Inverse Matrices And Encryption

Inverse Matrices And Encryptionencryption Has Existed Since The Dawn O

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 create 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 create 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, filling it 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 as a comma-separated list.

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: __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __

Encoded message from your partner: __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __ , __

Decoded Message: __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __ __

Paper For Above instruction

The process of using matrices for encryption and decryption is a practical application of linear algebra principles in modern cryptography. It illustrates how mathematical concepts like matrix inversion can be used to secure information by transforming plaintext into ciphertext and vice versa. This method, often called matrix-based cipher, demonstrates the practical use of invertible matrices in real-world encryption systems.

In this method, a message is first translated into numerical form, with each character assigned a number based on its position in the alphabet. Spaces are represented by zeroes and the sequence of numbers is then arranged into a matrix according to a predetermined structure, such as filling down columns in a 5x5 matrix. The encoding process involves multiplying this message matrix by a coding matrix, which must be invertible to allow decoding later. The product matrix's entries are then read in the same column-wise order to produce the encoded message.

This matrix multiplication ensures that the message is transformed into a form that appears random and unreadable without the inverse matrix. The security of this method relies heavily on the invertibility of the matrix. If the matrix does not have an inverse, the message cannot be decoded, compromising the purpose of encryption. Selecting a matrix with integer entries and a non-zero determinant is crucial for the encryption-decryption process to work effectively. In actual cryptographic systems, such processes are often enhanced with additional security layers, but the fundamental concept remains rooted in linear algebra.

Decoding the message involves reversing the process. The recipient, knowing the original coding matrix, calculates its inverse using linear algebra techniques. The encoded message, arranged in a matrix form, is then multiplied by this inverse matrix, reverting the data back into its original numerical form. These numbers are then translated back into letters, completing the messaging cycle. This method highlights the power of matrices not just for mathematical abstractions but for real security applications in digital communication.

However, despite its elegance, matrix-based encryption is not entirely foolproof. It can be vulnerable to various cryptanalytic attacks if the matrices or the encoding process are poorly chosen or inadequately secured. Contemporary cryptography employs more complex algorithms that incorporate multiple layers of encryption, but understanding matrix-based encryption offers valuable insight into the fundamental mathematical principles underlying cryptographic methods.

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