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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February 9, 2012
Dr. Kristen Abernathy
Early cryptographers encoded messages using transposition ciphers (rearranging the letters) or substitution ciphers (replacing letters with other letters).
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.
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.
Let’s choose our message to be
ATTACK AT DAWN
and we’ll choose for our encoding matrix
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.
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:
We can now encode our message by multiplying our 3x3 encoding matrix by the 3x5 matrix formed from the vectors formed from the message:
Using Mathematica, we see the product of the encoding matrix and our message is:
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 *
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
C O N G R A T U L A T I O N S * Y O U * D I D * I T * *
Come up with your own secret message and trade with your neighbor!