Post processing operations on pseudo-random generators

Caio Nogueira

10/24/23, 1:49 PM

I am struggling to solve this proof.

The goal is to prove that $H \circ G$, which is a composite function $H(G(s))$ can be a pseudo-random generator under some conditions on $H$, given that $G$ is also a pseudo-random generator. Note that $H$ is length-preserving.

However, I can't find a way to prove (using the definition of a pseudo-random generator) that if $H$ is bijective, deterministic, can be computed in polynomial time and doesn't convey any information regarding the original input $s$, $H \circ G$ is also a pseudo-random generator.

I would appreciate some help on this topic.

0 + 7

pseudo-random-generator

Mark

10/24/23, 2:20 PM

Assume this isn't the case --- i.e. $H\circ G$ is *not* a PRG under your assumptions. What does this imply about $G$?

Caio Nogueira

10/24/23, 3:05 PM

@Mark I guess that would mean that G cannot be a PRG. However, I can't find a way to prove this contradiction with the formal definition of a PRG

Mark

10/24/23, 4:59 PM

What does it mean for something to not be a PRG?

Caio Nogueira

10/24/23, 5:08 PM

@Mark Informally speaking, if $G$ is not a PRG, then it would be possible for an adversary to distinguish between its output and a truly random string.

Mark

10/24/23, 5:10 PM

Exactly! So assume $H\circ G$ is not a PRG, i.e. there is an adversary $\mathcal{A}_{H\circ G}$ that distinguishes its output. Can you use this to create an adversary $\mathcal{A}_G$ against $G$?

Caio Nogueira

10/24/23, 5:18 PM

I think that if $H$ is deterministic, then it would be possible for A to win over $G$

Mark

10/24/23, 5:21 PM

If you think so, try to prove it. Roughly, (formally) build $\mathcal{A}_G$ however you can, then see if the assumptions you require on $H$ are implied by the conditions you were given as part of the problem statement.

Elon Musk

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

EN: Post processing operations on pseudo-random generators

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.