Existence of PRGs satisfying the following weaker definition

Generic

6/27/24, 2:54 AM

Consider the following definition. Let $p(n) > n$ be a polynomial, and $G_n: \{0, 1\}^n \rightarrow \{0, 1\}^{p(n)}$ a PRG. Moreover, given $x \leftarrow \{0, 1\}^n$, we say $S$ is a length $n-1$ contiguous substring of $G(x)$ if $S = s_0\cdots s_{n-2}$ with $s_i \in \{0, 1\}$ is a contiguous substring of $G(x)$.

Then we say a PRG $G$ is "$n-1$ subsequence secure" if for all PPT adversaries $\mathcal{A}$, and any contiguous subsequence (substring) $S$ of $G(x)$ with $|S|=n-1$, there exists a negligible function $\mu$ such that $$ |Pr_{x \leftarrow \{0, 1\}^n}[\mathcal{A}(S)=1] - Pr_{r \leftarrow \{0, 1\}^{n-1}}[\mathcal{A}(r)=1]| \leq \mu(n). $$ Do we have any PRGs that could provably satisfy this definition (unconditionally)? I ask about provably so because this definition is strictly weaker than the standard definition of a PRG.

1 + 0

provable-security

one-way-function

pseudo-random-generator

Score:1

Crypto

Daniel S

6/27/24, 8:15 PM

If we start with a primitive linear feedback shift register of length $n$, it defines a "full cycle" generator where we can take any non-zero initial fill and the state will cycle through all non-zero fills. We can augment this with an additional zero for the feedback bit in the case 1000..000 and define the feedback back for the 000...000 state to be one and then cycle through all states. The set of all possible fills would then give a uniform distribution on all possible substrings and would satisfy the criterion with $\mu=0$

It would however be a rubbish PRNG.

+ 0

Elon Musk

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

EN: Existence of PRGs satisfying the following weaker definition

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