"Invalid/missing hash was encountered" for documents

Reviewing our logs recently we discovered that we were getting an unusually large number of messages to the tune of
ERROR MediaRequestProtection: An invalid/missing hash value was encountered.

As we all know, Sitecore 7.5 added the concept of a hash which is required when resizing media items, so naturally we assumed that some images were being included incorrectly on a page somewhere, or the secret was incorrectly configured.  However, on closer inspection we realised that it wasn't actually images at all, but documents (which certainly aren't being resized, and surely don't need a hash).

Documents (mostly PDFs) were being uploaded to the media library, and then in a rich text field somewhere on the site, an author was linking to the document using insert link -> media item.

Upon reaching out to Sitecore, they were quick to reply that this was actually a bug in the current version of Sitecore (8.1 update 2 - 160302).  So if you're having a similar issue and want to clean out your logs a bit, you can contact support for the patch, and/or use the reference number 438674.