Symmetric and asymmetric keys

A symmetric cipher uses one key for both directions, so both sides must hold that exact key before the conversation starts. If they have never met and have no private channel, agreeing on the key is the hard part, because sending it over the network hands it to anyone watching. This is the key distribution problem, and asymmetric keys were invented to solve it.

A private key file usually holds both halves, so the public half can be derived from it at any time with -pubout. The result is the key you hand out freely. The reverse is not possible: nothing can derive the private half from the public half, which is what makes the pair safe to split. Running this command never reveals the private key.

RSA can only encrypt a short message, shorter than the key itself, and it is slow compared with a symmetric cipher. So it is used to carry a short thing, and that short thing is a fresh symmetric key for the session. The bulk of the data then travels under the fast symmetric cipher. TLS does exactly this: an asymmetric handshake, then symmetric traffic.

The public half can lock a message to you and can check a signature you made, and it cannot do the reverse, so giving it away costs you nothing. The private half is the ability to read your mail and to sign as you, so it stays on your device behind a passphrase. Keeping the public half secret has no security benefit and simply stops people from contacting you privately.