On the Maximum Toroidal Distance Code for Lattice-Based Public-Key Cryptography

2026-01-13

AI Summary

This work aims to reduce the decryption failure rate (DFR) of lattice-based public-key encryption schemes such as NIST’s ML-KEM. The encryption encoding problem is modeled as selecting \(2^\ell\) points in the discrete \(\ell\)-dimensional torus \(\mathbb{Z}_q^\ell\) to maximize the minimum \(L_2\)-norm torus distance. To this end, we propose Maximum Torus Distance (MTD) codes, which significantly outperform existing Minal codes and maximum Lee distance codes for \(\ell > 2\), and reveal intrinsic connections to the \(D_4\) and \(2E_8\) lattices. Experimental results under Kyber parameters demonstrate that MTD codes achieve the largest known torus distances for \(\ell = 4\) and \(\ell = 8\), yielding lower DFRs than current approaches, while matching performance at \(\ell = 2\).

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We propose a maximum toroidal distance (MTD) code for lattice-based public-key encryption (PKE). By formulating the encryption encoding problem as the selection of $2^\ell$ points in the discrete $\ell$-dimensional torus $\mathbb{Z}_q^\ell$, the proposed construction maximizes the minimum $L_2$-norm toroidal distance to reduce the decryption failure rate (DFR) in post-quantum schemes such as the NIST ML-KEM (Crystals-Kyber). For $\ell = 2$, we show that the MTD code is essentially a variant of the Minal code recently introduced at IACR CHES 2025. For $\ell = 4$, we present a construction based on the $D_4$ lattice that achieves the largest known toroidal distance, while for $\ell = 8$, the MTD code corresponds to $2E_8$ lattice points in $\mathbb{Z}_4^8$. Numerical evaluations under the Kyber setting show that the proposed codes outperform both Minal and maximum Lee-distance ($L_1$-norm) codes in DFR for $\ell>2$, while matching Minal code performance for $\ell = 2$.

Problem

• lattice-based cryptography
• decryption failure rate
• toroidal distance
• public-key encryption
• post-quantum cryptography

Innovation

• Maximum Toroidal Distance
• Lattice-based Cryptography
• Decryption Failure Rate
• D4 lattice
• E8 lattice