Session 9, Part D:
Working with Algebraic Structures

In This Part: Units Digits Equations | Cryptography

 This system of "units digit arithmetic" may seem like abstract nonsense -- when would you need to compute just with units digits? In fact, different algebraic systems arise in all kinds of applications. Note 9 In the past few activities, you have been looking at a modular system. The "mod 10" system means you divide by 10 and take the remainder -- in other words, take the units digit. We'll now focus on an application involving another modular system: enciphering and deciphering messages. First, notice that you can assign each letter of our alphabet a number from 0-25: One of the oldest known substitution ciphers (a code where one letter stands for another) is the one reportedly used by Julius Caesar himself: To get the ciphered letter, add 3 to the original letter, or "plaintext." In symbols, this is: C = P + 3. Note 10 Problem D6 Encipher your name using Caesar's cipher.

 Problem D7 Shelly didn't know how to encipher the "y" in her name? What should she do?

 Problem D7 suggests that the algebraic description C = P + 3 is not quite right. We need some way to describe "wrapping around" so that the answers are always between 0 and 25. The solution? A modular system! Here's a new rule: To get the ciphered letter, add three to the original letter or "plaintext," then take the remainder when you divide by 26. In symbols, this is: C = P + 3 (mod 26).

 Problem D8 Decipher this message, which was created using Caesar's code. Explain how you did it. Note 11

 Problem D9 Here's a new rule: C = 3P + 2 (mod 26) Use this rule to encipher a secret word (at least five letters long) for a partner. Note 12

 Problem D10 Trade words with your partner and decipher their secret rule. Explain how you did it.

 Problem D11 Can you find a rule that would undo this cipher? That is, can you find a and b so that P = aC + b (mod 26) is an equation that "undoes" C = 3P + 2 (mod 26)?

 Modular systems for enciphering messages are not just for fun and games. It's essential that a secret message be hard to decipher if you're not the intended recipient, but easy to decipher if you are the intended recipient. Algorithmic ciphers are much better than "code books" because people can remember the algorithm, so it can't be lost or stolen. Modern cryptography, based on these modular systems -- using blocks of letters instead of single letters, exponential functions, and very large prime numbers -- is what's used these days to keep your credit card number safe when you purchase something on the Internet!

 Video Segment In this video segment, Ari Juels of RSA Security describes the methods and applications of modular arithmetic to modern cryptography. You can find this segment on the session video, approximately 21 minutes and 32 seconds after the Annenberg Media logo.

 The cryptography problems, D7 and D8, are adapted from Mathematical Methods in High School, by the Center for Mathematics Education. Copyright 2000 Education Development Center, Inc. This material is used by permission of Education Development Center, Inc.

Next > Part E: Summing Up

 Session 9: Index | Notes | Solutions | Video