Question: Let E k : {0, 1} 128 {0, 1} 128 be an ideal block cipher with message space {0, 1} 128 , ciphertext space {0,

Let E_k : {0, 1} ¹²⁸ {0, 1}¹²⁸ be an ideal block cipher with message space {0, 1}¹²⁸, ciphertext space {0, 1}¹²⁸, and key space {0, 1}¹²⁸. Let x and y denote bitstrings of length 128. Which of the following hash functions H_i : {0, 1}²⁵⁶ {0, 1}¹²⁸, if any, are preimage resistant? Explain.

(a) H₁(x, y) = E_y(y) x.

(b) H₂(x, y) = E_y(x).

(d) H₄(x, y) = E_x(y) y.

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