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Let’s solve these problems step by step.

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### **Problem 1**: Hill Cipher

**(a) Encrypting "In pulvere vinces"**

To solve this, we will:

1. Convert the message into numerical values using a simple alphabet mapping (e.g., A=0, B=1, ..., Z=25).
2. Use block size 3 and apply the given matrix \(A = \begin{bmatrix} 5 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 4 \end{bmatrix} \).
3. Compute modulo 29 as instructed.

**(b) Decrypting "UWVIKHACQVGI"**

1. Convert the ciphertext back to numbers.
2. Find the inverse of matrix \(A\) modulo 29.
3. Multiply by the ciphertext blocks and reduce modulo 29.

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### **Problem 2**: Hill Cipher Decryption by Eve

1. Known plaintext: Bob always starts with “Hello.”
2. Break the ciphertext using block length 2.
3. Recover the encryption matrix using the plaintext-ciphertext relationship and decrypt the rest.

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### **Problem 3**: Finding \(A^{-1}\) using Gauss-Jordan

Given:
\[ A = \begin{bmatrix} 2 & 3 & 0 \\ 1 & -2 & -1 \\ 2 & 0 & -1 \end{bmatrix} \]

Apply Gauss-Jordan elimination to find the inverse.

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### **Problem 4**: Prove \((AB)^{-1} = B^{-1}A^{-1}\)

- Using the associative property of matrix multiplication and the definition of the inverse:
  \[ (AB)(B^{-1}A^{-1}) = I \]

Would you like detailed calculations for any of these problems? Let me know where you'd like to begin!

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### Related Questions:
1. How does the Hill Cipher ensure security with modular arithmetic?
2. What is the importance of matrix invertibility in Hill ciphers?
3. How do we compute modular inverses in Hill Cipher matrices?
4. How is plaintext size handled in encryption schemes like Hill Cipher?
5. Why does matrix multiplication follow the order for inverses \((AB)^{-1} = B^{-1}A^{-1}\)?

**Tip**: Understanding modular arithmetic is crucial when working with cryptographic schemes.

Using the Hill cipher with block size 3 and encryption matrix, encrypt and decrypt messages, and find matrix inverses.

Step-by-step solutions for encrypting and decrypting messages using the Hill Cipher with matrix operations and modular arithmetic. Suitable for students studying cryptography and linear algebra.

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