Decryption of Ciphertext 'HI' using Hill Cipher

Hill Cipher decryption requires multiplying the ciphertext vector by the inverse of the key matrix modulo 26.

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Given

• Ciphertext: HI
• Key Matrix: $K = \begin{bmatrix} 5 & 3 \\ 4 & 3 \end{bmatrix}$

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Step A: Convert Ciphertext to Numbers

• H = 7, I = 8
• Ciphertext vector: $C = \begin{bmatrix} 7 \\ 8 \end{bmatrix}$

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Step B: Find the Inverse of Key Matrix (mod 26)

i. Calculate Determinant:

$$\det(K) = (5 \times 3) - (3 \times 4) = 15 - 12 = 3$$

ii. Find Multiplicative Inverse of det mod 26:

We need $3^{-1} \mod 26$, i.e., find $x$ such that $3x \equiv 1 \mod 26$

$$3 \times 9 = 27 \equiv 1 \mod 26$$

So, $3^{-1} \mod 26 = 9$

iii. Find Adjugate Matrix:

$$\text{adj}(K) = \begin{bmatrix} 3 & -3 \\ -4 & 5 \end{bmatrix}$$

iv. Compute Inverse Matrix mod 26:

$$K^{-1} = 9 \times \begin{bmatrix} 3 & -3 \\ -4 & 5 \end{bmatrix} \mod 26$$

$$K^{-1} = \begin{bmatrix} 27 & -27 \\ -36 & 45 \end{bmatrix} \mod 26 = \begin{bmatrix} 1 & 25 \\ 16 & 19 \end{bmatrix}$$

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Step C: Decrypt by Multiplying $K^{-1} \times C \mod 26$

$$P = \begin{bmatrix} 1 & 25 \\ 16 & 19 \end{bmatrix} \times \begin{bmatrix} 7 \\ 8 \end{bmatrix} \mod 26$$

$$P_1 = (1 \times 7) + (25 \times 8) = 7 + 200 = 207 \mod 26 = 207 - 7(26) = 207 - 182 = 25$$

$$P_2 = (16 \times 7) + (19 \times 8) = 112 + 152 = 264 \mod 26 = 264 - 10(26) = 264 - 260 = 4$$

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Step D: Convert Numbers to Letters

• 25 = Z
• 4 = E

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Result

$$\boxed{\text{Plaintext} = \textbf{ZE}}$$