It’s called the $600 million challenge.

Decrying an out-of-control deficit, Governor Paterson is ordering state lawmakers to return to Albany for an emergency “economic session” next month and brainstorm ways to come up with that much money through “savings initiatives.”

Fortunately for the legislators, the governor didn’t specify what he meant by “initiatives,” leaving them with plenty of room for interpretations that don’t entail actual budget cuts.

Lawmakers interviewed said they have quite a few ideas for how to raise the $600 million: Democrats in the Assembly are partial to an income-tax hike on millionaire residents, while Senate Republicans say they would prefer a crackdown on Indian reservation tax evasion and an expansion of gambling operations.

For both houses, cutting spending — and running the risk of inflaming voter discontent in an election year — doesn’t appear to be under serious consideration. Lawmakers said they would rather hunt for change under the cushions than saw apart the couch.

“The millionaire’s tax, that’s my perfect resolution,” a Democratic assemblyman of the Bronx, Ruben Diaz, said. “Nobody is hurt too bad, and you get more than $600 million.” Mr. Paterson has said increasing taxes on rich is a “last resort.”

Said a Republican senator of Brooklyn, Martin Golden: “Before you cut money from schools, you want to get the tax dollars from the Indian reservations, that’s for sure.” By law, New York can collect such taxes on cigarette sales to non-Indians, but the state has chosen not to enforce it.

Another Republican senator, William Larkin of Orange County, said New York “ought to jump-start those video lottery terminals at Aqueduct,” adding that the state has taken too long to select a gaming operator to run casinos at the Queens thoroughbred racetrack.

Some lawmakers say they shouldn’t pass any legislation until after voters go to the polls in November, ostensibly so they have enough time to verify the Paterson administration’s calculations and economic forecasts.

“We’d like to confirm that information with our advisers,” a Democratic assemblyman of Brooklyn, Joseph Lentol, said.

Albany is on track to hike spending next year by 11% but is expecting tax and fee revenues to rise by only 2%, leaving a shortfall of $6.35 billion in the 2009-10 fiscal year. More than half of the gap is a product of increases in education and Medicaid spending. Public school aid itself is on pace to rise by $2 billion, or more than 10%.

Mr. Paterson, who yesterday was in Washington, D.C., speaking before the National Press Club, told reporters that “there’s a new sheriff in town,” saying New York is “going to make these cuts as soon as they need to be made.”

Lawmakers said they didn’t think there should be any rush.

“We’ve been having tough budgets and multibillion-dollar deficits in New York since 1990,” a Democratic assemblyman of Manhattan, Richard Gottfried, said. “I think we have over the years cut quite a lot. The notion that New York State’s budget is laden with things that can be painlessly dispensed with is mythical.”