Early methods created a 1-1 correspondence between characters. Such schemes were in use in ancient times (e.g. by Julius Caesar!).

Decryption amounts to finding the inverse of the mapping.

One of the earliest methods for encoding information involved what is called symmetric encryption, that is using the same key to encode and decode information. The DES (Data Encryption Standard) is used by the Federal government.

The major difficulty with this method is that you have to transfer the key between sender and receiver, which is generally not secure.

To resolve this problem, in 1976 Martin Hellman and Whitfield Diffie proposed a public-key encryption system.

Assymetric encryption is based on two keys, rather than one. The public key can be distributed to anyone. The private key is secret.

If I want to encrypt a message to you, I take your public key, and my private key, and encode the message. I can then send it to you, and only you can decode it!

If I encrypt something with my private key, it can be un-encrypted using my freely available public key, but by no other key! In this way, I create a "digital signature." You can be sure that it has been sent by the user whose public key you have.

However, what if I distribute a public key, claiming that it came from someone else. I then send a message using my private key, and the recipieent thinks that it came from some one else. We have to guarantee that the public keys belong to the right people. This is the function of certificate authorities. They issue a public/private key combination after validating that the requester is who they say they are.

Here is an example of what you have to go through to get Thawte's 128 bit "super-certificate."

VeriSign is another blue-chip certificate authority.

You can see a list of trusted CA's in IExplorer by going to Tools -> Internet Options... -> Content -> Certificates ... -> Trusted Root Certificate Authorities

In Netscape and Mozilla, it can be found under Edit -> Preferences -> Privacy & Security -> Certificates -> Manage Certificates ... -> Authorities

The PGP (Pretty Good Privacy) algorithm is widely used to encrypt e-mails. The source code is available, if you sign a licensing agreement. There is also an international PGP site.

The mathematics of the RSA algorithm.

Here is an example of using RSA to generate a public/private key using prime numbers.

Real time encryption of ethernet traffic can be accomplished via SSL (secure sockets layer).

There are free, public, certificate authorities such as www.cacert.org.

• Bruce Schneier's Applied Cryptography, second edition
• Introduction to Cryptography
• Overview of PGP
• Math 470 Communications and Cryptography
• Math 673 Information, Secrecy, and Authentication I
• Math 674 Information, Secrecy, and Authentication II.
• RSA Code C code for decryption (limited to small primes)
• Java Applet which demonstrates encryption.
• RSA Challenge - prime factorization contest (win $10,000 - $200,000!)
• Prime Factorization Algorithms
• The Quadratic Sieve algorithm
• The Number Field Sieve algorithm.
• More information on Cryptography from RSA Corp.
• Visualization of Primes in 2D and 3D.