By Michael Georgy

CAIRO, Oct 5 (Reuters) - Egyptian authorities warned the Muslim Brotherhood on Saturday against staging violent protests and tightened security in cities and strategic installations after clashes on Friday killed at least four people.

Supporters of deposed Islamist President Mohamed Mursi on Friday mounted their boldest demonstrations since troops crushed their protest camps demanding his reinstatement on Aug. 14.

Both opponents and supporters of the Brotherhood have called for mass protests on Sunday, when the country plans to celebrate the anniversary of an Egyptian attack on Israeli forces in the Sinai during the 1973 war.

"The Ministry of Interior asserts its determination on confronting violence and infringements of the law by Muslim Brotherhood supporters," a ministry statement said.

"Security has been stepped up on highways, in all cities and at important installations. The Ministry of Interior warns against attempting to spoil the 6th of October commemoration."

On Saturday afternoon, about 1,000 Mursi supporters tried to reach the Rabaa al-Adawiya mosque area in northeast Cairo, where security forces crushed one of their protest camps in August. All but about 50 were turned back by police, who fired tear gas, security sources said.

Interior Minister Mohamed Ibrahim said in a meeting with aides that security forces would not tolerate any attempts to block roads or attempts to "spread chaos", the state news agency reported.

"The ministry will deal with the utmost firmness and decisiveness with any of those practices, and confront any lawlessness," it quoted him as saying.

"EVIL ELEMENTS"

The military boosted its presence around Tahrir Square - where hundreds of thousands of Egyptians demonstrated during the revolt that toppled autocrat Hosni Mubarak in 2011 - after clashes on Friday in several cities.

Political tensions have gripped Egypt and hammered the economy since the army ousted Mursi in July, installed an interim government and drew up a political road map it promised would bring fair elections.

Prime Minister Hazem el-Beblawi said in a statement to the nation on Saturday that "evil elements" still posed a danger but had lost much of their power, a reference to Islamist militants.

Beblawi said the road map was "taking its natural course" and that he hoped it would conclude soon. He said the economy was starting to improve and "there were clear signs and reassuring indicators".

Authorities have cracked down hard on the Brotherhood, which won every election since Mubarak's fall but became unpopular during Mursi's rule, with many Egyptians accusing him of trying to acquire sweeping powers and mismanaging the economy, allegations he denies.

The Brotherhood accuses the military of staging a coup and sabotaging democracy by removing Mursi, the country's first freely-elected president.

On Aug. 14, Egypt's military-backed authorities smashed the two pro-Mursi sit-ins in Cairo, with hundreds of deaths, and then declared a state of emergency and imposed a curfew. Many of the Brotherhood's leaders have been arrested since.

Attacks by Islamist militants in the Sinai, which borders Israel, have risen sharply since Mursi was toppled and fears of a wider Islamist insurgency have grown.

In September, a Sinai-based militant group claimed responsibility for a failed suicide bombing against the interior minister in Cairo.