New Maximum Toroidal Distance Codes Aim to Lower Decryption Failures in Lattice‑Based Encryption

Global: New Maximum Toroidal Distance Codes Aim to Lower Decryption Failures in Lattice‑Based Encryption

On January 13, 2026, researchers Shuiyin Liu and Amin Sakzad submitted a study proposing a maximum toroidal distance (MTD) code designed to improve lattice‑based public‑key encryption. The work targets a reduction in decryption failure rates for post‑quantum schemes, notably the NIST‑standardized ML‑KEM (Crystals‑Kyber). By selecting 2^ℓ points within the discrete ℓ‑dimensional torus ℤ_q^ℓ, the authors aim to maximize the minimum L₂‑norm toroidal distance between points.

Background on Lattice‑Based Encryption

Lattice‑based cryptography underpins many emerging post‑quantum protocols because of its resistance to attacks by quantum computers. Decryption failures, measured by the decryption failure rate (DFR), remain a critical performance metric for schemes such as Kyber.

Maximum Toroidal Distance Code Concept

The proposed MTD code treats the encryption encoding problem as a geometric selection task. By maximizing the smallest L₂‑norm distance on the torus, the construction seeks to create a more robust point set that tolerates noise during decryption.

Specific Constructions for Different Dimensions

For ℓ = 2, the authors demonstrate that the MTD code aligns with a variant of the Minal code presented at IACR CHES 2025. For ℓ = 4, a construction based on the D₄ lattice is introduced, achieving the largest known toroidal distance in that dimension. When ℓ = 8, the code corresponds to 2E₈ lattice points within ℤ₄⁸.

Performance Evaluation Against Existing Codes

Numerical simulations conducted under the Kyber parameter set indicate that the MTD codes outperform both the Minal and maximum Lee‑distance (L₁‑norm) codes for ℓ > 2, while matching Minal performance for ℓ = 2. The reported improvements focus on lower DFR values across tested configurations.

Implications for Post‑Quantum Cryptography

If adopted, the MTD constructions could enhance the reliability of lattice‑based key‑encapsulation mechanisms, potentially influencing future revisions of NIST’s post‑quantum standards.

Future Directions

The authors suggest extending the approach to higher dimensions and exploring alternative lattice families to further increase toroidal distances. Additional empirical testing on hardware implementations is also proposed.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.