The two bombers who carried out the attack in Dimona on Monday came from the Gaza Strip, sources in the Strip told Ynet.
One of the bombers was identified as Mussa Arafat, a Khan Younis residents from the Abu Ali Mustafa Brigades, the military wing of the Popular Front for the Liberation of Palestine.
"The Palestinian Authority expresses its full condemnation of the Israeli military operation this dawn in (the Palestinian village of) Qabatiya just as it condemns the operation that took place today in Dimona," a statement from Abbas's office said.
The al-Aqsa Martyrs' Brigades, Fatah's military wing, the Abu Ali Mustafa Brigades, and the United Resistance Brigades have claimed responsibility for Monday's terror attack in Dimona, Hizbullah's television station al-Manar reported.
Abu al-Walid, a senior al-Aqsa Brigades official, said in a phone conversation with al-Manar, "We succeeded to carry out the attack today and we take responsibility for it, along with the Abu Ali Mustafa Brigades and the United Resistance Brigades, in response to the occupation and the killing of children and innocent people in the West Bank and the Strip."
According to the report, the first suicide bomber was a Fatah member, while the second one was sent by the other two organizations. It was also reported that the two terrorists were residents of the Gaza Strip.
Hamas referred to the Dimona attack as "a heroic act", which constitutes "a natural response to the crimes of the occupation."
Military sources have been warning over the past few weeks, since the Rafah border was breached, that terrorists leaving the Gaza Strip into the Sinai Peninsula would try and infiltrate Israel in order to carry out a terror attack.
There is no ground obstacle at the Israel-Egypt border and the plan to build a fence has not been budgeted over the past few years.
As a first step, the IDF decided to temporarily close Road 10 along the Israel-Egypt border to civilian traffic.
Roee Nahmias and Reuters contributed to this report