More than 300 of former Secretary of State Hillary Rodham Clinton’s emails — or 5.1 percent of those processed so far — have been flagged for potential secret information, the State Department reported to a federal court Monday as the political furor continued to grow for the Democratic presidential candidate and her aides.

A top senator has demanded that Mrs. Clinton’s attorney detail the steps he used to protect classified information in her emails, which he kept on a flash drive at his office. ABC News reported this weekend that the Colorado company that helped run her server said there may be a full backup server, which might have many of the messages she said she deleted.

In the new court filing, the State Department said it is getting back on schedule for publicly releasing the Clinton emails after falling more than 1,000 pages behind in July, when the need to screen messages for secret information overwhelmed the department.

Now, the screening process is running smoothly, with five security agencies involved in the review. They have been through 20 percent of Mrs. Clinton’s messages, the department said. Officials told the court that they have found 305 messages — about 5.1 percent — that needed to be referred to the security agencies to determine whether they did, in fact, have secret information that needed to be redacted before public release.

“We’re taking this very seriously,” department spokesman John Kirby told reporters, though he refused to call the revelations of classified emails on Mrs. Clinton’s server “troubling” at this point.

The department’s next challenge will come later this week. A federal judge has scheduled a hearing for Thursday, when officials will have to detail the steps they have taken to try to track down all of the messages, and any other electronics that might still hold messages, from Mrs. Clinton and two top personal aides, Huma Abedin and Cheryl Mills, who served in the department with her.

Judicial Watch, a conservative public interest law firm that has sued to get a look at those communications, said the department has refused to say how thorough its search has been despite Judge Emmet G. Sullivan’s demands for more information.

“It appears as though the declarant made no effort whatsoever to find out what electronic devices the former head of the agency and two of her closest advisors used to conduct official government business for four years and where these electronic devices may be located or if they are still in existence,” Judicial Watch said in a filing asking for the hearing.

Judge Sullivan is one of several federal judges who, responding to open-records requests, have forced the State Department to be more forthcoming with Mrs. Clinton’s communications.

A Fox News poll released over the weekend found a public decidedly skeptical of Mrs. Clinton, with 58 percent saying she “knowingly lied” when she claimed her emails didn’t contain classified information and 54 percent thinking she put national security at risk.

Mrs. Clinton rejected the use of a State.gov email account during her tenure and instead issued herself an account on a server that she kept at her home in New York. She has since acknowledged that was a bad idea.

Prodded by the investigation into the 2012 terrorist attack in Benghazi, Libya, she returned to the State Department in December some 30,000 messages that she deemed government business. She said she deleted 32,000 other messages that she deemed private.

More than 60 of those messages have now had information classified, and more than 300, out of about 6,000 that have been screened so far, have been sent to intelligence agencies for review to determine whether they have information that needs to be redacted before release.

Mrs. Clinton has insisted that she never sent any classified information from her account at all and that none of the messages she received had information that was marked classified at the time — though some of it has since been designated.

“I was permitted to, and used, a personal email and, obviously in retrospect, given all the concerns that have been raised, it would have been probably smarter not to,” she told Iowa Public Radio last week. “But I never sent nor received any classified email, nothing marked ‘classified.’ And I think this will all sort itself out.”

Internal watchdogs have contradicted Mrs. Clinton’s account, saying messages clearly contained classified information, even if it wasn’t marked as such, and should have been kept more secure than on her own server.

More recently, Mrs. Clinton’s attorney, David E. Kendall, revealed that he kept the emails in electronic form on a flash drive stored at his office.

Sen. Chuck Grassley, Iowa Republican and chairman of the Senate Judiciary Committee, said Monday that raised all sorts of questions about security — even after July, when the State Department gave Mr. Kendall a safe in which to keep the flash drive.

Mr. Grassley said it’s questionable whether the safe was an acceptable level of security for information that now has been deemed top secret, and demanded to know what level of security clearance Mr. Kendall has, who issued it to him, and what level of clearance his associates who also had access to the flash drive possessed.

“Many intelligence community personnel, uniformed personnel and the American people may be at risk when classified material is not properly secured,” the senator wrote in his letter to Mr. Kendall, dated Friday. “Accordingly, it is very important that the Judiciary Committee fully understand the events that have transpired.”

Mr. Kendall’s office said he is out of the country. A message left with another law firm employee to whom calls were referred was not returned Monday.

Meanwhile, the State Department continues to try to make headway in producing the Clinton emails, insisting that the process is running smoothly and they are back on track after falling behind Judge Sullivan’s schedule for making all of the emails public.

The reviewers have screened about 20 percent of the 30,000 emails that Mrs. Clinton returned to the department, which means if the rate of potentially secret information remains steady, more than 1,500 messages will have to be sent to intelligence community agencies, known in government as “IC,” to screen out classified information.

“Out of a sample of approximately 20 [percent] of the Clinton emails, the IC reviewers have only recommended 305 documents — approximately 5.1 [percent] — for referral to their agencies for consultation,” the Obama administration said in court papers.

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