May I ask a very basic question about public and private keys?

I am a signal processing engineer and I understand Galois fields, particularly GF-2. We call these "PN Sequences" or "linear-feedback shift register sequences" (LFSR) or "Maximum Length Sequences" in digital signal processing.

I understand what a primitive polynomial is and most of the properties of LFSR sequences. Like I know that the bit-reversal of a primitive polynomial is also a primitive polynomial. And I understand that the LFSR must go through all bit patterns, except all zeros, before repeating.

My question is precisely how are the public and private keys determined in public-key encryption methods? My crude (and possibly mistaken) understanding is that a private party uses some algorithm to find two independent primitive polynomials with a lotta bits (like 128 or more). One of those primitive polynomials will be their secret private key and the product (in the GF-2 sense) of the two primitive polynomials is the public key. Is that correct?

If it's not correct, can you educate me a little?