Proof for secure stream cipher implies secure PRG

cryptolearner

12/29/23, 4:09 AM

I am self studying "A Graduate Course in Applied Cryptography" by Boneh-Shoup. I am not sure if my proof for the following problem in the book is correct. The problem asks to prove that if a stream cipher is semantically secure, then the underlying PRG is secure. Please let me know if my attempt works. If my solution is incorrect, I would appreciate a hint for how to approach the problem.

Let $G$ be the underlying PRG of the stream cipher $\mathcal{E}$ and let $\mathcal{B}$ be an adversary of G. We construct an adversary $\mathcal{A}$ attacking the semantic security of the corresponding stream cipher and we relate $\mathcal{A}$'s advantage to $\mathcal{B}$'s advantage.

As in the semantic security game, $\mathcal{A}$ must send messages $m_0, m_1$ to the stream cipher challenger. $\mathcal{A}$ chooses $m_0$ to be a string of zeros and samples $m_1$ at uniform random from the message space.

The challenger responds choosing $b$ at random from $\{0,1\}$, $s$ at random from the PRG seed space and computing $G(s)$. He proceeds by sending $G(s) \oplus m_b$ to $\mathcal{A}$.

Now $\mathcal{A}$ invokes $\mathcal{B}$ by playing the role of challenger to $\mathcal{B}$ in the PRG security game. He forwards $G(s) \oplus m_b$ to $\mathcal{B}$ and outputs whatever $\mathcal{B}$'s response is.

Let experiment zero be the case when $m_b$ is the string of zeros (ie b = 0). Let $W_0$ be the event where $\mathcal{B}$ outputs $1$ in experiment zero.

Let experiment one be the case when $b = 1$.Let $W_1$ be the event where $\mathcal{B}$ outputs $1$ in experiment one.

Note that $W_0$ is precisely the event where $\mathcal{A}$ outputs one in experiment zero and $W_0$ is precisely the event where $\mathcal{A}$ outputs one in experiment one. So we need to calculate $$|Pr[W_0] - Pr[W_1]|.$$ Since $m_1$ is randomly generated, the distribution of $m_1 \oplus G(s)$ is uniform random. By construction, the distribution of $m_0 \oplus G(s)$ is the same as the distribution of the PRG. So this quantity is precisely $\mathcal{B}$'s advantage.

Therefore for every PRG adversary $\mathcal{B}$ there exists a semantic security adversary $\mathcal{A}$ with $$PRGAdv(B,G) = SSAdv(A, \mathcal{E}).$$ But since the stream cipher is secure, we know $SSAdv(A, \mathcal{E})$ is negligible which tells us the PRG is secure.

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puppet puppet

12/29/23, 8:31 AM

Proof looks convincing to me :)

Small nits:

Note that $W_0$ is precisely the event where $\mathcal{A}$ outputs one in experiment zero and $W_0$ [sic] is precisely the event where $\mathcal{A}$ outputs one in experiment one.

I think the second random variable here should be $W_1$, not $W_0$.

The challenger responds choosing $b$ at random from $\{0,1\}$, $s$ at random from the PRG seed space and computing $G(s)$.

This phrasing makes it seem like the encryption challenger chooses a random key/seed and bit upon receiving the encryption query consisting of two chosen messages. Usually in security games related to encryption, the challenger chooses a random key and bit once at startup, and then uses these values across every $(m_0, m_1)$ encryption query. For one-time semantic security (i.e. only needing to encrypt a single message per key), which is the relevant security definition here, this distinction doesn't matter; there is only one $(m_0, m_1)$ encryption query that the challenger responds to. It does matter for multiple-message semantic security (i.e. potentially more than one message encrypted per key), as the challenger must respond to potentially many encryption queries $(m_0, m_1)_0, (m_0, m_1)_1, \ldots$ using the same key and bit.

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EN: Proof for secure stream cipher implies secure PRG

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