JERUSALEM, Oct 17 (Reuters) - A major Israeli attack on Gaza would not curb growing extremism in the Palestinian enclave, with the ruling Islamist group Hamas itself struggling to quell radicalism, a senior Israeli official said on Wednesday.

Voicing concern about a recent influx of increasingly potent weaponry into the Gaza Strip, the director of Israel's Ministry of Strategic Affairs told reporters that international pressure was needed to try to put an end to militancy.

Yosef Kuperwasser said more than 800 rockets and mortar rounds had been fired into southern Israel since the start of the year, with organisations such as the Islamic Jihad taking over from Hamas as the main perpetrators of these attacks.

Israel has regularly targeted Palestinian militants in the coastal territory, and on Saturday killed two men identified as the most senior al Qaeda affiliates in Gaza.

But in a rare admission of the limitations of Israeli firepower, Kuperwasser said military might would not solve the root causes.

"If worst comes to worst, we can (launch) a much wider operation in Gaza. But that is not going to really solve the problem," he said.

"There is a wide and deep problem of hate indoctrination that produces more and more terrorists all the time."

Hamas has governed Gaza since 2007. It rejects permanent peace with Israel and the two sides fought a three-week war from December 2008 into January 2009. An estimated 1,400 Palestinians and 13 Israelis died in the conflict.

Israel maintains a tight blockade on the territory.

Kuperwasser said that the Hamas leaders had tempered their desire to strike at Israel, aware of the heavy cost the general Palestinian populace was paying in the violence. But he said the smaller groups were far less bothered by such concerns.

"Most of the activity is coming now not from Hamas," he said, adding that Islamic Jihad was receiving a relatively bigger share of the weapons that had "been pouring into Gaza".

GROWING ARSENAL

He said that since the downfall of Libyan leader Muammar Gaddafi last year, huge quantities of arms from the north African country had been smuggled into Gaza, adding to homemade weaponry and continued arms supplies from Iran.

In a sign of how this arsenal was growing, Israel has said Gaza militants shot a shoulder-fired Strela missile at an Israeli helicopter last week. It did not hit the target.

Kuperwasser said Israel had assumed these missiles had been in Gaza for some years, but confirmed that this was the first time one had been fired. "Maybe because they have so many right now, they changed their policy about using them," he said.

The ministry director, a former intelligence officer, said Hamas itself was worried about the gathering strength of other armed factions, including ultra-conservative Salafist militants, but did not know how to deal with them.

He said this hesitancy was partly "a reflection of Hamas's assessment of how strong these guys are. They are not very comfortable confronting them ... They are not sure that if their people get the order to do it, they will really do it."

He added that Islamist extremism was flourishing in the enclave, which is home to some 1.6 million people - 44 percent of whom are aged 14 or under, according to the CIA handbook.

"Radicalism is gaining power ... This atmosphere drives people to extremist ends," he said. (Editing by Jon Hemming)