[Math Lair] Cryptography

Math Lair Home > Topics > Cryptography

Communication has always been important to humanity, but sometimes it's important to keep communications secret from prying eyes. This is where codes and cyphers come in.

A cypher is a message that has been encrypted by making changes or substitutions to the letters in the original message. The original message, called the "plaintext," is encrypted by the sender into the "cyphertext," which the receiver then decrypts.

One of the earliest cyphers known is called the Caesar cypher or the three-shift system. As the name implies, it was used by Julius Caesar 2,000 years ago. To encrypt a message, each letter in the plaintext is replaced with a letter three letters later on in the alphabet. The last three letters of the alphabet are replaced with letters at the start of the alphabet. So, keeping in mind that there were only 23 letters in the Latin alphabet 2,000 years ago, the letters would be translated as follows:

So, if Caesar wanted to send the message "Veni, Vidi, Vici," he would send "ZHQMZMGMZMFM." The receiver would simply shift each character three characters backwards to get the original message. A cypher like this is called a substitution cypher, as one letter is substituted for another.

A cypher like this can easily be broken. In the ninth century, the Arabic mathematician and philosopher Al Kindi wrote a book about cracking cyphers using frequency analysis. Certain letters, or pairs of letters, appear more often than other pairs. For example, the most common letter in English is the letter E. A piece of cyphertext encrypted using this cypher would contain the letter H far more than any other letter. It wouldn't be too difficult to conclude that H maps to E. You could continue the process with other letters until the message is decrypted.

Even though a cypher such as the Caesar cypher is not too complex, until rather recently cyphers were not much more sophisticated than that. During World War I, to crack enemy cyphers, the British primarily employed linguists, not mathematicians.

The need for cyphers that were much more difficult to break first occurred after World War I, as wireless communication became more and more popular. With wireless communication, it's much easier for someone to eavesdrop on conversations. This led to the development of cypher machines such as the Enigma machine, made in Germany. The Enigma was first used in the field of commerce (the German Navy, surprisingly enough, was not interested in the machine initially). The Enigma machine had several rotors; every time a key was pressed and a letter encoded, one or more of the rotors would rotate by a step, changing the substitution alphabet at every letter. Breaking this cypher would require a fair bit of mathematical knowledge.

This page will be continued later.

Sources used (see bibliography page for titles corresponding to numbers): 47.