Re-write paper to make it truly double blind

Check out double-blind peer review policies from the venue you'll be publishing in. Here's an example from Elsevier. In a nutshell, you want to prevent any reader from immediately connecting the paper to your identity by e.g. not listing your affiliation, anonymizing self-references, removing funding sources, and so on; but you're not aiming to "beat Google".

I would think that this would be an unusual practice, at least in the field of mathematics. Almost all journals would allow for publication of results previously presented by an author in his/her dissertation. There may need to be appropriate attributions made, and you likely could not place the published version of the paper in your dissertation, but if your dissertation is already completed it would seem this would not be an issue.

I published two papers based on my dissertation in abstract algebra and character theory. The topic was such that someone could Google the titles of the papers and usually find my dissertation as well. The two texts are not identical, but they are certainly similar enough that an alert human could quickly determine the authorship of the papers given my dissertation, even without definitively knowing I wrote both papers.

Some questions I have:

If the co-collaborator does not know enough to re-write the paper in "new language," why is he even being given authorship?

Is the topic specific enough that the paper could not be somewhat feasibly re-written in places in order to make it slightly different?

Based on the fact that this paper is in your dissertation, I assume you are the principal/first author?

A low-quality answer, but: in mathematics, my field, spending time to obscure the authorship of a paper is misguided and un-necessary: established people have a viewpoint and style that would be unmistake-able, for example.

More generally, the goal of work/research is not anonymity, but progress in our collective understanding. Notably, this involves people, not anonymous entities. Referees can barely be anonymous, and it's even harder for authors to be anonymous. This cannot possibly be a high priority...