GAZA (Reuters) - Palestinian security forces widened their control in the Gaza Strip on Friday under orders from President Mahmoud Abbas to prevent attacks on Israelis in another important move toward reviving Middle East peacemaking.

But in a sign of challenges ahead for Abbas, Hamas appeared to have made a strong showing in the first-ever municipal elections held in the Gaza Strip, seen as a test of strength between the Islamic militant group and the new leader.

Building on a northern Gaza deployment a week ago, some 2,000 Palestinian paramilitary police began to fan out across the southern part of the territory, turning back the clock to the days of security cooperation and peacemaking with Israel.

Some of the uniformed men, armed with assault rifles, took up positions between Jewish settlements on occupied land, often hit by mortar bombs and rockets launched by militants, and Palestinian towns.

Others were headed to Gaza´s border with Egypt, where Israel has conducted punishing raids in Palestinian residential areas to root out weapons-smuggling tunnels and militants.

Over the past week, violence has dropped sharply in the Gaza Strip, where Abbas is trying to arrange a cease-fire in the run-up to a planned Israeli withdrawal this summer.

Israeli Prime Minister Ariel Sharon said on Thursday conditions were right for a "historic breakthrough" on peace after measures taken by Abbas to bring calm.

He said in a speech that if Palestinians worked to "fight terror" after more than four years of bloodshed, then Israel could move forward with a U.S.-backed peace "road map" meant to lead to a Palestinian state.

BUSH PRAISES ABBAS

President Bush, in an interview published on Friday in The New York Times, said Abbas has shown strength in the early phases of his administration.

"He is sending assurances that he will put a 100-percent effort into protecting the people on both sides of the issue from terrorists. And to me, that´s an impressive start," Bush said.

In a sign of a new U.S. push for peace after Yasser Arafat´s death, officials said Secretary of State Condoleezza Rice would visit the region early next month on the first trip of her new job.

Palestinians said that in order to help Abbas, elected on Jan. 9 to succeed Arafat as president, Israel agreed in principle to pull back troops from West Bank cities reoccupied during a 4-year-old uprising and to free hundreds of prisoners.

A senior Israeli official confirmed such a pullout was on the cards but said the government would not approve it until the Palestinians "come up with a proper plan" for implementation.

The prisoners´ issue would be on the agenda at Abbas´s expected summit with Sharon next month, the official said. A Palestinian official said the talks may take place on Feb. 8.

In what appeared to be a setback for Abbas, unofficial election results in the Gaza Strip indicated Hamas had won seven of 10 municipalities in Thursday´s vote.

The Independent Palestinian Center for Policy and Survey Research said its exit polls showed Hamas was likely to take three of the four biggest districts in the vote, including one where Abbas´s Fatah movement had been expected to win.

Voter turnout topped 80 percent, Palestinian officials said. (Additional reporting by Allyn Fisher-Ilan in Jerusalem and Wafa Amr in Ramallah) (© Reuters 2005 01/28/05)