Al Jazeera's Ayman Mohyeldin, speaking from Jerusalem, said there had been concerns in Israel that the recent border breach between Gaza and Egypt had made southern Israeli towns more vulnerable to attacks.
In a telephone interview, Abu Fouad, a spokesman for the Brigades, said the attackers sneaked into Egypt after the border breach, then crossed into Israel using unspecified "private contacts".
He also claimed the group had more fighters inside Israel waiting to strike.
Just hours after the suicide attack, three members of the Popular Resistance Committees were wounded in an Israeli air raid in the Gaza Strip.
Reporting from the scene on Monday, David Chater, Al Jazeera's correspondent in Israel, said: "The police have moved people away because they are trying to take the explosive vest off one of the men."
He said Ehud Barak, Israel's defence minister, had visited the scene of the bombing.
A unit of the Brigades in Gaza, identifying itself as the "Army of Palestine", later released a video showing the two attackers in a farewell message.
Wearing military-style fatigues and clutching an assault rifle, Loai al-Aghwani, 20, from Gaza City said in his videotape that he hoped his participation in the attack would "restore dignity to the Palestinian people".
A video broadcast from the scene in Dimona, aired later on Israeli television, showed a close-up of the face of the second attacker who was shot dead after the bomb went off.
The body and facial features shown on the latter video appeared larger and older than al-Aghwani on the "Army of Palestine" video.
"The alternative to the blockade, bloodshed and violence is peace with the Palestinians"
Yellow Fatah flags flew outside his Gaza home on Sunday as al-Aqsa Martyrs Brigades members fired in the air in tribute to him.
Al-Aghwani's mother, Ibtissam, held up a picture of him as a young teenager, while male relatives scolded her for crying, saying she should be proud.
The Brigades in Gaza denied al-Aghwani and the other bomber, Musa Arafat, 23, from the Gaza town of Khan Younis, reached Israel from Egypt.
But Arafat's mother said her son had telephoned her from the Egyptian town of el-Arish.
Ibtissam too said her son had gone to Egypt three times after the border was breached, and she last saw Loai on Wednesday afternoon.
He gave no indication that he was about to embark on a suicide mission, she said.
In Gaza, Ayman Taha, a Hamas spokesman, rejected suggestions that the bombing will hurt Hamas' chances of reopening Gaza's border with Egypt.
"The suicide bombings were there before the closures and the resistance used every opportunity to make these glorious acts," he told the Associated Press news agency.
"They show the Palestinians can respond to the enemy and their crimes."
|Separately, Israeli troops killed two Islamic|
Jihad fighters in Jenin on Monday [AFP]
Mahmoud Habbash, minister of social affairs in the Ramallah-based Palestinian government, told Al Jazeera that he held Israel responsible for the "violence and escalation".
Jibril Rajoub, a member of Fatah's revolutionary council and a former senior security official, said the only way for Israel to prevent attacks was to make peace with the Palestinians.
He said: "We hope the Israelis will realise that the blockade, killings, raids and what is going on every day in Gaza Strip and the West Bank will not bring them security, peace or stability.
"The alternative to the blockade, bloodshed and violence is peace with the Palestinians."
The Israeli government vowed to "continue to fight terrorism by all necessary means".
"Terror organisations have again shown their true face ... they strike civilian population centres with the intention of killing innocent civilians," Arye Mekel, a foreign ministry spokesman, said.
Earlier on Monday, Israeli troops killed two fighters belonging to Islamic Jihad near the West Bank city of Jenin.
Islamic Jihad announced over loudspeakers that Ahmed Abu Zeid and Imad Zakarneh, its local leaders, were killed in the incident.