Normally "private-key cryptography" is the antonym of "public-key cryptography", which is to say, conventional symmetric-key cryptography such as AES or Enigma. That's what "private-key cryptography" redirects to on Wikipedia. This article is about Astra breaking a message encrypted with a private-key cryptographic algorithm, but one that was already known to be weak. I'm guessing that if you had messages encrypted with weak public-key algorithms it could break those too.

But they said new forms, I think that implied not the ones humans already discovered/designed