In the agreement, Mr. Hastert acknowledged that he had given Individual A a total of $1.7 million before his indictment. After being questioned by banking authorities about large withdrawals in 2012, Mr. Hastert had begun to structure the cash withdrawals into smaller amounts to get around reporting regulations, gathering $952,000 in at least 106 bank transactions for under $10,000, the plea agreement says.

Two people briefed on an F.B.I. inquiry said this year that the money had gone toward covering up claims of sexual misconduct with a male student decades ago, but no details of the misconduct or about the identity of Individual A emerged in court proceedings or documents. By avoiding a trial, legal experts said, the details of any misconduct were likely to remain secret. Mr. Hastert, who was a coach and high school teacher from 1965 until 1981 in Yorkville, about 60 miles west of Chicago, was not charged with sex crimes, which are covered by statutes of limitation.

“The one thing this defendant clearly didn’t want to have was — essentially — his day in court,” said Jeffrey Cramer, a former federal prosecutor here who is not involved in the case. “He doesn’t want the reason for the structuring to come out. It seems that he spent $1.7 million or so to keep this secret, and his motivation now would seem to be keeping that secret.”

In Chicago, where corruption trials for public officials are painfully common, Mr. Hastert was only the latest in a series of officials to appear in this courthouse. Still, Dick Simpson, a political scientist and former Chicago alderman whose analysis of convictions since 1976 suggests that this is the most corrupt metropolitan region in the United States, said the remaining unknowns around Mr. Hastert’s case were unusual.

In the case of Jesse L. Jackson Jr., the former Democratic congressman who pleaded guilty in 2013 to fraud, details were made public down to the campaign money he spent on a fedora that once belonged to Michael Jackson, on stuffed animals from Build-A-Bear and on an elk head from Montana. In the case of Rod R. Blagojevich, the former Democratic governor who is serving time in federal prison for corruption, recordings of his phone calls, in all their salty language, were played at his trial.