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

2026-03-28

AI Summary

This work addresses the challenge of the lack of efficient attacks on k-sparse Learning With Errors (LWE) and Learning Parity with Noise (LPN) decisional problems in high-modulus settings. By generalizing the Kikuchi method to arbitrary modulus q, the paper establishes the first unified attack framework tailored for high-modulus sparse LWE and LPN instances. Central to this approach is the construction of a Kikuchi graph, which enables two novel strategies: a spectral-norm-based distinguisher leveraging the adjacency matrix, and an algebraic distinguisher exploiting nontrivial closed walks together with polynomials derived from edge labels. The proposed method achieves a new trade-off between sample and time complexity that improves upon existing techniques, substantially enhancing the efficiency of attacks against high-modulus sparse LWE and LPN problems.

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.

Problem

• Sparse LWE
• Sparse LPN
• decisional problem
• sample-time tradeoff
• high modulus

Innovation

• Sparse LWE
• Sparse LPN
• Kikuchi graph
• spectral norm
• sample-time tradeoff