A former head of the U.S. Justice Department’s criminal division who resigned amid fallout from Congress’s 2012 investigation into the “Fast and Furious” arms-trafficking sting has become an adviser to a bitcoin company.

San Francisco-based BitFury Group, which provides digital-currency computing and transaction-processing services commonly known as bitcoin mining, said Wednesday that Jason M. Weinstein would join the firm’s strategic advisory board. Mr. Weinstein will help the company develop some of the unique security capabilities...