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Are these distributions computationally indistinguishable ?

$f:\{0,1\}^n \to \{0,1\}^n $

$\{X_n\}_{n\in N}$ : uniform distribution for function which $f(0^n)=0^n$ and for other function probability is zero.

$\{Y_n\}_{n\in N}$ : uniform distribution for function which $f(0^n) \neq 0^n$ and for other function probability is zero.

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    $\begingroup$ What is meant by "under function which [...]"? $\endgroup$
    – dkaeae
    Dec 22, 2018 at 10:14
  • $\begingroup$ @dkaeae I mean, for all functions$ f (0^n) = 0^n$, the distribution of $\{X_n\}$ is uniform and for other functions distribution $\{X_n\}$ is non-uniform , and the probability of choosing these (other) functions is zero. $\endgroup$ Dec 22, 2018 at 15:22
  • $\begingroup$ @lkowalcz , thank you. If $\{Y_n\}_{n \in N}$ be a uniform distribution , then $\{X_n\} $ and $\{Y_n\}$ are indistinguishable? I think it's true. Is it true? $\endgroup$ Dec 23, 2018 at 8:34

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It looks like you're asking the following:

Consider the following families of distributions over functions from $\{0,1\}^{n} \to \{0,1\}^{n}$:

$\{X_n\}, \text{ where } X_n \text{ is the uniform distribution over the set of functions } f : \{0,1\}^{n} \to \{0,1\}^{n} \text{ such that } f(0^{n}) = 0^{n} $

$\{Y_n\}, \text{ where } X_n \text{ is the uniform distribution over the set of functions } f : \{0,1\}^{n} \to \{0,1\}^{n} \text{ such that } f(0^{n}) \neq 0^{n} $

Are these families of distributions computationally indistinguishable?

I claim they are computationally distinguishable, assuming that each function can be evaluated on input $0^{n}$ efficiently. Consider the following distinguisher, which on input $f$ computes $f(0^{n})$ and outputs $X$ if $f(0^{n}) = 0^{n}$ and $Y$ otherwise. By the definition of $\{X_n\}$, if $f$ is drawn from this family, this distinguisher will output $X$. By the definition of $\{Y_n\}$, if $f$ is drawn from this family, this distinguisher will output $Y$.

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