The problems in the image cover a variety of topics related to Hill ciphers, matrix operations, and modular arithmetic. I'll give a summary of each problem and the approach to solve them.

Problem 1 (4 marks) - Hill Cipher Encryption & Decryption

(a) Encrypting "In pulvere vinces":

1. Convert the phrase to numerical equivalents using A = 0, B = 1, …, Z = 25.
2. Group letters into blocks of 3 (as the matrix $A$ is 3x3).
3. Matrix Multiplication: Use the given encryption matrix $A = \begin{pmatrix} 5 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 4 \end{pmatrix}$ and multiply each block of three letters with this matrix.
4. Modulo Operation: Since it’s mod 29, apply this to keep results within the alphabet.

(b) Decrypting "UWVIKHACQVGI":

1. Convert the ciphertext back to numerical form.
2. Calculate the inverse of matrix $A$ modulo 29.
3. Multiply each block of 3 characters by this inverse matrix, then convert back to letters.

Problem 2 (3 marks) - Hill Cipher Message Decryption (Unknown Matrix)

In this problem, Eve intercepts an encrypted message and knows:

• Block length is 2.
• Plaintext starts with "Hello".

Steps:

1. Use the known plaintext-ciphertext pairs to set up a system of equations to determine the encryption matrix.
2. Once the matrix is determined, calculate its inverse modulo 29.
3. Use this inverse matrix to decrypt the intercepted message.

Problem 3 (2 marks) - Matrix Inversion using Gauss-Jordan

For the matrix $A = \begin{pmatrix} 2 & 3 & 0 \\ 1 & -2 & -1 \\ 2 & 0 & -1 \end{pmatrix}$ use the Gauss-Jordan elimination to find $A^{-1}$.

Problem 4 (2 marks) - Matrix Inverse Property

Show that for two invertible matrices $A$ and $B$, $(AB)^{-1} = B^{-1} A^{-1}.$ This can be proven by considering the properties of inverses and the associative property of matrix multiplication.

Would you like a detailed solution to any of these problems or specific steps for any of them?

Follow-Up Questions:

1. How do we find the inverse of a matrix modulo a number like 29?
2. What is the importance of block size in Hill cipher encryption?
3. How can we verify the correctness of the decrypted message in Problem 2?
4. What are some common methods to break a Hill cipher if the matrix is unknown?
5. Can any matrix be used for Hill ciphers, or are there specific requirements?

Tip:

For modular matrix operations, always ensure the determinant is coprime with the modulus (e.g., 29) to guarantee an inverse exists.