Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.07266

Quadratic Extractors

Gautam Chandrasekaran·Jesse Goodman·David Zuckerman
View original source ↗Read source document ↗Export metadata ↓

Abstract

A quadratic source is a random variable $X\sim\mathbb{F}_2^n$ generated by applying $n$ quadratics to a hidden uniform string. Previously, the best-known explicit extractor for this model required min-entropy roughly $k=n-\sqrt{\log n}$. In this paper, we give an explicit construction for min-entropy $k=\mathrm{polylog}(n)$. Even for the special case of $2$-local sources, where each bit of $X$ depends on two hidden uniform bits, achieving an entropy requirement below $k=\sqrt{n}$ was an important open problem. This is because such sources provably cannot be reduced (in a black-box manner) to our most powerful general-purpose seedless extractor, known as a sumset extractor. Our new extractor bypasses this barrier, providing the first explicit construction to go beyond sumset extraction.

Explore related subjects

cs.CCcs.CRmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gautam Chandrasekaran, Jesse Goodman, David Zuckerman. 2026-11-05. Quadratic Extractors. https://arxiv.org/abs/2611.07266

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↗