By Shadia Nasralla

CAIRO (Reuters) - Egypt has tightened control of crossings from the Sinai after an Islamist militant group based in the peninsula said it tried to kill the interior minister in Cairo last week, the state news agency reported on Monday.

The group Ansar Bayt al-Maqdis claimed responsibility on Sunday for last Thursday's suicide bombing aimed at Interior Minister Mohamed Ibrahim. It promised more attacks in revenge for a crackdown on Egypt's Islamists, raising fears that militant violence in the Sinai could spread across the country.

The Egyptian military on Saturday launched a major assault on militants in North Sinai, killing or wounding at least 30 people in clashes close to the Palestinian Gaza Strip.

State news agency MENA said that Ibrahim decided to boost security surveillance and tighten control of crossings from the Sinai to other Egyptian regions in conjunction with a broad security campaign in the peninsula.

Sporadic violence continued on Monday, when one supporter of deposed Islamist President Mohamed Mursi's Muslim Brotherhood was killed and 10 people were injured in clashes between soldiers and supporters of the group in North Sinai, state television said.

Security sources later said two people were killed in the clashes.

In two separate incidents in central Sinai, three soldiers were injured when gunmen opened fire, according to MENA, while security sources said two soldiers were killed in attacks by gunmen in Sinai.

Islamist militancy has risen sharply in the often-lawless region adjoining Israel and the Gaza Strip, and elsewhere in Egypt, since the army deposed Mursi two months ago following mass protests against him.

Last Thursday's daylight attack was the most spectacular so far. A suicide car bomber blew himself up next to Ibrahim's convoy as he left his Cairo home for work in an armored limousine. The bomber, a passerby and an unidentified person were killed and more than 20 others were wounded.

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Security officials said they were assessing the threat posed by Ansar Bayt al-Maqdis, which is said to have 700 to 1,000 members and is considered the second-largest jihadist group in Sinai behind Salafiya Jihadiya, which has an estimated following of around 5,000 members.

Some officials doubt Ansar Bayt al-Maqdis is capable of mounting attacks outside Sinai.

Security officials told Reuters they were especially concerned that the group, whose name means "supporters of Jerusalem", will use stolen government vehicles for car bombs.

ATTACK ON TAMARUD CO-FOUNDER

In another development, unidentified gunmen on Monday opened fire on the car of the co-founder of a movement that helped bring down Mursi. The Tamarud movement's website said Mahmoud Badr's car was stolen in the attack on the outskirts of Cairo and that he was unharmed.

It also said the gunmen seized papers related to the committee entrusted with amending the constitution that Mursi had signed into law. Badr is a member of the committee.

Mohamed Haykal, another founding member of Tamarud, told the website of state newspaper Al-Ahram that "thugs" were responsible for the incident and that it was not a political attack against Badr or Tamarud. He added that the papers did not contain information of great importance.

Badr was a vocal supporter of the crackdown on the Muslim Brotherhood following Mursi's removal from power.

The Tamarud movement said it secured 22 million signatures for a petition calling on Mursi to step down. It mobilized mass protests against his rule that prompted the army to install a new government.

Ansar Bayt al-Maqdis counts Grad rockets among its weapons and produces homemade bombs used against Egyptian security forces and soldiers, security sources said. Rocket-propelled grenades are often fired at buildings housing security forces.

Last year, the same group claimed responsibility for rocket attacks launched on Israel from Sinai. It has also claimed responsibility for at least 10 attacks in the past two years on a gas pipeline linking Egypt, Israel and Jordan.

Sinai's eastern border with Israel and Gaza is a particularly sensitive one. Israel made its concerns known when jihadist groups expanded into a security vacuum left by the fall of Egypt's veteran autocrat Hosni Mubarak in 2011.

Since the army toppled Mursi - and especially since security forces killed hundreds of supporters of the Muslim Brotherhood when they smashed protest camps in Cairo on August 14 - there have been online calls from radicals to abandon attempts to achieve Islamist rule by democracy, and instead step up violence.

(Editing by Will Dunham)