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Inverse Matrices and Encryption

Encryption has existed since the dawn of warfare and power. The Egyptians used it as early as 1900 BCE. The idea is simple. You have information that you only want certain specific people to be able to see. Modern encryption is extremely powerful. Our phones come with encryption built in so that others can't take our information. However, some simple information is open for the world to see. An email is like a postcard as it passes through the Internet. Anyone monitoring the traffic can read your message. We would like to be able to keep things private at times.

Here we will use matrices to encode and decode messages.

For face-to-face classes, we will pair up in class.

For online classes, the instructor will be your partner and will provide the other half.

First, you need to make a message that is between 17 and 25 characters, spaces and letters only (Please keep it clean and don't share, except in Blackboard!)

Message: __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __

Next, you will need to make a 5 x 5 coding matrix. Keep it simple and use integers. Remember, integers include negatives and 0 if you want. You will want to make sure that the inverse of your matrix exists.

You now need to encode your message. First, you need to express your message numerically. Under each letter in your message above, put the corresponding number for its place in the alphabet, like 1 for A, 2 for B, and so on. Put a 0 for any spaces in the middle or at the end of the message to fill out all 25 spaces. We want to put our original message into a matrix, putting the numbers in order to fill out the matrix by going down the first column, then the second, etc.

To encode your message, multiply the matrices . Next, take the numbers out of your product, again going down the columns, and list the numbers below comma separated.

Encoded Message:

__ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __

You now have the information you need to trade with someone. Fill in your A matrix and your encoded message on the next page. For face-to-face, you will then hand that page to your partner, who will decode it. You will take your partner's message and A matrix and decode that one. Make sure when you get back your page to write down your partner's A matrix, encoded message, then decoded message. "How do you decode it?" you might be thinking. Take the encoded message, and put the numbers in a B matrix, again 5 x 5 going down the columns. Multiply , then pull out the numbers of that product, again going down the columns. Change those numbers back into letters, where , , etc. as before.

For online students, your instructor is the other half of your pair. Here is your instructor's encoded message and A matrix:

Encoded Message: 35, -225, 203, 79, 139, 14, -158, 195, 172, 167, 238, -42, 51, 158, 181, 257, -113, 90, 147, 160, 94, -105, 58, 28, 70

Your information:

Encoded Message:

__ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __

Your partner's information:

Encoded Message:

__ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __

Decoded Message:

__ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __