EXCLUSIVE

The hacker group that claims responsibility for exposing private financial information about Michelle Obama and many other celebrities has issued an ultimatum to the U.S. Government -- either declare war on North Korea or they will detonate bombs at the White House ... TMZ has learned.

We're told the group -- which claims responsibility for swatting Rihanna, Diddy, Ryan Seacrest, Justin Timberlake, Paris Hilton, Chris Brown, Tom Cruise and others -- contacted authorities through 911 and threatened to detonate 8 devices around the White House perimeter if President Obama does not declare war within 24 hours.

There was a tweet that went out on the Associated Press' twitter account saying, "Two Explosions in the White House and Barack Obama is injured." The White House quickly discredited this as an "errant tweet." We're told the hackers are claiming credit for hacking AP's twitter and posting the message.