Security Analysis of Integer Learning with Errors with Rejection Sampling

2025-12-08

AI Summary

This work systematically evaluates the practical security of integer learning-with-errors (ILWE)-based digital signature schemes—such as CRYSTALS-Dilithium—under rejection sampling in small-parameter settings. Method: We propose a novel technique to directly construct ILWE instances from valid signatures, eliminating reliance on side-channel information. To model the ILWE problem, we introduce real-valued matrix approximations of modular polynomial arithmetic and formulate solving as a linear least-squares optimization. We further design an efficient, large-scale numerical solver for this formulation. Contribution/Results: Experiments confirm that, under standard parameter sets, the attack does not breach the provable security bounds, thereby validating the feasibility of small-parameter ILWE-based signatures. Our modeling and analysis framework provides a reproducible, side-channel-free quantitative assessment tool for deploying post-quantum cryptographic primitives—particularly in resource-constrained domains such as intelligent transportation systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

At ASIACRYPT 2018, a digital attack based on linear least squares was introduced for a variant of the learning with errors (LWE) problem which omits modular reduction known as the integer learning with errors problem (ILWE). In this paper, we present a theoretical and experimental study of the effectiveness of the attack when applied directly to small parameter ILWE instances found in popular digital signature schemes such as CRYSTALS-Dilithium which utilize rejection sampling. Unlike other studies which form ILWE instances based on additional information obtained from side-channel attacks, we take a more direct approach to the problem by constructing our ILWE instance from only the obtained signatures. We outline and introduce novel techniques in our simulation designs such as modular polynomial arithmetic via matrices in $mathbb{R}$, as well as algorithms for handling large sample sizes efficiently. Our experimental results reinforce the proclaimed security of signature schemes based on ILWE. We additionally discuss the implications of our work and digital signatures as a whole in regards to real-world applications such as in Intelligent Transportation Systems (ITS).

Problem

• Analyze ILWE attack effectiveness on digital signatures
• Develop novel simulation techniques for ILWE analysis
• Reinforce security of ILWE-based signature schemes

Innovation

• Modular polynomial arithmetic via real matrices
• Algorithms for efficiently handling large sample sizes
• Direct ILWE instance construction from signatures only