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 Hi : {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).

(c) H₃(x, y) = E_y(x) y.

(d) H₄(x, y) = E_x(y) y.