An Introduction to Cryptography

An Introduction to Cryptography

TEA fellowsFebruary 9, 2012

Dr. Kristen Abernathy

Applications of cryptography include:

•ATM cards

•Computer passwords

•Electronic commerce

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

Examples:

•Loleh tuedssnt

• Transposition cipher

• Lzsg hr etm

• Substitution cipher

Key

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?

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”

Let’s choose our message to be

ATTACK AT DAWN

and we’ll choose for our encoding matrix

“ATTACK AT DAWN”

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 DAWN”

Assigning these numeric values, our message becomes

A T T A C K * A T * D A W N1 20 20 1 3 11 27 1 20 27 4 1 23 14Since we are using a 3x3 matrix, we break our message into a collection of 3x1 vectors:

“ATTACK AT DAWN”

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!

• Log on as “visitor”

• Password is “winthrop”

• Open the program “Wolfram Mathematica 8”

• Click on the option: (Create New) Notebook

“ATTACK AT DAWN”Using Mathematica, we see the product of the encoding matrix and our message is:

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

and multiply by the inverse of the encoding matrix…

The string of numbers we would send in our message is:17, 40, 144, 32, 14, 57, -4, 21, 191, -89, 5, 124, -3, 41, 242

Using our key:

we can decode the message

1 20 20 1 3 11 27 1 20 27 4 1 23 14 27A T T A C K * A T * D A W N *

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

YOUR TURN!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 27C 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!

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