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An Introduction to Cryptography. TEA fellows February 9, 2012 Dr. Kristen Abernathy. Applications of cryptography include: ATM cards Computer passwords Electronic commerce.

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An Introduction to Cryptography

TEA fellows

February 9, 2012

Dr. Kristen Abernathy


Early cryptographers encoded messages using transposition ciphers (rearranging the letters) or substitution ciphers (replacing letters with other letters).

Key

  • Examples:

  • Loleh tuedssnt

    • Transposition cipher

  • Lzsghretm

    • Substitution cipher


Encryption is the transformation of data into some unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

Decryption is the reverse of encryption; it is the transformation of encrypted data back into some intelligible form.


Where does math fit in
Where does math fit in? unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

We can use matrix algebra to encrypt data!

If we use large matrices to encrypt our message, the code is extremely difficult to break.

However, the receiver of the message can simply decode the data using the inverse of the matrix.


Attack at dawn
“Attack at dawn” unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

Let’s choose our message to be

ATTACK AT DAWN

and we’ll choose for our encoding matrix


Attack at dawn1
“Attack at dawn” unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

We’ll assign a numeric value to each letter of the alphabet:

We’ll also assign the value 27 to represent a space between two words.


Attack at d awn
“Attack at unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data. dawn”

Assigning these numeric values, our message becomes

A T T A C K * A T * D A W N

1 20 20 1 3 11 27 1 20 27 4 1 23 14

Since we are using a 3x3 matrix, we break our message into a collection of 3x1 vectors:


Attack at dawn2
“Attack at dawn” unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

We can now encode our message by multiplying our 3x3 encoding matrix by the 3x5 matrix formed from the vectors formed from the message:


Computer time
Computer time! unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

  • Log on as “visitor”

  • Password is “winthrop”

  • Open the program “Wolfram Mathematica 8”

  • Click on the option: (Create New) Notebook


Attack at dawn3
“Attack at dawn” unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

Using Mathematica, we see the product of the encoding matrix and our message is:


The string of numbers we would send in our message is: unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data.

17, 40, 144, 32, 14, 57, -4, 21, 191, -89, 5, 124, -3, 41, 242

In order to decipher this code, we need to re-form our 3x1 vectors:

and multiply by the inverse of the encoding matrix…


When we multiply my the inverse of the encoding matrix, we get the vectors:

Using our key:

we can decode the message

1 20 20 1 3 11 27 1 20 27 4 1 23 14 27

A T T A C K * A T * D A W N *


Your turn
Your Turn! get the vectors:

With the same key as before:

and the encoding matrix:

decode the message:

211, 605, 310, 1355, 246, 1970, 692, 379, 204, 1136, 488, 259, 318, 2125, 730, 1493, 349, 2632, 953, 1641, 162, 1466, 350, 977, 406, 1905, 712, 1977


3 15 14 7 18 1 20 21 12 1 20 9 15 14 19 27 25 15 21 27 4 9 4 27 9 20 27 27

C O N G R A T U L A T I O N S * Y O U * D I D * I T * *


Now you give it a try
Now you give it a try! 27 9 20 27 27

Come up with your own secret message and trade with your neighbor!


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