Part A: Monoalphabetic vs Polyalphabetic — Which is More Vulnerable?

Monoalphabetic cipher is more vulnerable than polyalphabetic cipher because it uses a single fixed substitution alphabet, making it easily breakable by frequency analysis.

Justification:

• In monoalphabetic cipher, each plaintext letter is always mapped to the same ciphertext letter throughout the message.
• This preserves the statistical frequency of letters (e.g., 'E' is the most common letter in English).
• An attacker can perform frequency analysis — comparing letter frequencies in ciphertext with known language frequencies to crack the cipher.
• In polyalphabetic cipher (e.g., Vigenère cipher), multiple substitution alphabets are used in rotation, which flattens the frequency distribution and makes frequency analysis much harder.

Conclusion: Monoalphabetic cipher is significantly more vulnerable because its one-to-one mapping leaks statistical patterns of the plaintext.

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Part B: Weak Keys in DES

Weak keys in DES are keys that produce identical subkeys for all 16 rounds, making encryption and decryption the same operation (i.e., $E_K(E_K(P)) = P$).

Types of Weak Keys:

• DES has 4 weak keys and 12 semi-weak keys.
• The 4 weak keys are:

Weak Key (Hex)	Pattern
0000000 0000000	All zeros
FFFFFFF FFFFFFF	All ones
0000000 FFFFFFF	First half zeros, second half ones
FFFFFFF 0000000	First half ones, second half zeros

• Semi-weak keys are pairs of keys where encrypting with one key is equivalent to decrypting with the other: $E_{K1}(P) = D_{K2}(P)$.
• These keys are dangerous because they reduce the effective security of DES.

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Part C: Round Operation in IDEA

IDEA (International Data Encryption Algorithm) is a block cipher that operates on 64-bit plaintext blocks using a 128-bit key over 8.5 rounds (8 full rounds + 1 output transformation).

Round Operation:

Each round takes a 64-bit input divided into four 16-bit sub-blocks: $X_1, X_2, X_3, X_4$ and uses 6 subkeys ($K_1$ to $K_6$) per round.

Three Algebraic Operations Used:

• Multiplication modulo $2^{16} + 1$ (denoted ⊙)
• Addition modulo $2^{16}$ (denoted ⊞)
• Bitwise XOR (denoted ⊕)

Steps in One Round:

• a. $X_1$ ⊙ $K_1$ → result $R_1$
• b. $X_2$ ⊞ $K_2$ → result $R_2$
• c. $X_3$ ⊞ $K_3$ → result $R_3$
• d. $X_4$ ⊙ $K_4$ → result $R_4$
• e. $R_1$ ⊕ $R_3$ → fed into MA (Multiplication-Addition) structure
• f. $R_2$ ⊕ $R_4$ → fed into MA structure
• g. In MA: the two inputs are combined using $K_5$ and $K_6$ with ⊙ and ⊞ operations to produce two 16-bit outputs.
• h. The MA outputs are XORed with all four sub-blocks.
• i. The inner two sub-blocks are swapped (except in the last round).

Output Transformation (Half Round 9):

• Uses 4 subkeys with ⊙ and ⊞ operations on the four sub-blocks to produce the final 64-bit ciphertext.

Key Insight: The strength of IDEA comes from mixing three incompatible algebraic operations from different algebraic groups, making cryptanalysis extremely difficult.

Conclusion: IDEA's round operation achieves confusion and diffusion by combining multiplication, addition, and XOR — operations that are algebraically incompatible — ensuring high security.