Cryptography and Advanced Ciphers

Summary:
This lesson is an extension of one on basic cryptography. The main idea is to show that messages may be encoded and decoded using different numeric representations of the letters in the alphabet. As the encryption method gets more difficult, the students can see that there are further advances which may be taken to obtain the highest levels of security with documents.


Subject:

·  Math: ___
TEKS: Please write number and description.

·  7.1A – compare and order integers and positive rational numbers.

·  7.2C – use models to add, subtract, multiply and divide integers and connect the actions to algorithms.

·  7.2F – select and use appropriate operations to solve problems and justify the selections.

·  7.13A – identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics.

Grade Level:

·  Target Grade: ___ 8

·  Upper Bound: ___ 11

·  Lower Bound: ___ 6

Time Required: 1-2 for lecture and activity. (>50 minutes)

Activity Team/Group Size: 1-2

Materials:

·  One worksheet per team/student (depends on size of class)

·  One cipher wheel per team/student/station (depends on how the stations are set up)

Reusable Activity Cost Per Group [in dollars]: Cost of paper and ink for worksheets and cipher wheels.
Expendable Activity Cost Per Group [in dollars]: Cost of paper for worksheets.

Learning Objectives:

·  Integer addition/subtraction/multiplication/division

·  Introduction to matrices (just shown, not discussed in detail)

·  Mathematical applications in real-world setting (data hiding, etc.)

Lesson Introduction / Motivation:
There is a link to a YouTube video on the second slide show of an individual getting pulled on a BMX bike by a van. The bicyclist hits a ramp and flies through the air towards a building in an attempt to land on the roof. He does not make it far enough and hits the edge of the roof and falls two or three stories down to the ground. Leads into a necessity for mathematics in an everyday setting. The fourth slide also gives an example of passing notes in class and how if the students didn’t want the teacher to find out, they should encode the information somehow. From here, the lesson may continue in the direction of integers and their application to cryptography.

Lesson Plan:
Go over the PowerPoint slideshow with the students. This may take one full class period. Once the students have been given the background on what cryptography is and the necessity to keep information safe, the activity may be given. Some messages are already included in the file entitled station_assignments. The answers to these encoded messages are:

  1. HOW DO YOU KNOW (shift key = 2)
  2. YOU WIN (shift key = -4)
  3. ENCRYPTION (shift key = 2)
  4. MATH CLASS (shift key = 6)
  5. GREAT WORK (shift key = -1)
  6. Student’s create their own

Place the station assignments at various locations throughout the room or show them on a projector for larger classes so students remain in their seats. Have the students solve for the shift key at each station. Then, turn the cipher wheel in the appropriate direction the appropriate number of spaces. Record the number associated with each letter in the encoded message. Look at the top of the worksheet, and replace the recorded number with its corresponding letter. This will be the decoded message.

Lesson Closure:
The basic idea is that students should get better with their manipulation of algebraic expressions and integer addition and subtraction. They should see a real-world application of integers and mathematics.
Assessment:

See if students are able to create their own code on the last station. There are examples of encoding and decoding messages on the worksheet. See if it is possible to decode their example or if anyone else can. Then, have a discussion with all of the students about the topic and see if they can point out anywhere else they may see this as being important.

Vocabulary / Definitions:

·  Modulus - a quantity by which two given quantities can be divided to yield the same remainders.

·  Integer - one of the positive or negative numbers 1, 2, 3, etc., or zero.

·  Cryptography - the science or study of the techniques of secret writing, esp. code and cipher systems, methods, and the like.

·  Cipher - a secret method of writing, as by transposition or substitution of letters, specially formed symbols, or the like.

Background and Concepts for Teachers:

·  Cryptography has been used ever since people have been able to write. This is because many people were not educated and could not read. Therefore to keep things a secret all educated people needed to do was write things down.

·  Julius Caesar used cryptography to encode his messages by using a shift on his alphabet by a value of 3.

·  The basic principle behind this whole lesson is that cryptography is used when we want to keep things secure. Many of the ways in which this is done is by using integer shifts/multiples of the numeric representation of the alphabet to represent items so they remain protected.

·  The slideshow attached has more details on many of the subjects and goes through some examples.

Prerequisites for this Lesson: (Optional)

·  Math

Lesson Scaling:

Easier or harder problems may be used to solve for the shift keys. Also, more challenging codes may be used and more difficult expressions may be used.

Multimedia Support and Attachments:

·  Cipher wheel on last slide of power point

·  Cryptography lesson

·  Cipher worksheet

Authors:
Graduate Fellow Name: Jason Wardlaw
Teacher Mentor Name: Caroline Jones
Undergraduate Fellow Name: N/A
Date Submitted: 9/29/2008
Date Last Edited: 9/29/2008

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