Work each of the following problems from the textbook, Introduction to Cryptography with Coding Theory: Second Edition by Trappe and Washington.. For each problem, show your work and justify your answer, whether or not the question specifically requests this.

Problem 2: Affine Cipher

Textbook, problem 2.13.4.

Problem 3: Vigenère Cipher

Textbook, problem 2.13.10.

Problem 4: Hill Cipher

Textbook, problem 2.13.15.

Problem 5: LFSR Machine

Textbook, problem 2.13.21.

Problem 6: Chaining Modes

For each of the standard chaining modes ECB, CBC, CFB, OFB, and PCBC, describe how to recover from the loss of a ciphertext block, and state explicitly how many message blocks become unrecoverable as a result of the loss.

Problem 7: Birthday Paradox Calculation

Write a computer program to compute p_m, the probability that two random subsets of size m drawn from a universe of size 100 have a non-empty intersection. Use your program to find the smallest values of m for which p_m ≥ 1∕2 and for which p_m ≥ 3∕4.

Problem 8: Meet-in-the-Middle Attack

Let E,D be the encryption and decryption functions for a symmetric cryptosystem with key space . Assume the plaintext and ciphertext spaces are the same. Let EE,DD be the encryption and decryption functions for the doubled verion as described in Lecture 5.

Eve is carrying out a known plaintex attack on the doubled system. Suppose she knows a pair (m,c) where c = EE_(k1,k2)(m), but she does not know k₁ or k₂. She computes two sets:

1. Explain why X_m ∩ X_c∅.
2. How might these sets help her break the doubled system?
3. Construct a symmetric cryptosystem and find a pair (m,c) for which ∣X_m ∩ X_c∣ ≥ 2, or show why that is not possible.