Let $F:\{0,1\}^n\times \{0,1\}^m \to \{0,1\}^l$ be a PRF. I want to show that $G(-)=F(-,x_0)$ is a PRG for every $x_0\in \{0,1\}^m$.

Proof attempt:

Let $D$ be an efficient distinguisher against $G$ in the PRG experiment (see below). We aim to show that $D$ has negligible advantage.

Consider the following efficient distniguisher $D'$ in the RPF experiment. Given access to a function $\mathcal O$ which is either a random function $\{0,1\}^m\to \{0,1\}^l$ or $F_k$, then $D'$ outputs precisely the accept/reject choice of $D$ on the string $\mathcal O(x_0)$.

By construction $D'$ wins if and only if $D$ accepts. By assumptions, since $F$ is an RPF, the former event has probability bounded above by $\frac{1}{2} + \text{negl}(n)$ for some negligible $\text{negl}$, and hence the same is true for the latter event.

My questions:

1. Is the proof correct?
2. Is the level of rigor sufficient?

Additional information: Here are the two experiments I was referring to (taken from Katz/Lindell).

Let $F:\{0,1\}^n\times \{0,1\}^m \to \{0,1\}^l$ be a PRF. I want to show that $G(-)=F(-,x_0)$ is a PRG for every $x_0\in \{0,1\}^m$.

Proof attempt:

Let $D$ be an efficient distinguisher against $G$ in the PRG experiment (see below). We aim to show that $D$ has negligible advantage.

Consider the following efficient distinguisher $D'$ in the PRF experiment. Given access to a function $\mathcal O$ which is either a random function $\{0,1\}^m\to \{0,1\}^l$ or $F_k$, then $D'$ outputs precisely the accept/reject choice of $D$ on the string $\mathcal O(x_0)$.

By construction $D'$ wins if and only if $D$ accepts. By assumptions, since $F$ is a PRF, the former event has probability bounded above by $\frac{1}{2} + \text{negl}(n)$ for some negligible $\text{negl}$, and hence the same is true for the latter event.

My questions:

1. Is the proof correct?
2. Is the level of rigor sufficient?

Additional information: Here are the two experiments I was referring to (taken from Katz/Lindell).

Let $F:\{0,1\}^n\times \{0,1\}^m \to \{0,1\}^l$ be a PRF. I want to show that $G(-)=F(-,x_0)$ is a PRG for every $x_0\in \{0,1\}^m$.

Proof attempt:

Let $D$ be an efficient distinguisher against $G$ in the PRG experiment (see below). We aim to show that $D$ has negligible advantage.

Consider the following efficient distniguisher $D'$ in the RPF experiment. Given access to a function $\mathcal O$ which is either a random function $\{0,1\}^m\to \{0,1\}^l$ or $F_k$, then $D'$ outputs precisely the accept/reject choice of $D$ on the string $\mathcal O(x_0)$.

By construction $D'$ wins if and only if $D$ accepts. By assumptions, since $F$ is an RPF, the former event has probability bounded above by $\frac{1}{2} + \text{negl}(n)$ for some negligible $\text{negl}$, and hence the true is for the latter event.

My questions:

1. Is the proof correct?
2. Is the level of rigor sufficient?

Additional information: Here are the two experiments I was referring to (taken from Katz/Lindell).

Let $F:\{0,1\}^n\times \{0,1\}^m \to \{0,1\}^l$ be a PRF. I want to show that $G(-)=F(-,x_0)$ is a PRG for every $x_0\in \{0,1\}^m$.

Proof attempt:

Let $D$ be an efficient distinguisher against $G$ in the PRG experiment (see below). We aim to show that $D$ has negligible advantage.

Consider the following efficient distniguisher $D'$ in the RPF experiment. Given access to a function $\mathcal O$ which is either a random function $\{0,1\}^m\to \{0,1\}^l$ or $F_k$, then $D'$ outputs precisely the accept/reject choice of $D$ on the string $\mathcal O(x_0)$.

By construction $D'$ wins if and only if $D$ accepts. By assumptions, since $F$ is an RPF, the former event has probability bounded above by $\frac{1}{2} + \text{negl}(n)$ for some negligible $\text{negl}$, and hence the same is true for the latter event.

My questions:

1. Is the proof correct?
2. Is the level of rigor sufficient?

Additional information: Here are the two experiments I was referring to (taken from Katz/Lindell).