# RSA Asymmetric-Key Algorithm

RSA is a well-known asymmetric-key algorithm. It uses the factoring of large numbers into large primes as its one-way function. RSA has an interesting property: If the private key is first applied to a message and then the public key is applied to the result, the original message is obtained. Alternatively, if the public key is applied to a message and then the private key is applied to the result, again the original message is obtained.

Thus, RSA can be used for both encryption and digital signatures. In encryption and decryption, the public key is used to encrypt data, and the private key is used to decrypt data. For digital signatures, the private key is used to digitally sign, and the public key is used to verify signatures.

# RSA for Key Encryption

To encrypt using the asymmetric keys, you apply the recipient's public key, which is known to you because it's public, to the message. The holder of the private key can apply their key to recover the message. It is secret from everyone else because the private key is, well, private.

In practice, messages are not typically encrypted directly with an asymmetric key. The data size is limited, based on the size of the key. Breaking up the message into smaller pieces is possible but impractical because asymmetric-key operations are typically very slow.

The usual pattern is to encrypt a symmetric key with the asymmetric public key, send the encrypted symmetric key to the recipient, and then encrypt the message with that symmetric key. The recipient decrypts the symmetric key using their private key and then decrypts the message with the much faster symmetric-key algorithm.

# RSA for Digital Signatures

A digital signature is similar to an HMAC but has additional properties. To create an asymmetric-key digital signature, the signer applies their private key to a message. The verifier applies the public key to the signature to recover and verify the message.

A digital signature permits the recipient to know that a message has integrity and is authentic, qualities that an HMAC also possesses. An asymmetric-key digital signature goes further:

• Because the verification key is public, multiple parties can verify the signature. With an HMAC, the verification key is a shared secret, and only a holder of the shared secret can verify the message.

• Because a private key is used to generate the signature, only the sender (the holder of the private key) could have generated the signature, and the recipient can prove it to a third party. With an HMAC, a shared secret is used, and both the sender and the recipient know the shared secret. The recipient can verify that the signature was generated by the sender, but the recipient can't prove this to a third party, because, for all the third party knows, the recipient could also have generated the signature.

As with asymmetric-key encryption, the digital signature isn't typically applied directly to the message, because the message would be limited based on the key size.

The usual pattern is to digest the message and apply the private key to the smaller digest. The verifier applies the public key to the signature to recover the signed digest and compares that digest to one calculated from the message. This works because it is infeasible for an attacker to construct a second message with the same digest.

RSA is not the only asymmetric-key algorithm. Elliptic curve cryptography (ECC) is gaining popularity and is included in the latest specification.