Their popularity had been in decline since 2006, but Blecher said that appeared to be changing. "Since the fighting restarted, it is back up again," he said. "The Palestinians see Israel getting a taste of their own medicine, and it is Hamas giving them the dose, so that boosts their popularity."

What is the international community's view?

Hamas is considered a terrorist organization by the United States, Israel, Britain and the European Union.

In the wake of the 2008-2009 Gaza conflict, in which more than 1,400 people were killed -- including 700 civilians in Gaza and three civilians in Israel -- the United Nations Goldstone Report found that rocket attacks by Hamas constituted war crimes, and may have amounted to crimes against humanity. The Goldstone Report also found that Israeli forces had committed war crimes and possible crimes against humanity during the conflict.

Human Rights Watch, too, has accused Hamas -- and Israeli authorities and other Palestinian armed groups in Gaza -- of war crimes, of violating international humanitarian law, and of meting out cruel and inhuman treatment to detainees.

What is its relationship with Fatah?

Like Hamas, Fatah was once dedicated to launching attacks on Israel, as part of what they saw as the Palestinians' struggle for liberation. However, Fatah became part of the PLO in the 1960s, and in 1993, Fatah and the PLO recognized Israel's right to exist and renounced violence.

Fatah, which controls the West Bank, is considered more moderate, and while Hamas is outlawed and viewed as a terrorist group, Fatah enjoys the support of Western nations including the United States.

Fatah and Hamas were estranged for several years after Fatah lost its majority in the Palestinian Parliament to Hamas in 2006. The rift was partially healed when the two groups signed a reconciliation deal in May 2011, linking the one-time rivals in their goal to establish a Palestinian state.

The unity agreement was met with mixed reactions from the international community -- former U.S. President Jimmy Carter insisted "the accord could lead to a durable cease-fire." However, Netanyahu called the deal "a tremendous blow to peace, and a great victory for terrorism."

What would Hamas want out of a peace deal?

Hamas wants to see an end to attacks on Gaza, and to see the territory opened up to the world, with restrictions at its border crossings eased, an end to the maritime blockade, and the reduction of the buffer zone around Gaza, which has seen one third of its agricultural land placed out of the reach of Gazan citizens.

"Will Hamas insist on every last one of those things? No," Blecher said. "What they want to see is the opening up of Gaza in a broad sense, so if they can come up with a package of measures that includes some of those things, but not all, I think they would live with that."

But what they probably wouldn't be happy with would be the promise of a few tentative moves, with the promise of bigger concessions at a later date.

"For the first 15 or so years of the peace process, the idea was that it was done step-by-step, incrementally, building trust as it went along, but that failed, and from Hamas' perspective, if they agree to a limited 'Stage One' deal, it's not clear they'll ever get to 'stage two,' so from their point of view, they want it all tied up in a bigger package," Blecher said.