I finally worked out what was happening, or at least the cause of it.

For years now I have been using the Rainbow portal on my localhost at work as a personal portal (aside from my personal webpage which is DNN).  So when I started to try to transfer all my links and so on to DNN on my localhost, I would have both Rainbow and DNN open at the same time (and be editing both, too).  Some word or other I read in the forum seemed to suggest that there was  problem with having multiple instances of DNN open for editing, and it occurred to me that DNN and Rainbow share a common heritage: IBuySpy.  So this might mean that they are closely enough related to make having both open for editing at the same time.  Accordingly, I tried editing the DNN, really putting it to use, without Rainbow being in use, and worked on it pretty much steadily for quite some time without experiencing a single problem.  Then I fired up Rainbow, without logging in, and could still edit DNN with no problem.  As soon as I signed in to Rainbow, however, the next time I tried editing DNN I got the cryptographic exception.  I did a lot of verfication of this, and am now convinced that was my problem.

Wow.