Twenty people were reported killed in Touma village, while the Sheikh Zuwaid area to the west was also hit.

The strikes came after security checkpoints were allegedly attacked by gunmen in the town of al-Arish, leaving a number of people injured.

After that attack - the deadliest and most brazen against Egyptian troops in this border region for decades - Israeli forces said they killed some of the militants who broke through into Israel.

There has been a heavy military build-up around al-Arish, correspondents report, and Egypt's Rafah border crossing to Gaza has been indefinitely closed as security forces hunt the remaining attackers.

Egypt is also reported to have begun sealing off the illicit smuggling tunnels into Gaza - reportedly sending prices of key goods soaring in Gaza.

This is the first time Egypt has fired missiles in Sinai since the 1973 war with Israel, when it attempted to recapture the Sinai peninsula, security officials told Associated Press.

Analysis

image of Jeremy Bowen Jeremy Bowen BBC Middle East editor

Egypt's President Mursi promised he would re-impose full control on Sinai, which has been increasingly lawless since the revolution in Egypt last year.

The area in Sinai where the borders of Egypt, Israel and Gaza meet has become a hotbed of smuggling - through the tunnels into Palestinian Gaza. It is a place where jihadists are said to be operating, with an eye on attacking Israel.

Violence here could threaten the peace treaty between the two countries. That's why Israel has told Egypt to put its house in order, rather than taking action itself.

But the Egyptian raid on what it called "terrorists" won't change much on its own. Sinai is a poor and neglected region, with independent-minded Bedouin tribes. It is an inviting place for groups that want to use the lack of central control to attack across the border into Israel.

Egyptian military presence in Sinai is limited and requires Israeli approval under the terms of the 1979 peace treaty between the nations which returned Sinai to Egyptian control.

Tensions are very high in the area, where Islamic extremists are said to have gained a foothold in recent months, taking advantage of the security vacuum left after former President Hosni Mubarak was ousted last year, reports the BBC's Yolande Knell in Sinai.

A Sinai army commander told Reuters news agency the army had received information that there were many militants in Touma.

"We have succeeded in entering Touma, killed 20 terrorists and destroyed three armoured cars belonging to terrorists. Operations are still ongoing," he told Reuters.

State news agency Mena said "intensive security campaigns" had been launched against "terrorist hotbeds" in southern al-Arish, Sheikh Zuwaid and Rafah.

It said that during the air raids "terrorists" had used rockets and rocket-propelled grenades to try to shoot down the military aircraft "but they failed".

The air raids came hours after security checkpoints were attacked in the main regional town of al-Arish, sparking clashes with security forces.

Locals said rounds of gunfire could be heard just before midnight and telephone lines and the Internet were cut off.

At least four people - including police officers and a civilian - were wounded in those attacks.

The Egyptian soldiers killed in Sunday's attack were buried on Tuesday in a funeral marked by angry calls for vengeance.

Some protesters chanted slogans against the Muslim Brotherhood, and according to witnesses, tried to assault Prime Minister Hisham Qandil.

SINAI

Huge desert peninsula of about 60,000 sq km (23,000 sq miles) with a population of only a few hundred thousand
Distinct identity from rest of Egypt; populated by native Bedouin and a substantial minority of Palestinian extraction
Of great strategic importance, linking Egypt to the Middle East, and site of Egyptian border with Israel and only non-Israeli border with Gaza
Under Israeli occupation from 1967 to 1982, it remains under special security regime mandated by 1979 peace treaty, which significantly restricts Egypt's freedom of military action and requires deployment of multinational peacekeeping force
There has been a surge in jihadist activity in the past decade, but analysts say discriminatory treatment by Cairo underpins local unrest

Both Israeli and Egyptian officials blamed Sunday's attack on Islamist militants - though Egypt's Muslim Brotherhood, the group to which President Mohammed Mursi belongs, accused the Israeli spy agency Mossad of being responsible.

On Tuesday, Israel handed Egypt six "completely charred" bodies it says are some of the militants behind Sunday's attack on the Egyptian soldiers, a medical official in al-Arish told AFP news agency. The bodies have not yet been identified.

Israel signalled its approval of Egypt's strikes, with senior defence official Amos Gilad telling Israel Radio on Wednesday that Egypt was determined to "impose order in Sinai because that is their responsibility... If they don't remove and uproot [the threat], it will continue to strike".

The rising violence in the area is a test of credibility for the government of President Mursi, correspondents say.

Although it is clear that Israel has approved the build-up of troops around al-Arish, Israel has historically been reluctant to see a large increase in Egyptian troops close to its border.