Publishing open-source code similar to unfinished master's work
As long as your new project is entirely original and does not infringe any patents, there is no issue with releasing it.

It is highly unlikely your professor or old institution would possess any patents for a simple program. It's impossible for me to state anything with certainty as I do no know the nature of the software, but software patents are already fairly uncommon and legally questionable. You could look up the concept and professor's name somewhere like Google Scholar to be sure.

The primary concern would be if you re-used source from the time you were at the institution. This depends on the agreements you made with the university; if you are reusing components, you should review what you signed related to your master's program and possibly contact the institution's legal department for advice. You could of course cut out such components as well, which is what I would suggest.

The fact you learned something you later applied does not give those who taught it to you any legal or moral rights over the work. Especially considering it sounds like you did all the leg work in the first place.

If you are worried in an academic context of accusations of plagiarism, you could simply acknowledge where the concept came from / previous work in a README file. If you are not infringing patents or copyrights, there is no reason not to.