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Let : { 0 , 1 } × { 0 , 1 } → { 0 , 1 } F:{0,1} n ×{0,1} n →{0,1} n be a secure PRF (i.e. a PRF where the key space, input space, and output space are all { 0 , 1 } {0,1} n ) and say = 128 n=128. Which of the following is a secure PRF (there is more than one correct answer): 1 point ′ ( ( 1 , 2 ) , ) = { ( 1 , ) when ≠ 0 2 otherwise F ′ ((k 1 ​ ,k 2 ​ ), x)={ F(k 1 ​ ,x) k 2 ​ ​ when x  ​ =0 n otherwise ​ ′ ( , ) = reverse ( ( , ) ) F ′ (k,x)=reverse(F(k,x)) where reverse(y) reverses the string y so that the first bit of y is the last bit of reverse(y), the second bit of y is the second to last bit of reverse(y), and so on. ′ ( , ) = { ( , ) when ≠ 0 0 otherwise F ′ (k, x)={ F(k,x) 0 n ​ when x  ​ =0 n otherwise ​ ′ ( , ) = ( , ) ∥ 0 F ′ (k,x)=F(k,x) ∥ ∥ ∥ ​ 0 (here ∥ ∥ ∥ ∥ ​ denotes concatenation) ′ ( , ) = ( , ) [ 0 , … , − 2 ] F ′ (k,x)=F(k,x)[0,…,n−2] (i.e., ′ ( , ) F ′ (k,x) drops the last bit of ( , ) F(k,x)) ′ ( , ) = ⨁ F ′ (k, x)=k⨁x

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To determine which of the given options represent a secure PRF, we need to consider the properties of a secure Pseudo-Random Function (PRF). A secure PRF must exhibit pseudorandomness, meaning that it should be computationally indistinguishable from a truly random function.

Let's analyze each of the provided options:

1. \(F'(k_1, k_2), x) = \begin{cases) (1, F(k_1, x) \oplus k_2), & \text{when ] x \neq 0 \\ 2, & \text{otherwise) \end{cases}\)

- This function seems to XOR the output of \(F(k_1, x)\) with \ (k_2\) when \(x \neq 0\). If the XOR operation is reversible with \ (k_2\), it may not preserve the pseudorandomness property required for a secure PRF.

2. \(F'(k, x) = \text(reverse)(F(k, x))\)

- Reversing the output of the original PRF does not guarantee the preservation of pseudorandomness, as the reversed output may not exhibit the same randomness properties as the original output.

3. \(F'(k, x) = \begin{cases) (F(k, x), 0^n), & \text{when } x \neq 0 \\ n, & \text{otherwise} \end{cases}\)

Appending zeros to the output of the original PRF when \(x \neq 0\) does not necessarily maintain the pseudorandomness property, as the added zeros may introduce patterns that deviate from randomness.

4. \(F'(k, x) = F(k, x) II 0\)

- Concatenating a fixed value (in this case, 0) to the output of the original PRF does not ensure the preservation of pseudorandomness, as the fixed value can introduce regularities that compromise the randomness.

5. \(F'(k, x) = F(k, x)[0, ..., n-2]\)

- Truncating the last bit of the output from the original PRF might impact the pseudorandomness property, as removing bits can alter the distribution of the output and potentially introduce patterns.6. \(F'(k, x) = k \oplus x\)

- Performing an XOR operation between the key \(k\) and the input \(x\) does not guarantee the preservation of pseudorandomness, as the XOR operation may not sufficiently hide the underlying structure of the function.

After considering each option, it is essential to note that options 1, 2, 3, 4, and 5 may compromise the security of the PRF by potentially introducing patterns or regularities that deviate from true randomness. Option 6 also falls short in ensuring the pseudorandomness required for a secure PRF.

Therefore, none of the provided options can be considered a secure PRF based on the criteria of maintaining computational indistinguishability from a truly random function.

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