in the first Bitcoin implementations (even the first one) there was used a random generator to generate the private keys and out of these the public keys and addresses..
so, no magic algorithm but randomnes and for each transaction an own key..
but maybe i missed your question..
Thank you very much, yes, you correctly understood my question. I would like to speculate on this topic:
Those. when Satoshi began to mine, and every 2-5 minutes a new block appeared, and he received a reward for the proof of his work - she came on! a randomly generated wallet !, this wallet was generated by the customer himself, the "first miner" btc, right?
and these keys (private key) from this purse were automatically recorded somewhere? tell me more pls.