Public-Key Cryptography
A system of paired keys where one key can be shared openly and the matching secret key is kept private.
The private key is a large random number and the public key is derived from it by elliptic-curve multiplication, an operation that is easy to perform and infeasible to reverse, so publishing the public key reveals nothing about the secret. Anyone can use the public key to check a signature, while only the holder of the private key can create one, which is how a network verifies that a transfer was authorized without any account registry. Bitcoin and Ethereum use ECDSA over the secp256k1 curve for transactions, and Ethereum's consensus layer uses BLS signatures over BLS12-381 because they can be aggregated into one compact signature for thousands of validators. Addresses are derived from public keys, so the public half is an identifier rather than a secret, and because there is no issuer there is no reset or revocation.
In practice
A wallet proves ownership of an address by signing a challenge message with the private key, and the verifier recovers the public key from the signature and compares the derived address.
The common misunderstanding
Sharing an address or public key does not put funds at risk; it cannot be reversed into the private key and it grants no ability to spend.