1. Cryptography and Advanced Ciphers
Becoming Experts in a very secure field.
9/18/2008
TAMU NSF GK-12 PEER Program -

2. Why do we learn math anyway?
Why is math important?
Check this video out.
Looks like he miscalculated a little bit !! 
Video just used to get the students attention. The animation on this page gets the basic point across of the video. After showing the video this can lead into a discussion on what went wrong in the video. In turn, this shows the students that ‘MATH IS EVERYWHERE!!’

3. On to the meat of the material.

4. Why do we encode information?
Alright! He didn’t know what it said!!
I’ll send this note to Stacy.
I don’t know what this says. I guess it’s okay.
This slide is to show the students why it may be important to encode information, and/or an example of encoding information in their own context.
Is that a note?

5. What is encryption? Used to “disguise” information.
Used to “disguise” information.
Keeps others from knowing (sometimes) crucial information.
Can be done in many ways (electronically, using a cryptex, etc.).
encryption is a process of transforming information into a form that is unreadable to those who do not have a special knowledge
Begins to show why we encode information in addition to the previous slide on note passing.
The cryptex is used here as an example for those students who may have seen the movie The Da Vinci Code. For those that didn’t more information is available here: and .
Encoder

6. What is decryption Decryption is the reverse direction of encryption.
In other words, it is a process of making the encrypted message readable again

7. Why do we encode information?
Keeps others from knowing what’s going on.
Makes information secure.
Sometimes it’s cheaper to encode information.
Why else? . . .
The basic theme behind this slide is that encoded information is done for security. That’s the whole reason any of this was ever developed. Can talk about bank accounts, private information, etc. One way to lead in is if it is discussed not letting a teacher know what is written on a note that is passed from one student to another in a class. This tends to draw them into the idea of why encoding my be necessary.
In stating that it is sometimes cheaper to encode information, this has to do with the way images are typically encoded. Video files are larger than audio files (in terms of memory used) due to the necessity to encode audio and video information. Therefore, to help reduce storage space, sometimes it is easier to not save color and position information about every pixel. If someone does not move very much or there is a large amount of colors which are similar in a frame, then it is only necessary to store some of the information and know that the surrounding pictures will be the same color. Although the quality is reduced, the size of the file is reduced and therefore is ‘cheaper’ in terms of storage space.

8. Roots of encryption.
Earliest form of encryption was actually just writing on paper.
Could have been rearranging letters in a message.
Julius Caeser would replace a letter by a letter three positions down!!
Many years ago, only educated people knew how to read and write. Therefore, if anyone wanted to ‘encode’ a message they just needed to write it down. Can have a lead-in with the students about how many of them think they would be in school if it were 5,000 years ago. Would they be working in a field? Sewing? Planting crops? Selling items on the street? Etc. Then leads into the idea of encoding information by just writing it down.
Then, talks about how the Romans enjoyed knowledge and learning. When the Roman empire was at it’s peak, Caeser needed to send messages to all ends of the empire. Instead of visiting his generals throughout the land, he would send messages. In case these were intercepted by enemies, he would shift every letter over three places in the alphabet so that AD, BC, etc.

9. Some examples of encoding.
Having letters as numbers.
Representing larger numbers with other numbers.
Converting bases.
Mapping pixels and audio for data streams.
Etc.
A=1, B=2, etc.
100=10 x 10
DEC 25 = OCT 31 What does this one mean? Is it true?!
It should be noted that although here A=1, B=2, etc. it is important that the instructor point out that this is not the only starting point. As long as there is a known definition for these values, then anything is okay. The instructor can then point out that it is not necessary to start with A=1 (A could be 0, 2, 8, 17, , etc.). This way, there will be some correlation/preparation for the activity associated with the presentation.
The example of 100=10 x 10 is only used to get the students thinking differently. That although numbers are assumed to be just numbers, they are actually often combinations of other numbers. This may help with some of the explanation of the different types of encoding in the next slides (especially those which use the modulo function).
The third example about DEC 25=OCT 31 says that DECIMAL(base 10) 25 is equivalent to OCTAL(base 8) 31 when represented in BINARY(base 2).
DECIMAL 25BINARY
(0)25 + (1)24 + (1)23 + (0)22 + (0)21 + (1)20 =
OCTAL 31BINARY
3(0)22 + (1)21 + (1)20 = 011
1(0)22 + (0)21 + (1)20 = 001
011001

10. TAMU NSF GK-12 PEER Program - http://peer.tamu.edu
Types of Ciphers
A=1
A=1+N, N is an integer
A=(α×1)mod(26), α is an integer that is relatively prime with 26
A= (α×1 + N)mod(26)
Straight
Shift
Multiplication
Affine
Others
Different types of ciphers. The ‘mod’ function is discussed in the next slide. The basic idea is that we only care about the remainder when dividing by a number. More is explained on the following slide.
Relatively prime is one that denotes the two numbers do not have any common factors. It doesn’t necessarily mean that the multiplying factor has to be prime (e.g. 13 is a prime number but is not relatively prime with 26. Therefore, you can not use 13 as the multiplication factor).
9/18/2008
TAMU NSF GK-12 PEER Program -

11. What is the modulo function?!!
The mod function, as seen previously, is the remainder when two numbers are divided.
a mod(b) = r. a/b
Useful in many applications. (DSP, algorithms, oscillators, ADC’s, others).
Can even be used for card tricks!!!
7 mod(3) = 1
8 mod(3) = 2
9 mod(3) = 0
10 mod(3) = 1
etc.
The answer will always be less than the modulo number (i.e. for mod(3), the result (y) will always be 0 ≤ y < 3.)
9/18/2008
TAMU NSF GK-12 PEER Program -

12. Is Cryptography actually hard?
A short answer is yes.
Involves very complex math and complex algorithms.
Requires pattern recognition, higher order mathematics (matrices, discrete math, 3-D vector calculus, etc.)
What does the math look like?
Used to lead into matrices. Many of the students may think that cryptography in general is simple. This slide and the next are not intended to discourage the students, but rather used to show that cryptography can get more complex if necessary. The pdf file linked in the next slide goes through an example of encoding and decoding a message using matrices.
9/18/2008
TAMU NSF GK-12 PEER Program -

13. Have you heard of a matrix?

14. Matrices Matrices are used extensively in linear algebra.
Solve multiple equations with multiple variables at one time. (systems of equations).
Impact on many everyday accounts (computer programming, optics, microwave transmission, atmospheric sciences, etc.)
An advanced version of the algebra you all will take soon or are taking now.
Example of matrices in cryptography.

15. What are we doing today? We’ll be decrypting some secret messages.
May be challenging.
See the next page for the decoder we will be using . . .

16. TAMU NSF GK-12 PEER Program - http://peer.tamu.edu
This cipher wheel is useful when decoding messages, but is less obvious when encoding information. Follow the example on the worksheet to see how to use this wheel. The idea is to print these out beforehand and have them ready for the students. Print as many copies as needed. It is necessary to cut-out the inner wheel (the one with numbers) from the larger wheel (the one with letters) in order to use this properly.
To use this to encode information:
Select a message to encode.
Figure out what ‘shift’ will be put on the message (+3, -3, etc.)
Write down the letters using their numeric representation.
Turn the wheel in the opposite direction that the wheel says in the appropriate number of spaces (e.g. your shift is +3, turn the wheel to the LEFT 3 spaces). Remember, it is only backwards for encoding information. For decoding information, it works as shown.
Replace each number with the new letter associated with that number after the shift.
Example:
Message to encode – ‘Hello’
Shift – ‘+3’
Numeric representation before shift – ’ ’
Encoded message – ‘EBIIL’
Decoding a message using cipher wheel:
Solve for the shift key.
Write down the number associated with each letter in the encoded message.
Shift the inner wheel the appropriate amount in the correct direction.
Whatever letter is given in the encrypted message, replace it with the number associated after the shift.
Return the wheel to the starting point.
Write the letter corresponding to the number previously written for the decoded message.
Encoded message – ‘QEROPAXV’
Shift key – 9x=27x=+3
Numeric representation of encoded message before shift – ‘ ’
Numbers to letters after shift performed – ‘T-H-U-R-S-D-A-Y’
9/18/2008
TAMU NSF GK-12 PEER Program -