Senator Marco Rubio, a Republican, has said that the authorities decided to get around those concerns by issuing a general notice to all counties that an intrusion had occurred.

“I think they should be named,” Mr. DeSantis said.

That the governor was asked to sign a nondisclosure agreement to receive classified information is not unusual. The F.B.I. makes similar requests of police chiefs when discussing cases involving classified terrorism threats, for example. Typically, the F.B.I. does not release the names of hacking victims for privacy reasons.

Less clear is why information, the gist of which has already been made public in the Mueller report, would be deemed too sensitive for public disclosure. There was an immediate flood of questions on social media, with many wondering how the public could feel confident that the voter rolls had not been tampered with and that local elections supervisors had taken the necessary steps to prevent a similar hack in the future — without knowing which counties had been breached.

“This is not acceptable, to keep secret attacks on the most public of our political processes: our elections,” said Ion Sancho, a Democrat and the former elections supervisor of Leon County, which includes the state capital, Tallahassee.

On Sept. 30, 2016, while still in office, Mr. Sancho took part in a conference call among Florida elections supervisors in which, he said, the F.B.I. told local officials that no Russian hacking had taken place. The F.B.I. had asked elections supervisors to keep the content of the call confidential, but Mr. Sancho spoke about it at the time anyway.

“I’m not a member of a law enforcement agency — I’m a member of the elections profession,” he said on Tuesday. “I did talk about it, because it seemed to me we needed that kind of impetus if we’re going to prepare and ensure we’re going to protect our election system, which we still have not done.”

Not one of Florida’s 67 counties has admitted to being one of the hacking victims. The intrusion was first made public last year by former Senator Bill Nelson, a Democrat, who came under stinging criticism from his re-election opponent, former Gov. Rick Scott, who dismissed his claim as unfounded. The state, he said, had not been informed about any Russian hacking. Mr. Nelson said then that he had come across the information as a member of the Senate Select Committee on Intelligence — but like Mr. DeSantis now, could not reveal more.