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Let F : {0, 1}^k {0, 1}^m {0, 1}ⁿ be a secure PRF (pseudorandom function). Consider the following function families:

1. F₁ : {0, 1}^k {0, 1}^m {0, 1}²ⁿ specified for all x {0, 1}^m and all K {0, 1}^k by

F₁(K, x) = F (K, x) F (K, x)

2. F₂ : {0, 1}^k {0, 1}^2m {0, 1}²ⁿ specified for all x1, x2 {0, 1}^m and all K {0, 1}^k by

F₂(K, x₁ x₂) = F (K, x1) F (K, x₁ x₂)

Above, x y means concatenation of x and y, x means the bitwise complement of x, and is bitwise XOR.

Prove that F1 and F2 are not secure PRFs (pseudorandom functions).

Let F:{0,1}k{0,1}m{0,1}n be a secure PRF. Consider the following function families: 1. F1:{0,1}k{0,1}m{0,1}2n specified for all x{0,1}m and all K{0,1}k by F1(K,x)=F(K,x)F(K,x) 2. F2:{0,1}k{0,1}2m{0,1}2n specified for all x1,x2{0,1}m and all K{0,1}k by F2(K,x1x2)=F(K,x1)F(K,x1x2) Above, xy means concatenation of x and y,x means the bitwise complement of x, and is bitwise XOR. Prove that F1 and F2 are not secure PRFs