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How do public and secret keys work together in an asymmetric cryptosystem?

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A Vigenere cipher table displays the alphabet in both the top row and left column, with rows shifted to create a grid for encryption and decryption. Highlighted letters illustrate how to use the table. An explanation and example using the words cipher, key, and plaintext appears to the right. — cryptology In encrypting plaintext, the cipher letter is found at the intersection of the column headed by the plaintext letter and the row indexed by the key letter. To decrypt ciphertext, the plaintext letter is found at the head of the column determined by the intersection of the diagonal containing the cipher letter and the row containing the key letter.

In an asymmetric cryptosystem, also known as public-key cryptography, the transmitter and recipient use a matched pair of keys: one public and one secret. This system solves the key distribution problem by allowing users to publish an encryption key in an authenticated directory while keeping the corresponding decryption key private.12

The two keys work together based on the following principles:

Secrecy Channel: To send a private message, a sender uses the recipient's public key to encrypt the data. Because the system is mathematically designed so that only the corresponding secret key can decrypt it, the message remains secure even if the public key is intercepted.1
Authentication Channel: Conversely, if a user encrypts a message (or a digital digest) with their secret key, anyone with access to the corresponding public key can decrypt it. This confirms the identity of the sender, as only the holder of the secret key could have created the cipher.13

Because asymmetric encryption is computationally intensive, it is often used in hybrid systems to exchange a temporary session key, which then facilitates faster communication via symmetric encryption.3