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Pressure is mounting on Israel and Hamas to find a way to end the war in Gaza. Both sides have responded positively, if tentatively, to Egyptian proposals for a phased truce that would begin with a lull in fighting for a defined period (10 days, by some accounts). That interlude would then allow for the brokering of a more comprehensive cease-fire. But each side's goals from any truce remain antagonistic to the other's, and reaching an agreement that bridges the vast gap between them remains a herculean diplomatic challenge.

Even before the Israeli invasion began in late December, Hamas had offered to renew its six-month cease-fire with Israel on the condition that the border crossings from Egypt and Israel into Gaza be reopened. Those crossings have been closed as part of a strategy of imposing economic deprivation on the people of Gaza in the hopes that they would turn on Hamas; Israel remains reluctant to agree to reopen them as part of a cease-fire deal, since that would be claimed as a victory by Hamas. Hamas also insists on a full and immediate withdrawal of all Israeli forces from Gaza; Israel is reluctant to comply until mechanisms are in place to prevent Hamas from rearming.

Israel's declared purpose in launching Operation Cast Lead was to halt Palestinian rocket fire from Gaza and prevent Hamas from being able to rearm by smuggling weapons from Egypt. Israel remains committed, however, to a long-term goal of ending Hamas' control of Gaza, and it insists that the movement should gain no "recognition" or "legitimacy" as part of any truce — a tough call since Hamas is the key combatant on the Palestinian side. (See pictures of the Gaza ground war.)

So how will the Gaza conflict be resolved? Israel's dominant military stance puts its leaders in a position to decide how the hostilities will end. But those leaders remain locked in debate among themselves over the best way to do that. Here are the three most likely scenarios, each with different political consequences for the main players and the future of the conflict:

Scenario 1: Regime Change
Given Israel's long-term goal of ousting Hamas in Gaza, some key military and political leaders have urged that it expand the goals of its current operation and use its momentum to take control of Gaza City and decapitate Hamas. Most vocal in advocating this option is Likud leader Benjamin Netanyahu, the hawkish front runner in the race for Prime Minister, who will portray any outcome that leaves Hamas intact in Gaza as a failure — bad news for his chief rivals, Defense Minister Ehud Barak and Foreign Minister Tzipi Livni.

But the "regime change" option is even reported to have support from Prime Minister Ehud Olmert, who sees it as a way to restore the control over all Palestinian territories of his peace partner, Palestinian Authority President Mahmoud Abbas. Skeptics, including Barak and Livni, warn that pursuing regime change would require the Israeli military operation to continue for months, risking diplomatic isolation and a dramatic increase in casualties. And the Israeli security establishment is justifiably skeptical of the prospects for reimposing the already enfeebled Abbas on a hostile Gaza. Rather than boost his power, the latest confrontation has further marginalized Abbas. Even his future control over the West Bank has come into question.

Even if forced out of power, Hamas would maintain a resistance role that would prevent anyone else from governing the territory. (The organization is estimated to have close to 20,000 men under arms in Gaza, of whom Israel claims to have killed no more than 2.5% so far.) That would force Israel to reoccupy a territory from which it sought to separate in 2005. Still, Israeli leaders hope the military operation can deal a powerful enough blow to hobble Hamas. They still wish to see Abbas' authority reimposed as part of any truce. More realistically, perhaps, Arab mediators and the U.N. Security Council have urged that cease-fire plans restore reconciliation between Abbas and Hamas. Arab countries had previously brokered a national-unity government between the two, and Hamas remains the ruling party in the Palestinian Authority's legislature. But Israel has long insisted it will not deal with a Palestinian government that includes Hamas. (See pictures of life under Hamas in Gaza.)

Scenario 2: Long-Term Cease-Fire
Israel has insisted that a cease-fire be "sustainable" by ensuring that Hamas is unable to rearm itself. An actual disarming of Hamas' current militias is unlikely without a full-scale reoccupation of Gaza, which would involve tens of thousands more Israeli troops over many months. Anything less would leave Hamas as the dominant security presence inside Gaza. So Israel's priority will be to choke off the supply of rockets and mortar shells, which have been smuggled through tunnels from Gaza and fired at Israel. The Israelis want Egypt to police those tunnels under U.S. supervision. Egypt has been reluctant to take on the potential domestic political headache of having foreign troops policing the Gaza border on its soil and fears that Israel will seek to force Cairo to accept increasing responsibility for the territory — a role Cairo steadfastly refuses to play.

Egypt is reportedly proposing that an immediate truce, in which Israeli forces retain their current positions but advance no farther, be followed by negotiations of a full withdrawal and reopening of the crossings. Egypt will most likely agree to enhanced mechanisms for policing the smugglers' tunnels, but those tunnels were also Gaza's economic lifeline, and Egypt will insist they can be closed only if the legitimate crossings into Gaza are reopened to allow the flow of normal humanitarian and commercial traffic. That, of course, is what Hamas has been demanding, which will make Israel — and Egypt — uncomfortable. Neither wants to see the radical movement emerge from this confrontation with an enhanced status, but the scale of the humanitarian disaster wrought by Operation Cast Lead renders maintaining the economic blockade untenable. Hamas may claim vindication, but it will not be allowed to directly control the crossings itself, as it had demanded, and it will be forced to swallow many other compromises.

Policing the crossings on the Palestinian side will probably be the responsibility of the Palestinian Authority, although that will require new agreements between Hamas and President Abbas. Any cease-fire is likely to implicitly recognize Hamas' dominance as an inescapable reality in Gaza. Hamas will claim victory from any truce that results in the crossings being reopened, and its claim may well be echoed by Netanyahu on the campaign trail. After all, ending the current operation on the basis of a formal long-term truce in Gaza will codify Israeli-Hamas coexistence. That's why Israeli journalist Aluf Benn dubbed the conflict "Gaza's War of Independence," an allusion to the conflict 60 years ago in which Israel established its existence as an intractable political-military fact. (See pictures of 60 years of Israel.)

Scenario 3: The Guns Go Silent Without a Formal Truce
If the offensive cannot deal Hamas a death blow, Israel may see benefit in holding its fire, in line with the first phase of the Egyptian plan but not necessarily concluding a comprehensive cease-fire. It would simply maintain the halt to hostilities and even withdraw its forces on an open-ended basis. Israeli leaders saw Operation Cast Lead as an opportunity to restore Israel's "deterrent" power, which it believed had been damaged when it was fought to a draw by Hizballah in Lebanon in 2006. But the Gaza operation, with its almost 100-to-1 ratio of Palestinian to Israeli casualties, has issued a painful reminder of Israel's capacity and willingness to abandon restraints and rain devastation on the heads of all challengers.

By simply stopping its operation without a formal truce, Israel can claim to have re-established its "deterrent" on future rocket fire without "recognizing" Hamas' authority in Gaza. This option would allow Israel to avoid accepting any new restraints on its actions in Gaza. It would also bypass the need to deploy international forces, a move that would complicate any future offensive. Israel ended its 2002 offensive against militants in Jenin and other West Bank cities on its own terms, choosing where to remain deployed and continuing to raid those cities as deemed necessary. The six-month truce that maintained calm in Gaza from June until November last year was never formally codified — each side had its own interpretation of understandings reached with the Egyptian mediator, and there was no publicly agreed text or mechanism for monitoring or arbitrating disputes.

Some Israeli reports suggest that halting the offensive without an agreement is the option favored by Livni. And its prospects may be enhanced by the realization that negotiations over a formal cease-fire may take more than 10 days and may, in fact, not be resolved before Israel has elected a new government — possibly, one with little interest in a truce with Hamas. But even an unspoken truce would have to involve the opening of crossings to relieve the humanitarian catastrophe and would require mechanisms for monitoring the flow of goods into Gaza and smuggling via tunnels. In other words, even an unspoken cease-fire will require many of the features of a formal one. Hamas has also insisted that it won't accept another vague or open-ended cease-fire without defined timetables and verifiable goals, although its ability to hold out for its terms will be determined by the resilience of its forces on the ground. But Egypt and other regional players will press Israel to formalize the truce terms in order to prevent a recurrence of the horrors seen in Gaza over the past three weeks. (See pictures of heartbreak in the Middle East.)

Whichever of these three permutations defines the Gaza outcome, the likelihood is that Operation Cast Lead will not have ended the conflict between Israel and Hamas, but will instead have propelled it into a new phase.

• Tony Karon
• Analysis: As diplomatic efforts intensify, both Israel and Hamas remain determined to win the peace