Making Hard Problems Easier with Custom Data Distributions and Loss Regularization: A Case Study in Modular Arithmetic

TL;DR

This paper introduces novel training techniques involving custom data distributions and loss functions that significantly improve machine learning models' ability to perform modular arithmetic, enhancing their effectiveness in attacking complex LWE cryptographic problems.

Contribution

The paper presents a new approach to training ML models for modular arithmetic by using tailored data distributions and loss functions, enabling better performance on complex LWE tasks.

Findings

ML models can now recover secrets twice as hard as previous methods.

Techniques improve learning in problems like copy, recall, and parity.

Enhanced ML attack capabilities on advanced LWE settings.

Abstract

Recent work showed that ML-based attacks on Learning with Errors (LWE), a hard problem used in post-quantum cryptography, outperform classical algebraic attacks in certain settings. Although promising, ML attacks struggle to scale to more complex LWE settings. Prior work connected this issue to the difficulty of training ML models to do modular arithmetic, a core feature of the LWE problem. To address this, we develop techniques that significantly boost the performance of ML models on modular arithmetic tasks, enabling the models to sum up to $N=128$ elements modulo $q \le 974269$. Our core innovation is the use of custom training data distributions and a carefully designed loss function that better represents the problem structure. We apply an initial proof of concept of our techniques to LWE specifically and find that they allow recovery of 2x harder secrets than prior work. Our techniques also help ML models learn other well-studied problems better, including copy, associative recall, and parity, motivating further study.