I'm embarrassed to admit this, but I didn't. I guess it could go with whatever intractable scientific problems currently require large computational brute-force approaches [1]: fluid dynamics (including combined free/forced convective heat transfer), protein folding, weather, electron wave functions, ... .
The general approach would be:
1) Base a cipher on the difficult aspect of some scientific problem.
2) Publish the cipher.
3) Wait until someone finds a shortcut to decrypting the cipher without the private key.
4) Simplest explanation that fits the data!
Remaining problem: It's not enough for a scientific problem to be difficult, it must have a trapdoor aspect to it: something that, for the entire problem class, makes it easy to solve if you know it. I'd have to think about how to get that to work for the one domain I'm most familiar with (fluid dynamics).
[1] I guess they're not technically brute force, because they don't try a bunch of solutions in parallel, but you have to do a lot of grind to find the solution that fits all the constraints.
The one which nagged at me was whether cryptographic methods would have been useful in deducing the periodic table. I have no idea whether they would have helped, but it's a relatively simple pattern underlying a lot of data.
More generally, are there past problems where cryptographic methods would have helped?
Short version: Why can't cryptanalysis methods be carried over to science, which looks like a trivial problem by comparison, since nature doesn't intelligently remove patterns from our observations? Or are these methods already carried over?
Long version: Okay, I was going to spell this all out with a lot of text, but it started ballooning, so I'm just going to put it in chart form.
Here is what I see as the mapping from cryptography to science (or epistemology in general). I want to know what goes in the "???" spot, and why it hasn't been used for any natural phenomenon less complex than the most complex broken cipher. (Sorry, couldn't figure out how to center it.)
EDIT: Removed "(cipher known)" requirement on 2nd- and 3rd-to-last rows because the scientific analog can be searching for either natural laws or constants.