Explosions outside Cairo University killed two people including a police brigadier-general on Wednesday in what appeared to be a militant attack targeting security forces.
A group calling itself Ajnad Misr, or Soldiers of Egypt, claimed responsibility for the blasts, which follow a string of operations launched against police and soldiers since the army ousted Islamist president Mohamed Morsi in July.
The fast-growing insurgency threatens the security of the most populous Arab nation ahead of a presidential election in May - as well as the vital tourist industry.
Hours after the explosions in an upmarket area near the zoo in Giza, a high-level government security committee said it would present legislation "connected to combating terrorism" for the cabinet's approval on Thursday, without going into further details.
Egypt's government already has wide-ranging security powers. It has detained thousands of Morsi's supporters and declared his Muslim Brotherhood a "terrorist organisation".
Ajnad Misr issued a short statement saying it was targeting senior security officers guilty of mass killings - a likely reference to the government crackdown on militants.
Two bombs left among trees outside the university killed the police officer and wounded five other security forces who had been guarding the facility, the interior ministry said.
Shortly afterwards, a third blast killed one person, security officials said. Police found a fourth bomb in the area.
"We expect trouble for the long term. How can the police protect us when they can't even protect themselves? It is not possible," student Mohamed Abdel Aziz said outside the university campus.
People screamed and ran for safety after the attacks as panic spread on the streets and on campus, a Reuters witness said.
Video footage online showed a cloud of smoke hovering above a tree-lined roundabout. A loud blast is heard moments later.
Security forces clad in black uniforms move away from what appears to be the source of the blast, and then advance with weapons drawn.
Reuters could not independently verify the authenticity of the video, which was released by El-Youm el-Sabaa newspaper.
Egypt's presidential spokesman Ahmed al-Muslimani said: "Terrorist groups want Egyptian universities to be known for chaos and bloodshed instead of for modernity and civilization."
Bombings and shootings targeting the security forces have become commonplace in Egypt since the army deposed Morsi. The government this week put the death toll from such attacks at nearly 500 people, most of them soldiers and police.
Analysts predict that militants will escalate violence before the May 26-27 presidential election that is expected to be easily won by Abdel Fattah al-Sisi, the general who toppled Morsi after protests against the former leader's rule.
Widely seen as Egypt's de facto leader, Sisi enjoys backing from supporters who believe he can end the political turmoil and bring prosperity to the country.
But he is viewed by the Islamist opposition as the mastermind of a coup that ignited the worst internal strife in Egypt's modern history.
It will be the second time Egyptians have voted in a presidential election in less than two years.
But in contrast to the 2012 vote won by Morsi, this election follows a fierce government crackdown on dissent that has included both Islamists and secular-minded democracy activists.
Thousands of Brotherhood supporters have been detained and killed in mass protests and clashes with police since Morsi was deposed. Last week more than 500 were sentenced to death in a mass hearing condemned by rights groups and Western governments.
The Brotherhood, Egypt's best organized political party until last year, has been banned and driven underground.
The movement says it is committed to peaceful activism. Senior Brotherhood politician Amr Darrag condemned the violence at Cairo University on his Twitter account and said it showed the failure of the security forces to protect Egyptians.
Additional reporting by Yasmine Saleh; Writing by Michael Georgy