Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.07829

Multiplicative and Additive Extractors for Low Entropy Samplable Sources Under Reduced Assumptions

Marshall Ball·Eshan Chattopadhyay·Mohit Gurumukhani·Ronen Shaltiel·Yunya Zhao
View original source ↗Read source document ↗Export metadata ↓

Abstract

Trevisan and Vadhan [TV, FOCS'00] constructed (seedless) extractors for distributions samplable by poly-size circuits that have very high min-entropy, under the very strong complexity theoretic hardness assumption that $\mathsf{E}=\mathsf{DTIME}(2^{O(n)})$ requires size $2^{Ω(n)}$ circuits that have oracle access to $Σ_6^{\text{P}}$. In this paper we give three new explicit constructions of extractors for samplable distributions. First, we construct an extractor that extracts one bit, with constant error, for min-entropy $k=O(\log n)$, which matches the non-constructive probabilistic argument. This result uses hardness for nondeterministic circuits, and achieves the same parameters as the recent extractor of Oh and Shaltiel [OS, STOC'26] under an incomparable hardness assumption. In contrast to the extractor of [OS26], our extractor (and, similarly, the constructions below) also works for the more general class of distributions samplable with postselection (a model introduced by [BGDM, FOCS'23] that generalizes samplable and recognizable sources). Second, we construct an extractor that extracts almost all the entropy, with polynomially small error, for min-entropy $k=n^α$, where $α>0$ is an arbitrary constant. This result also uses hardness for nondeterministic circuits, and significantly improves upon the recent extractor of [Shaltiel, FOCS'25] which required a much stronger assumption. This extractor is also \textit{multiplicative}: a stronger notion suggested in [AASY, Comp. Complexity'16; Shaltiel, CCC'25] to make extractors for samplable distributions suitable for selecting keys for cryptographic protocols. Third, we construct an extractor that extracts almost all the entropy, with polynomially small error, for min-entropy $k=\mathrm{polylog}(n)$. This result uses hardness for circuits that have oracle access to $Σ_2^{\text{P}}$.

Explore related subjects

cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Marshall Ball, Eshan Chattopadhyay, Mohit Gurumukhani, Ronen Shaltiel, Yunya Zhao. 2026-11-05. Multiplicative and Additive Extractors for Low Entropy Samplable Sources Under Reduced Assumptions. https://arxiv.org/abs/2611.07829

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Quadratic Lower Bound for Noncommutative Circuits

We prove that every fan-in 2 noncommutative arithmetic circuit computing the palindrome polynomial $\mathrm{Pal}_{n,d}(X,Y)$ of degree $2d$ in $2n$ variables has size $Ω(nd)$, which is tight up to a constant. In particular, when $d = n$ we obtain an $Ω(n^2)$ lower bound, improving the previous lower bound of $Ω(n \log n)$ by Nisan and concurrent work of $Ω(n^{1.5})$ by Raz (FOCS 2026). The proof builds on and refines a previous work of the first author, by constructing for every gate a small set that spans a subset of the coefficients of the polynomial computed by that gate. We show that the structure of the set depends only on the subsets of multiplication gates along paths going from that gate to a leaf. From this description, we are then able to get an upper bound on this set based on the amount of product gates in the circuit. The proof is surprisingly short and simple to understand, considering that improving Nisan's $Ω(n \log n)$ lower bound for noncommutative circuits has been an open problem for decades.

cs.CC↗
arXiv2026

A Resolution of Friedgut's Conjecture on Influential Coalitions

We prove that, for every constant $\varepsilon>0$ and every function $f:Σ^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the output equal to $b$ with probability at least $1-\varepsilon$. The bound is independent of the alphabet size and also holds for monotone Boolean functions on $[0,1]^n$, resolving a conjecture of Friedgut (Combinatorics, Probability and Computing, 2004). Unlike the Boolean cube setting, where Kahn, Kalai, and Linial (FOCS, 1988) give a coalition bound of $O(n/\log n)$, no sublinear bound independent of the alphabet size was previously known. In collective coin flipping, our result gives the first sublinear bound on the number of bad players needed to force a fixed output with probability at least $1-\varepsilon$ in any one-round protocol with independent uniform messages, regardless of the message length. A key ingredient in our proof is an encoding that lets us relate the influence of a function on a product space to the $p$-biased influence of the encoded function. We then rely on a structure theorem of Hatami (Annals of Mathematics, 2012) for functions with small $p$-biased influence to bias the encoded function.

cs.CC↗
arXiv2026

Information Loss Is All You Need: $TC^0$ Pseudorandomness from Worst-Case Lattices

Can deterministic information loss yield post-quantum pseudorandomness with $TC^0$ evaluation? We show that deterministic rounding achieves this under quantum worst-case lattice hardness, with fixed polynomial parameters and no predetermined sample bound. We introduce learning with algebraic rounding (LAR), which applies an explicit integer rounding map to blocks of secret-dependent linear measurements and encodes the result in a lattice quotient. For one fixed family with polynomially bounded modulus and dimensions, deterministic labels on exactly uniform, secret-independent public matrices are computationally indistinguishable from independent uniform labels for every polynomial number of classical random examples. Our classical reduction from Gaussian LWE, combined with the quantum worst-case reduction to LWE, bases LAR on GapSVP and SIVP over general lattices at approximation factor $\widetilde{O}(q)$, where $q$ is the modulus. The proof uses source-independent affine transport: public affine maps sampled independently of the hidden errors transport the Gaussian error law close to uniform on the rounding map's zero-label fiber while preserving exactly uniform public matrices. For every fixed inverse-polynomial accuracy, we give a polynomial-time sampler for the same public parameter family. This yields the first post-quantum weak pseudorandom function with joint key-and-input evaluation in $TC^0$, secure against uniform quantum polynomial-time distinguishers given any polynomial number of classical uniform random examples. We also prove hardness of learning polynomial-size two-hidden-layer ReLU networks from noiseless uniform random examples, even for quantum polynomial-time improper learners.

cs.CC↗