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Attacks on Sparse LWE and Sparse LPN with new Sample-Time tradeoffs

Shashwat Agrawal, Amitabha Bagchi, Rajendra Kumar

Abstract

This paper extends the Kikuchi method to give algorithms for decisional $k$-sparse Learning With Errors (LWE) and $k$-sparse Learning Parity with Noise (LPN) problems for higher moduli $q$. We create a Kikuchi graph for a sparse LWE/LPN instance and use it to give two attacks for these problems. The first attack decides by computing the spectral norm of the adjacency matrix of the Kikuchi graph, which is a generalization of the attack for $q=2$ given by Wein et. al. (Journal of the ACM 2019). The second approach computes non-trivial closed walks of the graph, and then decides by computing a certain polynomial of edge labels in the walks. This is a generalization of the attack for $q=2$ given by Gupta et. al. (SODA 2026). Both the attacks yield new tradeoffs between sample complexity and time complexity of sparse LWE/LPN.

Attacks on Sparse LWE and Sparse LPN with new Sample-Time tradeoffs

Abstract

This paper extends the Kikuchi method to give algorithms for decisional -sparse Learning With Errors (LWE) and -sparse Learning Parity with Noise (LPN) problems for higher moduli . We create a Kikuchi graph for a sparse LWE/LPN instance and use it to give two attacks for these problems. The first attack decides by computing the spectral norm of the adjacency matrix of the Kikuchi graph, which is a generalization of the attack for given by Wein et. al. (Journal of the ACM 2019). The second approach computes non-trivial closed walks of the graph, and then decides by computing a certain polynomial of edge labels in the walks. This is a generalization of the attack for given by Gupta et. al. (SODA 2026). Both the attacks yield new tradeoffs between sample complexity and time complexity of sparse LWE/LPN.

Paper Structure

This paper contains 33 sections, 45 theorems, 82 equations, 1 figure, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Kikuchi Graphs for sparse LWE/LPN.
  4. The spectral method
  5. The $q$-ary cover method
  6. Preliminaries
  7. Sparse LWE and Sparse LPN.
  8. Kikuchi Graph for sparse LWE/LPN instances
  9. Product equations over $q$th roots of unity
  10. The Kikuchi graph for $k\hbox{-}\text{LIN}_q$
  11. Distinguishing by spectral method over Kikuchi Graphs
  12. Independent Summands of $\mathcal{K}_l(\mathcal{I})$
  13. An upper bound for the random instance
  14. A lower bound for the planted instance
  15. Distinguishing between a planted and a random instance
  16. ...and 18 more sections

Key Result

Theorem 1.1

(Informal) For dimension $n$, number of samples $m\geq \left(\frac{n}{l}\right)^{k}$, modulus $q$, width parameter $r< q$, and a tradeoff parameter $k/2 \leq l\leq n$:

Figures (1)

  • Figure 1: A chain in the graph of $\bm{X}_i$

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 4.1
  • Definition 4.2
  • ...and 88 more