arXiv subjects

Aparna Gupte

Publications and source records attributed to Aparna Gupte.

9 recordsLinked to original sources

arXiv2026

Exponentially Fewer-Server PIR from Sparser $S$-Decoding Polynomials

We show that under a plausible number-theoretic conjecture, for any constant $s$ there exists an $s$-server private information retrieval (PIR) protocol that on an $n$-bit database requires communication $\exp(O((\log n)^{1/s} (\log \log n)^{1-1/s}))$. Previous constructions attaining the same communication required $2^{O(s)}$ servers. Our number-theoretic conjecture is implied by existing conjectures, namely the generalized repunit conjecture and Schinzel's hypothesis H (either one of these conjectures would suffice alone). Our result builds on the ``matching vector family + $S$-decoding polynomials'' framework pioneered by Efremenko (STOC 2009) and recently refined by Ghasemi, Kopparty, and Sudan (STOC 2025). The main ingredient is a framework for constructing $S$-decoding polynomials with only $k+1$ nonzero coefficients modulo special products of $k$ primes, resolving an open problem posed by Ghasemi and Kopparty (ITCS 2026). By the lower bound shown by Ghasemi and Kopparty, this is the minimum achievable sparsity. We also empirically validate our construction and make our result unconditional for all $s \leq 15$. We also apply our techniques to regimes where $s$ grows with $n$, showing under a stronger variant of our number-theoretic conjecture that the communication complexity of $s$-server matching-vector PIR can be superpolynomially reduced from the previous state of the art for any $s \leq \exp(o(\sqrt{\log \log n/\log \log \log n}))$. The main result for $s = O(1)$ and its proof were discovered in a GPT-5.5 Pro conversation prompted by the authors.

cs.CC↗

arXiv2026

On Best-Possible One-Time Programs

One-time programs (OTPs) aim to let a user evaluate a program on a single input while revealing nothing else. Classical OTPs require hardware assumptions, and even with quantum information, OTPs for deterministic functionalities remain impossible due to gentle-measurement attacks (Broadbent, Gutoski and Stebila, 2013). While recent works achieve positive results for certain randomized functionalities, the fundamental limits and the strongest achievable security notions remain poorly understood. In this paper, we ask for a "best-possible" OTP that achieves the strongest one-time security achievable by any OTP construction. We first show that a generic best-possible one-time compiler cannot exist, even for classical randomized functionalities (assuming lossy encryption schemes exist). Given this impossibility, we introduce a natural subclass of one-time compilers called "testable one-time program" compilers, which output quantum states augmented with reflection oracles for these program states. We show that best-possible testable OTP compilers are achievable by (1) formulating a generalized Single-Effective-Query (SEQ) simulation security notion for quantum channels and show that SEQ security implies best-possible testable one-time security, and (2) constructing SEQ-secure OTPs for all quantum functionalities in the classical oracle model. This yields the first OTP for arbitrary quantum channels beyond classical randomized functionalities. Finally, we propose stateful quantum indistinguishability obfuscation (stateful quantum iO) -- quantum state obfuscation for stateful quantum programs. We show that (1) stateful quantum iO implies best-possible testable OTPs and (2) stateful quantum iO is also achievable in the classical oracle model. These results identify stateful quantum iO as a promising approach towards best-possible testable OTPs.

cs.CR↗

arXiv2025

Classical Obfuscation of Quantum Circuits via Publicly-Verifiable QFHE

A classical obfuscator for quantum circuits is a classical program that, given the classical description of a quantum circuit $Q$, outputs the classical description of a functionally equivalent quantum circuit $\hat{Q}$ that hides as much as possible about $Q$. Previously, the only known feasibility result for classical obfuscation of quantum circuits (Bartusek and Malavolta, ITCS 2022) was limited to circuits that always reject. On the other hand, if the obfuscator is allowed to compile the quantum circuit $Q$ into a quantum state $|\hat{Q}\rangle$, there exist feasibility results for obfuscating all pseudo-deterministic quantum circuits (Bartusek, Kitagawa, Nishimaki and Yamakawa, STOC 2023, Bartusek, Brakerski and Vaikuntanathan, STOC 2024), and all unitaries (Huang and Tang, FOCS 2025). We show that (relative to a classical oracle) there exists a classical obfuscator for all pseudo-deterministic quantum circuits. We do this by giving the first construction of a compact quantum fully-homomorphic encryption (QFHE) scheme that supports public verification of (pseudo-deterministic) quantum evaluation, relative to a classical oracle. To construct our QFHE scheme, we improve on the approach of Bartusek, Kitagawa, Nishimaki and Yamakawa (STOC 2023), which required ciphertexts that are both quantum and non-compact due to the use of quantum coset states and their publicly-verifiable properties. We introduce new techniques for analyzing coset states that can be generated ''on the fly'', by proving new cryptographic properties of the one-shot signature scheme of Shmueli and Zhandry (CRYPTO 2025). Our techniques allow us to produce QFHE ciphertexts that are purely classical, compact, and publicly-verifiable. This also yields the first classical verification of quantum computation protocol for BQP that simultaneously satisfies blindness and public-verifiability.

quant-ph↗

arXiv2025

Sparse Linear Regression and Lattice Problems

Sparse linear regression (SLR) is a well-studied problem in statistics where one is given a design matrix $X\in\mathbb{R}^{m\times n}$ and a response vector $y=Xθ^*+w$ for a $k$-sparse vector $θ^*$ (that is, $\|θ^*\|_0\leq k$) and small, arbitrary noise $w$, and the goal is to find a $k$-sparse $\widehatθ \in \mathbb{R}^n$ that minimizes the mean squared prediction error $\frac{1}{m}\|X\widehatθ-Xθ^*\|^2_2$. While $\ell_1$-relaxation methods such as basis pursuit, Lasso, and the Dantzig selector solve SLR when the design matrix is well-conditioned, no general algorithm is known, nor is there any formal evidence of hardness in an average-case setting with respect to all efficient algorithms. We give evidence of average-case hardness of SLR w.r.t. all efficient algorithms assuming the worst-case hardness of lattice problems. Specifically, we give an instance-by-instance reduction from a variant of the bounded distance decoding (BDD) problem on lattices to SLR, where the condition number of the lattice basis that defines the BDD instance is directly related to the restricted eigenvalue condition of the design matrix, which characterizes some of the classical statistical-computational gaps for sparse linear regression. Also, by appealing to worst-case to average-case reductions from the world of lattices, this shows hardness for a distribution of SLR instances; while the design matrices are ill-conditioned, the resulting SLR instances are in the identifiable regime. Furthermore, for well-conditioned (essentially) isotropic Gaussian design matrices, where Lasso is known to behave well in the identifiable regime, we show hardness of outputting any good solution in the unidentifiable regime where there are many solutions, assuming the worst-case hardness of standard and well-studied lattice problems.

cs.LG↗

arXiv2024

Quantum One-Time Programs, Revisited

One-time programs (Goldwasser, Kalai and Rothblum, CRYPTO 2008) are functions that can be run on any single input of a user's choice, but not on a second input. Classically, they are unachievable without trusted hardware, but the destructive nature of quantum measurements seems to provide a quantum path to constructing them. Unfortunately, Broadbent, Gutoski and Stebila showed that even with quantum techniques, a strong notion of one-time programs, similar to ideal obfuscation, cannot be achieved for any non-trivial quantum function. On the positive side, Ben-David and Sattath (Quantum, 2023) showed how to construct a one-time program for a certain (probabilistic) digital signature scheme, under a weaker notion of one-time program security. There is a vast gap between achievable and provably impossible notions of one-time program security, and it is unclear what functionalities are one-time programmable under the achievable notions of security. In this work, we present new, meaningful, yet achievable definitions of one-time program security for probabilistic classical functions. We show how to construct one time programs satisfying these definitions for all functions in the classical oracle model and for constrained pseudorandom functions in the plain model. Finally, we examine the limits of these notions: we show a class of functions which cannot be one-time programmed in the plain model, as well as a class of functions which appears to be highly random given a single query, but whose one-time program form leaks the entire function even in the oracle model.

cs.CR↗

arXiv2024

SGD and Weight Decay Secretly Minimize the Rank of Your Neural Network

We investigate the inherent bias of Stochastic Gradient Descent (SGD) toward learning low-rank weight matrices during the training of deep neural networks. Our results demonstrate that training with mini-batch SGD and weight decay induces a bias toward rank minimization in the weight matrices. Specifically, we show both theoretically and empirically that this bias becomes more pronounced with smaller batch sizes, higher learning rates, or stronger weight decay. Additionally, we predict and empirically confirm that weight decay is essential for this bias to occur. Unlike previous literature, our analysis does not rely on assumptions about the data, convergence, or optimality of the weight matrices, making it applicable to a wide range of neural network architectures of any width or depth. Finally, we empirically explore the connection between this bias and generalization, finding that it has a marginal effect on the test performance.

cs.LG↗

arXiv2024

How to Construct Quantum FHE, Generically

We construct a (compact) quantum fully homomorphic encryption (QFHE) scheme starting from (compact) classical fully homomorphic encryption scheme with decryption in $\mathsf{NC}^{1}$, together with a dual-mode trapdoor function family. Compared to previous constructions (Mahadev, FOCS 2018; Brakerski, CRYPTO 2018) which made non-black-box use of similar underlying primitives, our construction provides a pathway to instantiations from different assumptions. Our construction uses the techniques of Dulek, Schaffner and Speelman (CRYPTO 2016) and shows how to make the client in their QFHE scheme classical using dual-mode trapdoor functions. As an additional contribution, we show a new instantiation of dual-mode trapdoor functions from group actions.

quant-ph↗

arXiv2022

Continuous LWE is as Hard as LWE & Applications to Learning Gaussian Mixtures

We show direct and conceptually simple reductions between the classical learning with errors (LWE) problem and its continuous analog, CLWE (Bruna, Regev, Song and Tang, STOC 2021). This allows us to bring to bear the powerful machinery of LWE-based cryptography to the applications of CLWE. For example, we obtain the hardness of CLWE under the classical worst-case hardness of the gap shortest vector problem. Previously, this was known only under quantum worst-case hardness of lattice problems. More broadly, with our reductions between the two problems, any future developments to LWE will also apply to CLWE and its downstream applications. As a concrete application, we show an improved hardness result for density estimation for mixtures of Gaussians. In this computational problem, given sample access to a mixture of Gaussians, the goal is to output a function that estimates the density function of the mixture. Under the (plausible and widely believed) exponential hardness of the classical LWE problem, we show that Gaussian mixture density estimation in $\mathbb{R}^n$ with roughly $\log n$ Gaussian components given $\mathsf{poly}(n)$ samples requires time quasi-polynomial in $n$. Under the (conservative) polynomial hardness of LWE, we show hardness of density estimation for $n^ε$ Gaussians for any constant $ε> 0$, which improves on Bruna, Regev, Song and Tang (STOC 2021), who show hardness for at least $\sqrt{n}$ Gaussians under polynomial (quantum) hardness assumptions. Our key technical tool is a reduction from classical LWE to LWE with $k$-sparse secrets where the multiplicative increase in the noise is only $O(\sqrt{k})$, independent of the ambient dimension $n$.

cs.CR↗

arXiv2022

The Fine-Grained Hardness of Sparse Linear Regression

Sparse linear regression is the well-studied inference problem where one is given a design matrix $\mathbf{A} \in \mathbb{R}^{M\times N}$ and a response vector $\mathbf{b} \in \mathbb{R}^M$, and the goal is to find a solution $\mathbf{x} \in \mathbb{R}^{N}$ which is $k$-sparse (that is, it has at most $k$ non-zero coordinates) and minimizes the prediction error $\|\mathbf{A} \mathbf{x} - \mathbf{b}\|_2$. On the one hand, the problem is known to be $\mathcal{NP}$-hard which tells us that no polynomial-time algorithm exists unless $\mathcal{P} = \mathcal{NP}$. On the other hand, the best known algorithms for the problem do a brute-force search among $N^k$ possibilities. In this work, we show that there are no better-than-brute-force algorithms, assuming any one of a variety of popular conjectures including the weighted $k$-clique conjecture from the area of fine-grained complexity, or the hardness of the closest vector problem from the geometry of numbers. We also show the impossibility of better-than-brute-force algorithms when the prediction error is measured in other $\ell_p$ norms, assuming the strong exponential-time hypothesis.

cs.LG↗