Elementary Number Theory with Programming

9CRYPTOGRAPHY

We shall see how number theory plays a major role in encoding information, such as credit card numbers and banking transactions transmitted via the Internet.

INTRODUCTION AND HISTORY

In this chapter, we will examine several applications of number theory to cryptography, the science of encoding information. With the advent of electronic media such as the telephone and computer, it became necessary to encode military, diplomatic, corporate, and financial data. A simple credit card transaction, for example, transmitted from a store or bank to a financial institution must be encoded lest someone intercept the credit card number, expiration date, user name, and password. A more urgent example is afforded by the need to encrypt a message from the Pentagon to a Trident nuclear submarine containing instructions to relocate. The victory of the United States and Britain over Japan and Germany in World War II may be attributed in part to the breaking of the Japanese and German codes by American and British mathematicians and computer scientists such as Alan Turing. The United States used the Navajo language in their radio transmissions in the Pacific Theater—a language unknown to the Japanese.

Here is a bit of standard terminology. The message to be encrypted (the technical term for encoded) is called plaintext, while the encrypted text is called the ciphertext. The ciphertext must be decrypted (the technical term for decoded) in order to be recovered as plaintext.