Rights groups report a sharp increase of attacks by West Bank settlers on Palestinians, as well as rising right-wing violence against left-leaning Israelis.
But the ruling Kadima Party, which votes on a new leader Wednesday, may not buy into the former army chief's security policy.
Exit polls showed that Israeli Foreign Minister Tzipi Livni won with roughly 48 percent of the vote to take over the party led by Ehud Olmert.
Aryeh Deri, jailed for bribery in 2000, tries for a comeback in the Jerusalem mayoral contest as Israel reels from several high-profile political scandals.
French President Sarkozy, in Damascus this week, was also thought to be seeking Syria's help in dealing with Iran's nuclear program.