The military war over the Gaza Strip also is being fought through a war of words, with Hamas and Israel each lobbing sharp statements at the other.
As Israeli forces pushed across the border Saturday night, Hamas said the troops were approaching the trap it had prepared for them in Gaza that will turn the territory into "a graveyard" for them.
Meanwhile, an Israeli official, in an interview with FOX News, dismissed any "arrogant" talk by Hamas while affirming Israel's right to defend itself.
Israel launched its ground offensive into Gaza after nightfall. Defense officials have said about 10,000 soldiers have been massed along the border in recent days.
Hamas said it broadcast a message in Hebrew on Israeli military radio frequencies promising to kill and kidnap Israeli soldiers. The message warned troops they would "suffer mental illness from the horrors" its fighters would show them.
Hamas has long prepared for an Israeli invasion, digging tunnels and rigging some areas with explosives. Before moving the ground forces moved in, Israeli artillery shelled border areas, apparently to detonate hidden explosives and mines.
Israeli Welfare Minister Isaac Herzog said the aim of the offensive is to allow Israelis to return to normal life without threat of Hamas rocket attacks.
"Right now we've started a ground operation, which is aimed at preventing missile launchings against Israel from various sites in Gaza Strip," Herzog told FOX News. "It may take time. Hamas will try to show in their arrogant way that they are still around, and our aim is to protect our citizens like any normal society would do."
The Associated Press contributed to this report.