Asymmetric encryption relies on two related keys: a public key and a private key. The idea is that it's easy to perform certain calculations using the public key, but nearly impossible to reverse those calculations without the private key.

You mentioned a 'random number,' which in encryption typically refers to a nonce or initialization vector (IV). This ensures that even if the same data is encrypted multiple times, the output will be different each time (because the number, which is random, will be different). This randomness prevents certain attacks like replay attacks.

To touch on why random numbers are important: if you can regenerate the number (because it wasn’t actually random), then you’re vulnerable to replay attacks.

About the mathematical relationship between the keys: the private and public keys are mathematically related, but they can’t be used to derive one from the other. This makes it very difficult (essentially impossible) for someone to reverse the encryption process without access to the private key.

So they basically share the public key with you, as well as the random number, so that you can validate that the data’s serialized/encrypted representation is derived from the private key. This works even though you don’t have direct access to the private key.

When we say things are impossible, we’re really just saying we calculated how long it would take and nobody should live that long… perhaps even, the universe hasn’t been alive long enough to brute force decrypt some stuff via traditional computers.