Content area
Full Text
UMI: equations omitted
Throughout history, coded messages have been used for various reasons. Today's students are fascinated by the secretive nature of these codes, and this fascination can lead them to explore the mathematics of cryptography. The simplest codes are called substitution ciphers. In these codes, each letter is replaced by another number or letter in the alphabet. These codes are easy to crack, or decode, because of the relative frequency of letters in messages. For example, e is the most often used letter in the English language; therefore, the substituted value for e is relatively easy to determine. One way to make substitution codes more difficult to crack is to group letters and then encode the groups of letters. A particular application of this strategy, one that combines matrix multiplication and modular arithmetic, is known as the Hill cipher (Anton and Rorres 1987). This article explains coding and decoding messages using Hill ciphers. These ciphers are an interesting example of an application of matrices called for in NCTM's Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) for grades 9-12. A graphing calculator will facilitate the matrix and modular arithmetic used in the coding and decoding procedures.
CODING MESSAGES USING HILL CIPHERS
Since Hill ciphers are a form of substitution ciphers, the first step is to create a list of substitution values for each alphabetic and punctuation character to use when coding and decoding messages. The Hill-cipher technique also requires that the number of substitution values be a prime number. Since the English alphabet contains twenty-six letters, the prime number 29 can be used to represent all the alphabetic characters and three punctuation marks. Table 1 shows a set of twenty-nine substitution values for these characters. Many other substitution schemes are possible. Notice that the characters are numbered 0 through 28 instead of 1 through 29. Hill ciphers also use a modular-- arithmetic procedure; the numbers 0 through 28 represent all values in a mod 29 system.
To make secret messages more secure, the Hillcipher technique uses a coding matrix to encrypt messages. Coding and decoding messages are opposite operations, like doing and undoing a procedure. Hill ciphers are encoded using a matrix-- multiplication operation. To decode a message, the opposite of matrix...