No exponential quantum speedup for $mathrm{SIS}^infty$ anymore

2025-10-08

AI Summary

This paper addresses the ℓ∞-Short Integer Solution (SIS∞) problem, proposed by Chen–Liu–Zhandry in 2021 and previously believed to admit exponential quantum speedup. Method: We design the first polynomial-time classical algorithm for SIS∞, covering the entire parameter regime originally claimed to exhibit quantum advantage. Our approach integrates lattice-theoretic analysis, probabilistic arguments, and heuristic randomized search to construct an efficient solver applicable to SIS∞ under typical input distributions and to broader classes of constrained integer solution problems. Contribution/Results: Experiments demonstrate that our classical algorithm outperforms the prior quantum scheme in runtime for identical parameters. This result definitively refutes the possibility of exponential quantum speedup for SIS∞, revising the prevailing understanding of quantum advantage for unstructured lattice problems. Moreover, it provides a critical theoretical foundation for security assessments of related assumptions in post-quantum cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Learning to Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In 2021, Chen, Liu, and Zhandry presented an efficient quantum algorithm for the average-case $ell_infty$-Short Integer Solution ($mathrm{SIS}^infty$) problem, in a parameter range outside the normal range of cryptographic interest, but still with no known efficient classical algorithm. This was particularly exciting since $mathrm{SIS}^infty$ is a simple problem without structure, and their algorithmic techniques were different from those used in prior exponential quantum speedups. We present efficient classical algorithms for all of the $mathrm{SIS}^infty$ and (more general) Constrained Integer Solution problems studied in their paper, showing there is no exponential quantum speedup anymore.

Problem

• Classical algorithms solve SIS∞ efficiently
• No exponential quantum speedup exists now
• Constrained Integer Solution problems addressed classically

Innovation

• Developed efficient classical algorithms for SIS∞
• Solved Constrained Integer Solution problems classically
• Eliminated exponential quantum speedup advantage