with al-Qaida.
In the latest violence, gunmen opened fire late Tuesday night on three security checkpoints around el-Arish, the capital of North Sinai province, some 50 kilometers (31 miles) from the borders with Gaza and Israel. One of the attacks was on the checkpoint on the main highway between el-Arish and the town of Rafah on the Israeli border.
The shootings wounded six people, among them a military officer, two soldiers, two policemen and a civilian whose condition is critical, security officials told The Associated Press. The officials spoke on condition of anonymity because they were not authorized to talk to the press.
Helicopters carried out strikes using missiles in retaliation later, security officials said. They did not give further detail.
Bedouin resident Abdel Rahman Abol Malkhous says he saw attack helicopters overhead firing missiles about 30 kilometers (18 1/2 miles) east of El-Arish in the area known as Sheikh Zuwayed near the Rafah border crossing with Gaza.
The security officials said it was the first time the army has fired missiles in Sinai since Egypt's 1973 war with Israel, in which Egypt tried to recapture the then-Israeli held peninsula. Sinai reverted to Egyptian control under Cairo's 1979 peace treaty with Egypt. Israeli forces completed the withdrawal from the peninsula in 1982.
In the Sunday attack, Egypt's military said 35 militants were involved, opening fire with assault rifles and rocket propelled grenades on soldiers at a border checkpoint as they broke their daily fast for the holy month of Ramadan with a sunset meal.
They then commandeered at least one armored vehicle from the post and drove about 10 kilometers (6 miles) north and crashed through a security fence at the border crossing known in Israel as Kerem Shalom and in Arabic as Karam Abu Salem. The crossing at the northeastern tip of Sinai is the meeting point between Egypt, Gaza and Israel.
Israeli officials said the militants blew up one of two vehicles they seized from the troops against the security fence to enter Israel. Israeli aircraft struck the militants as they tried to break through.
Egyptian officials told the daily Al-Akhbar Wednesday the militants had nearly 400 kilograms (880 pounds) of explosives in the vehicle. A major military funeral was held for the slain troops Tuesday in Cairo.
Sinai has seen lawlessness and militant violence in the past, but it took a turn to the worst after the uprising that toppled Hosni Mubarak in February 2011. Amid the uprising, police and internal security forces all but disappeared from the streets across the country. In Sinai, militants have grown steadily bolder and better armed as weapon smuggling from Libya picked up in the wake of the revolt there.
Residents say the militants far better armed than the security forces on the ground, which have repeatedly come under attack by militants. Since Mubarak's ouster, some of the groups have distributed fliers in Sinai urging the forces to leave the peninsula because, they say, it will be declared an Islamic state.
Under the peace treaty with Israel, a large chunk of Sinai is to be demilitarized. But in 2005 and following Mubarak's ouster last year, Israel agreed to boost the number of troops in the area, although they remain lightly armed. The Sunday attack spurred renewed calls in Egypt to amend the 1979 treaty to allow for more troops and ammunition in Sinai.