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Public and private keys are two very large numbers that (through advanced mathematics) have a unique relationship, whereby information encrypted with one number (key) can only be decrypted with the other number (key) and vice versa. In order to leverage this characteristic for security operations, once two numbers are mathematically selected (generated), one is kept secret (private key) and the other is shared (public key). The holder of the private key can then authenticate themselves to another party who has the public key.
Alternatively, a public key may be used by one party to send a confidential message to the holder of the corresponding private key. With SSH, the identity key is a private key and authorized keys are public keys.