By announcing a cease-fire Saturday night, accepted by a pummeled Hamas on Sunday morning, Israel has achieved one of its Gaza war aims: end it before Barack Obama comes into office.

Obama’s reaction to a bloody, controversial exercise of Israel’s right to self-defense could not be predicted with precision. Most Israeli officials and politicians have had trouble understanding our 44th president and where he’s “coming from”—frankly having clung, practically until November 4, to the belief that America would not elect a black man with a Muslim middle name.

Israel and its many American friends wonder if there is good reason to suspect that Obama would find a lot to agree with in the general European view: that Israel has been abusing its US-provided military might.

As close as Israel and the United States have been, in their alliance that spans almost the entire six decades of the Jewish state’s existence, misunderstandings are rife.

American administrations have tried to influence Israeli elections, and more incredibly, tiny Israel sometimes feels it’s a potent factor in American votes for president.

Bill Clinton obviously favored Israel’s Labor Party prime ministers—Yitzhak Rabin, Shimon Peres, and Ehud Barak—as their efforts to achieve peace treaties with their Arab neighbors matched his own ambition to make history. Likud Party hawks, such as Benjamin Netanyahu, when he was prime minister, were out of step with Clinton’s vision, and relations were noticeably frosty.

The Bush-Cheney team, with their own world view, favored hard-liners such as Ariel Sharon, reluctantly going along with his shockingly dovish shift that removed all Israeli settlers and troops from the Gaza Strip three years ago.

As much as it’s a Washington parlor game to guess what the new Middle East policy is going to be—and Obama has assured us there will be significant new elements—no one is watching for clues more than the Israelis. Our new president intends to reach out to Iran and Syria, lifting a long-term American cone of silence, and by all meanings of the term he’ll present a new face to a vast region where the US is reviled rather than respected.

Israel and its many American friends wonder if there is good reason to suspect that Obama, when given free rein to air his views, would find a lot to agree with in the general European view: that Israel has been abusing its US-provided military might, has been careless in slaying more than 1,000 Palestinians to make a political point, and needs to be slapped sternly so that it will remove settlements and roadblocks that humiliate Palestinians and leave them little choice but terrorism.

Many of Obama’s Jewish supporters, as thrilled as anyone at his inauguration this week, say they feel completely confident that he will be pro-Israel. They do acknowledge that the “tone” will be different, contending that the Bush-Cheney approach did not help Israel achieve peace. Early signs are that Obama and his chief diplomat, Hillary Clinton, haven’t yet agreed on a special Mideast envoy, and on who, where, and when any senior American will travel to start showing our new face.

Some parts of history are repeating themselves. Israel has its general election soon, on February 10, and whoever leads the party that wins the most seats in the Knesset will be the man or woman who gets to be prime minister.

The Gaza war—as popular in Israel as it’s been reviled in the streets of London, Islamabad, and Jakarta—has been a boon to two of the current candidates: Tzipi Livni and Ehud Barak. Livni is the foreign minister who leads the fairly new Kadima Party, now that Prime Minister Ehud Olmert has been forced out by a corruption investigation. Barak, familiar as a tough general-turned-peacenik as prime minister, is now the defense minister in Olmert’s coalition and the Labor Party leader seeking a second shot at greatness.

Polls still indicate that Bibi Netanyahu may win. As head of the Likud Party—the late Menachem Begin’s right-wing creation, led later by Yitzhak Shamir and by the still comatose Sharon until he formed Kadima—Netanyahu has not been part of the Gaza war cabinet.

If the mini-war had turned into a disaster, Bibi’s hands would have been clean. If it continues to be perceived by most Israelis as an effective reassertion of Israel’s toughness and deterrence power, Netanyahu can point out that he constantly expressed his support—and that he even unsheathed his formidable unaccented English-language skills on foreign TV networks to stand up for Israel.

While the Obama administration can be expected to keep its mouth wisely shut on the subject of domestic Israeli politics, there can be little doubt that our new president, who vows to “hit the ground running” with new ideas for reconciliation in the Middle East, and his secretary of state—significantly, the other half of the globally minded Clinton couple—privately prefer ABB in Israel’s election: Anybody But Bibi.

Dan Raviv, a CBS News correspondent, is co-author of Every Spy a Prince: The Complete History of Israeli Intelligence and Friends In Deed: Inside the Israel-U.S. Alliance.