Egyptian troops fired warning shots over the frontier into Gaza on Tuesday after a bomb exploded on Gaza territory near an Egyptian army convoy, Egyptian security sources said, blaming Islamist group Hamas, which denied the charge.
In Alexandria, Egypt's second city, a man was killed in a bomb blast hours after two devices were discovered at Cairo airport and another went off in the centre of the capital without casualties, other security sources said.
Hamas, which governs Gaza, said Palestinian positions had come under fire from Egyptian soil with no justification.
"Fire was directed in a surprising, unjustified way and without any violation from the Palestinian side," said Eyad al-Bozom, a spokesman for Gaza's Hamas-run interior ministry.
Security concerns have deepened in Egypt since last Thursday, when Islamic State's Egyptian affiliate claimed responsibility for coordinated attacks that killed over 30 security personnel in the Sinai Peninsula.
Egypt is facing an Islamist insurgency based in Sinai, which borders the Palestinian Gaza Strip as well as Israel and Egypt's Suez Canal.
The Egyptian government has accused Hamas of funnelling weapons and fighters to the Sinai-based militants.
Palestinian authorities have made contact with Egypt to protest about Tuesday's shooting and have demanded an investigation, Bozom of the Gaza interior ministry said.
An Egyptian court last week banned Hamas' armed wing and listed it as a terrorist organisation, prompting Hamas to reject Egypt as a mediator between Israel and the Palestinians, a role it has played for decades.
Authorities have accused the Muslim Brotherhood of conspiring with Hamas against the Egyptian state, allegations both groups deny.
Egypt declared the Brotherhood a terrorist group and repressed it systematically since the army ousted one of its leaders, Mohamed Mursi, from the presidency in 2013.
Both groups deny accusations that they engage in terrorism.
Militants have struck population centres such as Alexandria and Cairo with greater frequency since Mursi was ousted, though the most common and deadliest violence is concentrated in Sinai.