To spell it out further: if the two processed outputs from different accounts are bit-by-bit identical then it is impossible to tell them apart. Ie they cannot contain identifying data to you.

They might still contain data that groups those multiple accounts together though. Eg everyone in X university gets the same watermark.

Or encode the current date or week into the pdf. They could then correlate their log data with the day or week of retrieval

Doesn't work if the watermark has a time-based component.

I guess I should have expanded: a watermark is useful to reduce the number of possibilities of who could have released it. Over time, the more information is gained, the closer you are to a singleton (the result of the intersection of sets). Whether that takes 1 or 10 watermarks does not make a real difference over time, especially when specialized websites release thousands of such PDFs.

So if you have a watermark that associate files with the time they have been downloaded, and download it from multiple networks etc. at the same time, and remove the diff from these versions, you'll still have the time-based watermark in the resulting file, so you'll leak when you downloaded it.

Well sure. Try not to download them at the same time.

Problem is you only have to fail once to leak information, and there are only a finite number of bits of information you can leak before your identity is known.

Of course it does. If two files are identical then by definition there is nothing in them that can be used to tell them apart.

There is still the metadata the operating system stores with the file, even when the file is 1:1 identical. When you use Apple AirDrop to send a file to a friend the OS remembers who sent the file to your friend. Now if the police ever looks at your friend‘s computer they can see who this file came from. Or your friend saves it onto a usb formatted with an Apple File System the metadata is still preserved.

Sounds like you have an operating system problem not a watermark problem.

Nit: This is a function of the file system, not the operating system.

I don't see why.

What if the files are still different?

https://en.wikipedia.org/wiki/Steganography

Then your tool is incomplete. Steganography works by encoding information into side channels, like precise positioning of words in the PDF, fonts, etc...

Your tool can be as crude as running the document through pdftotext and only keeping the text output. It can just throw away all these possible side channels that are not relevant to the actual content.

Others mentioned other more sophisticated side channels, as rewording sentences, but I'm sure authors would not welcome that.

If the bits are identical, where does the hidden information live?