You're being a mathematician. Be a cop instead.
Police manage to arrest people all the time for, say, murder, even
though mathematically there are lots of people who could have committed
the crime. Perhaps 10 different people have had to disclose shares of
the key to Inspector Lestrade. But only one has been observed
scribbling clocks in a reporter's NY Times, and hanging around parking
garages at weird hours.
Put another way, legal proof is not the same as mathematical proof. To
use geeky language, cops should, can, and will look at out of band
information. Maybe using secret-sharing will increase the pool of
suspects. But the increase is by less than you think, since even if
just one person had the key, many more people are likely to be aware of
*some* interaction. The court (or police) order isn't served directly
on precisely the right individual; rather, it's served on the company,
which brings in its general counsel, several layers of pointy-hairs,
etc., until the key-holder is located. (The CEO doesn't keep
operational keys in his or her safe; rather, it's likely to be some
bearded nerd in the operations dept. who hangs out on cypherpunks....)
To be sure, the order may not specify that the key is wanted, but
finding the right individual is hard without *someone* else knowing
what's going on.
--Steve Bellovin
