Clean install, but then I restored data, settings, etc. from old computer (using mint backup tool). I didn't have this problem at that time. I did add a PPA, (details at this post:)viewtopic.php?f=46&t=72841
and it messed everything up. But after I got things running again, I did a ppa-purge. Also dist-upgrade. I'm certain, though, that I installed some programs since then, and didn't have this error message come up. I did install 2 programs from .deb files: the latest versions of skype (to see if sound problems were fixed), and the latest Anki (again, to fix some problems in the repository version).
If mint 10 has the equivalent of the APT::Get::AllowUnauthenticated (must be set to 0 or everything is treated as unauthenticated) variable, I'm guessing that it somehow got set to 1 and I need to reset it, but where to find it???