Story highlights NEW: U.S. Secretary of State Hillary Clinton blames the attack on Kurdish separatists

U.S. Secretary of State Hillary Clinton blames the attack on Kurdish separatists Prime Minister Recep Tayyip Erdogan says 24 soldiers died, 18 were hurt

U.S. President Obama "strongly condemns" the attack on a close ally

It follows recent attacks claimed by Kurdish rebel groups

Twenty-four soldiers were killed and 18 injured during an attack early Wednesday morning in southeastern Turkey, Prime Minister Recep Tayyip Erdogan said.

Rockets were launched at security forces and military sites in the town of Cukurca, in Hakkari province, an official with the provincial governor's office and Turkish President Abdullah Gul said.

Gul blamed terrorism when he spoke about the attack during a televised address.

"Our determination is certain. Those who think that democratic improvements in Turkey are achieved as a result of terrorism are making a big mistake," Gul said. "It is our decision to continue the struggle against terrorism without giving any concessions."

He warned that "those who inflict this pain on us will endure pain many times over" themselves.

U.S. Secretary of State Hillary Clinton blamed the attack on militants from a Kurdish separatist group, the Kurdistan Workers Party, or PKK, as she gave her condolences to the families of those killed.

"The United States condemns in the strongest possible terms the recent attacks by the PKK in Turkey's Hakkari province," she said in a statement Wednesday.

"We will continue our strong cooperation with Turkey as we work to combat violent extremism in all its forms and safeguard the security of peace-loving people everywhere."

Erdogan said "wide-scale operations, including hot pursuit as defined by international law" were continuing in the region, as the military seeks those responsible.

Speaking to reporters in Ankara, he said the "Turkish state will breathe down the neck of those who secretly or openly support or aid terrorism."

He added: "Turkey always played an active role in the international fight against terrorism and it expects also the international community and all countries to give support and active cooperation in combating terrorism."

Urging the country's people to show unity, he said the process could be lengthy but other countries had defeated terrorism in the end.

U.S. President Barack Obama also strongly condemned what he called "an outrageous terrorist attack against Turkey, one of our closest and strongest allies," in a statement released by the White House.

"The United States will continue our strong cooperation with the Turkish government as it works to defeat the terrorist threat from the PKK and to bring peace, stability and prosperity to all the people of southeast Turkey," Obama said.

"The people of Turkey, like people everywhere, deserve to live in peace, security and dignity."

The attack comes amid escalating tensions between the government and elements of the country's Kurdish minority.

Two weeks ago, lawmakers voted to extend authorization for the Turkish military to carry out cross-border attacks against Kurdish rebels in northern Iraq. Turkish police also arrested more than 100 people across the country suspected of links to PKK rebels.

The Kurdish Peace and Democracy Party issued a written statement Wednesday, saying: "We are calling on both the government and the PKK to urgently stop the war without wasting one more second. Turkey's most urgent need is peace."

The Kurds represent the largest ethnic minority in Turkey. For decades, they were the target of repressive government policies, implemented by officials who sometimes referred to them as "mountain Turks."

Until just a few years ago, it was illegal to speak Kurdish on radio and television in Turkey. Under Erdogan, the government has tried to improve relations by launching a state Kurdish language TV station in 2009.

Some observers have sounded the alarm about escalating tension between Turkey and its Kurdish minority, warning it may reignite a conflict that has simmered since 1984 and claimed more than 30,000 lives.

Erdogan canceled a trip to Kazakhstan planned for Wednesday in the wake of the attack.

"We changed our plans after hearing about the very sad incident that happened early this morning. ... Our heroes were killed during a cowardly attack," said Deputy Prime Minister Bulent Arinc. He earlier said 25 soldiers were killed.