Well, the fact is the player builds itself the public key from the private key at the beginning of the game, somehow choosing itself the randomness used in the key generation process. I was hopnig this gives a bit of an idea that public key is derived from private key.

In that sense the public key is a "mix" of the private key. Then the cipher is the message plus a "mix" of the public key. By transitivity the cipher is the message plus a "mix" of the private key; and the private key is simple enough so that it can be "unmixed"...