arXiv subjects

Shashwat Agrawal

Publications and source records attributed to Shashwat Agrawal.

5 recordsLinked to original sources

arXiv2026

Spectral Method attacks Sparse LWE, Sparse LPN and Beyond

Given a set of $k$-sparse linear equations over a ring $R$, we give algorithms to determine whether the right-hand sides are random or have a secret assignment planted with noise. For a parameter $k/2\leq l\leq n$, we give a spectral method to solve this problem in $\widetilde{O}\left(\binom{n}{l}\lvert{R}\rvert^l\right)$ time except with probability at most $n^{-Ω(l)}$, provided the number of samples is roughly at least $\left(\frac{\lvert{R}\rvert n}{l}\right)^{k/2}$. This attack generalizes the Kikuchi method described by Wein et. al. (Journal of the ACM 2019) for $\mathbb{Z}_2$ to (commutative) rings of any finite size. We also give a simpler algorithm with better runtime than the spectral method and better sample complexity when $\lvert{R}\rvert=ω(n/l)$. As a consequence, we obtain new sample-time tradeoffs for the decision problem of sparse LWE, sparse LPN over higher modulus $q$, and in general the distinguishing random vs planted $\mathbb{Z}_q$-linear equations for a large class of noise distributions. Our results imply a tightness of the hardness claims of Jain, Lin, Saha (Annual International Cryptology Conference, 2024) for sparse LWE.

cs.CR↗

arXiv2026

Fast $k$-means Seeding Under The Manifold Hypothesis

We study beyond worst case analysis for the $k$-means problem where the goal is to model typical instances of $k$-means arising in practice. Existing theoretical approaches provide guarantees under certain assumptions on the optimal solutions to $k$-means, making them difficult to validate in practice. We propose the manifold hypothesis, where data obtained in ambient dimension $D$ concentrates around a low dimensional manifold of intrinsic dimension $d$, as a reasonable assumption to model real world clustering instances. We identify key geometric properties of datasets which have theoretically predictable scaling laws depending on the quantization exponent $\varepsilon = 2/d$ using techniques from optimum quantization theory. We show how to exploit these regularities to design a fast seeding method called $\operatorname{Qkmeans}$ which provides $O(ρ^{-2} \log k)$ approximate solutions to the $k$-means problem in time $O(nD) + \widetilde{O}(\varepsilon^{1+ρ}ρ^{-1}k^{1+γ})$; where the exponent $γ= \varepsilon + ρ$ for an input parameter $ρ< 1$. This allows us to obtain new runtime - quality tradeoffs. We perform a large scale empirical study across various domains to validate our theoretical predictions and algorithm performance to bridge theory and practice for beyond worst case data clustering.

cs.DS↗

arXiv2025

Lattice Based Crypto breaks in a Superposition of Spacetimes

We explore the computational implications of a superposition of spacetimes, a phenomenon hypothesized in quantum gravity theories. This was initiated by Shmueli (2024) where the author introduced the complexity class $\mathbf{BQP^{OI}}$ consisting of promise problems decidable by quantum polynomial time algorithms with access to an oracle for computing order interference. In this work, it was shown that the Graph Isomorphism problem and the Gap Closest Vector Problem (with approximation factor $\mathcal{O}(n^{3/2})$) are in $\mathbf{BQP^{OI}}$. We extend this result by showing that the entire complexity class $\mathbf{SZK}$ (Statistical Zero Knowledge) is contained within $\mathbf{BQP^{OI}}$. This immediately implies that the security of numerous lattice based cryptography schemes will be compromised in a computational model based on superposition of spacetimes, since these often rely on the hardness of the Learning with Errors problem, which is in $\mathbf{SZK}$.

cs.CC↗

arXiv2025

A New Rejection Sampling Approach to $k$-$\mathtt{means}$++ With Improved Trade-Offs

The $k$-$\mathtt{means}$++ seeding algorithm (Arthur & Vassilvitskii, 2007) is widely used in practice for the $k$-means clustering problem where the goal is to cluster a dataset $\mathcal{X} \subset \mathbb{R} ^d$ into $k$ clusters. The popularity of this algorithm is due to its simplicity and provable guarantee of being $O(\log k)$ competitive with the optimal solution in expectation. However, its running time is $O(|\mathcal{X}|kd)$, making it expensive for large datasets. In this work, we present a simple and effective rejection sampling based approach for speeding up $k$-$\mathtt{means}$++. Our first method runs in time $\tilde{O}(\mathtt{nnz} (\mathcal{X}) + βk^2d)$ while still being $O(\log k )$ competitive in expectation. Here, $β$ is a parameter which is the ratio of the variance of the dataset to the optimal $k$-$\mathtt{means}$ cost in expectation and $\tilde{O}$ hides logarithmic factors in $k$ and $|\mathcal{X}|$. Our second method presents a new trade-off between computational cost and solution quality. It incurs an additional scale-invariant factor of $ k^{-Ω( m/β)} \operatorname{Var} (\mathcal{X})$ in addition to the $O(\log k)$ guarantee of $k$-$\mathtt{means}$++ improving upon a result of (Bachem et al, 2016a) who get an additional factor of $m^{-1}\operatorname{Var}(\mathcal{X})$ while still running in time $\tilde{O}(\mathtt{nnz}(\mathcal{X}) + mk^2d)$. We perform extensive empirical evaluations to validate our theoretical results and to show the effectiveness of our approach on real datasets.

cs.DS↗

arXiv2022

Euclidean algorithm for a class of linear orders

Borrowing inspiration from Marcone and Montálban's one-one correspondence between the class of signed trees and the equimorphism classes of indecomposable scattered linear orders, we find a subclass of signed trees which has an analogous correspondence with equimorphism classes of indecomposable finite rank discrete linear orders. We also introduce the class of \emph{finitely presented linear orders}-- the smallest subclass of finite rank linear orders containing $\mathbf 1$, $ω$ and $ω^*$ and closed under finite sums and lexicographic products. For this class we develop a generalization of the Euclidean algorithm where the \emph{width} of a linear order plays the role of the Euclidean norm. Using this as a tool we classify the isomorphism classes of finitely presented linear orders in terms of an equivalence relation on their presentations using \emph{3-signed trees}.

math.CO↗