3.1.3 What is public-key cryptography?

  Traditional cryptography is based on the sender and receiver of a message knowing and using the same secret key: the sender uses the secret key to encrypt the message, and the receiver uses the same secret key to decrypt the message. This method is known as secret-key cryptography. The main problem is getting the sender and receiver to agree on the secret key without anyone else finding out. If they are in separate physical locations, they must trust a courier, or a phone system, or some other transmission system to not disclose the secret key being communicated. Anyone who overhears or intercepts the key in transit can later read all messages encrypted using that key. The generation, transmission and storage of keys is called key management; all cryptosystems must deal with key management issues. Secret-key cryptography often has difficulty providing secure key management.

Public-key cryptography was invented in 1976 by Whitfield Diffie and Martin Hellman in order to solve the key management problem. In the new system, each person gets a pair of keys, called the public key and the private key. Each person's public key is published while the private key is kept secret. The need for sender and receiver to share secret information is eliminated: all communications involve only public keys, and no private key is ever transmitted or shared. No longer is it necessary to trust some communications channel to be secure against eavesdropping or betrayal. Anyone can send a confidential message just using public information, but it can only be decrypted with a private key that is in the sole possession of the intended recipient. Furthermore, public-key cryptography can be used for authentication (digital signatures) as well as for privacy (encryption).

Here's how it works for encryption: when Alice wishes to send a message to Bob, she looks up Bob's public key in a directory, uses it to encrypt the message and sends it off. Bob then uses his private key to decrypt the message and read it. No one listening in can decrypt the message. Anyone can send an encrypted message to Bob but only Bob can read it. Clearly, one requirement is that no one can figure out the private key from the corresponding public key.

Here's how it works for authentication: Alice, to sign a message, does a computation involving both her private key and the message itself; the output is called the digital signature and is attached to the message, which is then sent. Bob, to verify the signature, does some computation involving the message, the purported signature, and Alice's public key. If the results properly hold in a simple mathematical relation, the signature is verified as genuine; otherwise, the signature may be fraudulent or the message altered, and they are discarded.