Proving that a PRG is predictable

Chris

3/14/24, 12:01 AM

I am attending the video lectures from Prof Dan Boneh. He gives the following example.

Let $G:\mathcal K\longrightarrow \Bbb Z_2^n$ be a PRG with the property that from the last $\frac{n}{2}$ digits of $G(k)$ we can easily compute the first $\frac{n}{2}$ digits of $G(k)$. We want to show that $G$ is predictable for some $i\in\{0,\dots,n-1\}$.

Well, it is clear that we should use the contrapositive of Yao's Theorem. That is, if we prove that $G$ is not secure, then $G$ is predictable.

To do so, we have to define a statistical test $B:\Bbb Z_2^n\longrightarrow \Bbb Z_2$, and show that the advantage $$\mathrm{Adv_{PRG}}(B,G):=|\mathrm{Pr}_{k\overset{R}{\leftarrow} \mathcal K}(B(G(k))=1)-\mathrm{Pr}_{k\overset{R}{\leftarrow} \mathbb{Z_2^n} }(B(r)=1)|\in [0,1] $$ is non-negligible.

So, my question is what the statistical test $B$ could be?

0 + 1

statistical-test

pseudo-random-generator

Daniel S

3/14/24, 7:28 AM

HINT: If someone came to you with $n$ bits from $G$, but was worried that they might have become garbled, how would you check for garbling?

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Proving that a PRG is predictable

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.