  In the postprocessing of quantum key distribution, the raw key bits from the
mismatched-basis measurements, where two parties use different bases, are
normally discarded. Here, we propose a postprocessing method that exploits
measurement statistics from mismatched-basis cases, and prove that
incorporating these statistics enables uncharacterized qubit sources to be used
in the measurement-device-independent quantum key distribution protocol and the
Bennett-Brassard 1984 protocol, a case which is otherwise impossible.
