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Improving ML Attacks on LWE with Data Repetition and Stepwise Regression

Alberto Alfarano, Eshika Saxena, Emily Wenger, François Charton, Kristin Lauter

Abstract

The Learning with Errors (LWE) problem is a hard math problem in lattice-based cryptography. In the simplest case of binary secrets, it is the subset sum problem, with error. Effective ML attacks on LWE were demonstrated in the case of binary, ternary, and small secrets, succeeding on fairly sparse secrets. The ML attacks recover secrets with up to 3 active bits in the "cruel region" (Nolte et al., 2024) on samples pre-processed with BKZ. We show that using larger training sets and repeated examples enables recovery of denser secrets. Empirically, we observe a power-law relationship between model-based attempts to recover the secrets, dataset size, and repeated examples. We introduce a stepwise regression technique to recover the "cool bits" of the secret.

Improving ML Attacks on LWE with Data Repetition and Stepwise Regression

Abstract

The Learning with Errors (LWE) problem is a hard math problem in lattice-based cryptography. In the simplest case of binary secrets, it is the subset sum problem, with error. Effective ML attacks on LWE were demonstrated in the case of binary, ternary, and small secrets, succeeding on fairly sparse secrets. The ML attacks recover secrets with up to 3 active bits in the "cruel region" (Nolte et al., 2024) on samples pre-processed with BKZ. We show that using larger training sets and repeated examples enables recovery of denser secrets. Empirically, we observe a power-law relationship between model-based attempts to recover the secrets, dataset size, and repeated examples. We introduce a stepwise regression technique to recover the "cool bits" of the secret.

Paper Structure

This paper contains 15 sections, 6 equations, 2 figures, 31 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Experimental settings
  4. Recovering higher Hamming weights with large sets of repeated examples
  5. Taming the cool bit noise: stepwise regression
  6. Overall secret recovery
  7. Scaling laws
  8. Discussion and conclusion
  9. BKZ-reduced dataset
  10. Synthetic dataset
  11. Cool and cruel embedding ablation
  12. Bias in angular embeddings and ways to circumvent it
  13. Stepwise regression algorithm
  14. Linear regression
  15. Ternary secrets

Figures (2)

  • Figure 1: Model parameters $N$ vs model based attempts $A$ for three secrets with different Hamming weights $h$.
  • Figure 2: Total training data $D$ and repetition $R$ vs model based attempts $A$ for one secret with Hamming weight $h=70$. Total distinct data can be computed as $D/R$.