Ehud Olmert, the Israeli prime minister, said: "We will study this incident ... and we will draw the conclusions in order to direct our security officials to continue their battle without respite against the terrorists and their commanders."
Jacky Rowland, Al Jazeera's Jerusalem correspondent, said: "Israel as a whole has been put on a high state of alert now."
Bruno Stein, Eilat's police commander, said the police believe there could be more bombers in the town.
He said: "Our assumption is that it's not one bomber, and there might be more bombers in Eilat right now."
Claims of responsibility
Islamic Jihad, al-Aqsa Martyrs Brigades and a previously unknown group calling itself the Army of Believers each said they carried out the attack.
Khaled al-Batsh, a senior Islamic Jihad leader, said the attack was "a natural response to the continued crimes by the Zionist enemy".
Spokesmen from Islamic Jihad and al-Aqsa Martyrs Brigades claimed joint responsibility for the attack.
In a phone call to AFP offices in Gaza, the spokesmen named the suicide bomber as Mohammed Faisal al-Siksek, a twenty-one-year-old from the Al-Shujaiyah neighbourhood of Gaza City. The group had earlier said that he was from the West Bank.
The Jihad spokesman said al-Siksek belonged to the Islamist group but neither organisation specified how the bomber had managed to get from the Gaza Strip to Eilat, access to which is strictly controlled by Israel.
Islamic Jihad is not one of the Palestinian actors subject to a Gaza ceasefire agreed upon in November between Israel and some Palestinian groups.
The group has demanded that any truce also cover the occupied West Bank.
Nour Odeh, Al Jazeera's correspondent in Gaza, said: "Eyes are mainly looking towards the Islamic Jihad and a new offshoot of the Al-Aqsa Martyrs Brigades.
"Senior Islamic Jihad movement figures have been killed or detained in the past few weeks, and so Islamic Jihad in particular has threatened [a] response.
|The bomber who struck the Eilat cafe-bakery|
may have come in through Jordan [Reuters]
"[Palestinians] are expecting reprisals from Israel. So far we know that a ceasefire of sorts has been in place in the Gaza Strip between Palestinians and Israelis ... As for the West Bank, no such ceasefire exists.
"Some Palestinian groups may say that if the attacker came from the West Bank he would have been under no obligation to observe a ceasefire. Reprisals may come in Gaza, in the form of targeted assassinations [against Palestinian activists]."
Ismail Radwan, a Hamas spokesman, said the attack was a "natural response to the occupier's crimes against our people", although Hamas itself, which formed a Palestinian government last March, has not claimed a suicide bombing within Israel for nearly two years.
The attack is the first suicide bombing in Israel since last April when a suicide bomber killed 10 people in Tel Aviv.