by Tom Abraham and Connie Aphonephanh
INTRODUCTION TO CRYPTOGRAPHY
This lesson is intended to teach ninth and tenth grade high school algebra students the mathematics behind cryptography.
To spark students’ interest in cryptography, one example in history of a cryptography scandal will be provided to the students via the PBS Nova website (http://www.pbs.org/wgbh/nova/venona/). The students will also research the history of cryptography. The students will be divided into groups and each group will provide an example in history of a cryptography scandal. Each group will give their presentation to the class. After presentations, lessons on cryptography will begin. To start the long-term unit on codes and code-breaking, the question “How do you code a message so it is easy to decode and difficult to crack” will be posed to the students. The lessons will familiarize students with commonly used cryptography terms, guide students through encoding and decoding messages as well as introduce various encryption techniques such as Caesar cipher and public key encryption. To assess students’ level of understanding, the students can test their ability with the PBS Nova online activity (http://www.pbs.org/wgbh/nova/decoding/faceoff.html). As students gain more confidence in coding and decoding, their final project would be to develop their own secret codes and have their fellow classmates try to crack the codes.
What would the world look like if the Allies had not won World War II? Breaking the Axis military and diplomatic codes was an essential step in winning the war. How does an imbedded CIA agent in an enemy country convey information back to agency headquarters in Washington, D.C. without being revealed? Once students get introduced to the important roles codes and encryptions have had in the past and are having today in the security of our nation and once they can understand the underlying math, they will be able to better appreciate the field of mathematics. This appreciation will lead to better reasoning and problem solving abilities and a better understanding of how math is applied in actual situations. Students will be able to reason and use logic at higher levels and greater conceptual understanding of mathematics.
Students will be able to:
Imagine students who are truly excited about math, students who love math, and students who are quite adept at reasoning and problem solving. For too long have my students asked me about the usefulness and purpose of the mathematics they are learning in algebra. For far too long have we seen students give up on difficult problems that require reasoning and problem solving. We believe that a long-term unit on codes and code-breaking will both increase students’ interest in mathematics and improve their reasoning skills.
We are not out to change the world or completely overhaul the way math is taught across the country, but we do want to energize our students and instill in them the same passion for mathematics that we have, the same understanding of its wide applications. This unit is a good place to start in order to achieve those goals. This unit will begin to expose students to the world of spies, the Central Intelligence Agency, and the National Security Agency. Students will get the opportunity to come up with their own codes and attempt to break codes ranging in difficulty. They will see the math behind some very common forms of encryption – those used to encode secret messages and those used in online financial transactions everyday. Students will also be able to see how encoding/encryption have affected us throughout history, like how the U.S. Navy’s Signals Intelligence Division (SIGINT) broke Japanese military codes during World War II to how the NSA and CIA work to keep the United States one step ahead of its enemies today. Movies like the James Bond and Mission Impossible series and television shows like Alias and 24 have already sparked a great deal of interest in the secret world of spies and codes.
Once students understand the necessity of mathematics in such a crucial field, we believe that they will develop a greater appreciation for the subject. They will be able to see that a “dry” and “boring” subject can actually be exciting when they are able to put theory into practice. This is crucial for my students to better their reasoning abilities and problem solving skills.
The lessons for this project were taken from the textbook Mathematics: Modeling Our World: Course 1 and from the Nova website at pbs.org. Other resources for this unit were taken from “Cryptography in High School Mathematics” (http://www.gvsu.edu/math/enigma/School/Welcome.htm) and from “The Code War” (http://www.beyonddiscovery.org/content/view.article.asp?a=3420). The project is designed to introduce students to the mathematical concepts like congruencies and modular arithmetic, factoring composite numbers into prime factors, linear equations, and matrix multiplication.
Simply put, cryptography is the process of converting and deciphering information in different forms. The simplest form of encoding data is the shift cipher. All the letters in a word are replaced by those occurring before or later on in the alphabet. For example, converting “CAT” to “ECV” uses the shift “+ 2” cipher where the second letter to follow the original letter replaces it. A shift “– 2” cipher uses the second letter before the original letter. Assigning numbers to the alphabet (A = 1, B = 2, etc.), this form of encryption can be represented by the linear equation, c = p + 2, where c is the newly coded letter and p the original letter. A more complicated cipher is the stretched shift cipher. This type of cipher first assigns a value to each letter as before. Then these numbers are plugged into a linear equation, such as c = 4p + 3, where p is the number representing the original letter and c is the number representing the coded letter. Then using modular twenty-six arithmetic for numbers above twenty-six, a message is coded. A yet even more complex cipher is the keyword cipher. The keyword cipher uses a secret keyword known only to the sender and recipient of the message. If the keyword “BIG” was used for example, the sender would code the message in sets of every three letters. The first letter of the first trio would need to be shifted forward by one (since “B” is one letter after “A”), the second letter would need to be shifted forward by eight (“I” is eight letters after “A”), and the third letter would need to be shifted forward by six (“G” is six letters after “A”). So the word “CAT” is coded as “DIZ” and “BUTCHER” would be “CCZDPKS.” Frequency analysis is used to crack messages. Students will be shown how this can be an effective tool to cracking intercepted messages. The most commonly occurring letter in the English language is “E” and the second most commonly occurring letter is “T.” Sets of three-letter words that frequently occur are probably “THE.”
The conceptual understanding needed to understand encryption is in many cases the same math that is taught in high school. With some introduction into linear equations, number theory, modular arithmetic, and matrix calculations, students in even an Algebra 1 class can effectively code, decode, and crack many secret massages. Resources and tools for teaching this subject were once very limited but are not so anymore. The vast ocean of web resources and textbooks like Mathematics: Modeling Our World provide a great deal of ideas and assistance teach what was once thought to be too complicated to teach in high school and still be of real value to students with real world applications.
b.1 (D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.
b.1 (E) The student interprets and makes inferences from functional relationships.
b.3 (B) Given situations, the student looks for patterns and represents generalizations algebraically.
c.1 (C) The student translates among and uses algebraic, tabular, graphical or verbal descriptions of linear functions.
To see if goals are being reached, a series of assessments such as presentations, tests, and classroom assessment techniques will be given. Each type of assessment will provide written feedback, which can be kept for future reference. The feedback will gauge students’ level of understanding as well as misconceptions. Thus, the level of difficulty of the lessons can be changed to meet the needs of the students.
First type of assessment, presentations, will allow students to incorporate history, writing, and public speaking with mathematics. Presentations include researching, writing up a summary of the research, and giving a five minute presentation with visuals. This will accommodate students whose main strengths do not lie in mathematics, but like doing research, writing, or doing public speaking. Tests will monitor students’ level of understanding. Questions on the test will vary from multiple choice questions, true/false questions to free-response questions. The type of questions varies to accommodate different learning styles. Feedback from the test will see whether students are about to encode and decode messages as well as differentiate between the various encryption methods. Other types of assessments are classroom assessment techniques such as probing background knowledge, filling in productive study time logs, developing concept maps, and documenting step-by-step solutions to problems are assessments will gauge students’ level of understanding. These assessments are just a few types of assessments that are included in our unit project.
Students will be required to create a demonstration poster for their final presentations. These posters should include what type of encryption the students have chosen, the math behind it, and where all this sort of encryption can be used. The posters should have both text and graphics, be visually pleasing, and convey a great deal of information quickly and concisely.