One of the two bombings in the town of Rafah brought down a two-story building housing the local branch of military intelligence, while the other struck an army checkpoint.
The near-simultaneous attacks nudged the violence in the strategic Sinai Peninsula closer to a full-blown insurgency, compounding Egypt's woes at a time when the country is struggling to regain political stability and economic viability more than two years since longtime autocrat Hosni Mubarak was toppled in a popular uprising.
Mohammed Ibrahim, the minister, escaped unharmed, but the blast caused extensive damage in the area. An al-Qaida-inspired group based in Sinai later claimed responsibility for that bombing.
Today's attack on the intelligence building in Rafah collapsed the entire structure and buried an unspecified number of troops under the rubble, two security officials said, speaking on condition of anonymity because they were not authorised to talk to the media.
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The security officials said the blast at the intelligence building also badly damaged five houses nearby. The wounded included 10 soldiers and seven civilians, three of them women. Egypt's official MENA news agency later reported that following the attacks, authorities ordered the closure of the Rafah border crossing, which links Egypt to the Gaza Strip.
There was no immediate claim of responsibility for today's attack. Militants in Sinai, some with links to al-Qaida, have been targeting for months Egyptian forces in the strategic peninsula bordering Gaza and Israel.
After Mubarak's ouster, Morsi became the country's first democratically elected president in 2012 but he was deposed in July by the military after days of massive street protests against him.