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

Shuiyin Liu, Amin Sakzad

TL;DR

The paper addresses decryption failure rate and ciphertext expansion in lattice-based public-key encryption, focusing on Kyber. It proposes maximum toroidal distance (MTD) codes by selecting $2^{\ell}$ points in $\mathbb{Z}_q^{\ell}$ to maximize the minimum $L_2$-norm toroidal distance, thereby reducing DFR. A general Good Toroidal Distance (GTD) construction via lattices is developed for any $\ell$, with explicit realizations: $\ell=2$ as a variant of the Minal code, $\ell=4$ using the $D_4$ lattice, and $\ell=8$ using the $2E_8$ lattice; these codes show improved DFR performance in Kyber settings, outperforming Minal and MLD for $\ell>2$ and matching Minal for $\ell=2$. The work also demonstrates practical decoding considerations, achieving lower ciphertext expansion rates and enabling efficient encoding/decoding at small dimensions, with a fast lattice CVP decoder available for larger $\ell$ via the $E_8$ structure. An open question remains to design fast-decodable MTD codes for dimensions beyond $\ell=8$.

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 $\ell = 2$.

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

TL;DR

The paper addresses decryption failure rate and ciphertext expansion in lattice-based public-key encryption, focusing on Kyber. It proposes maximum toroidal distance (MTD) codes by selecting points in to maximize the minimum -norm toroidal distance, thereby reducing DFR. A general Good Toroidal Distance (GTD) construction via lattices is developed for any , with explicit realizations: as a variant of the Minal code, using the lattice, and using the lattice; these codes show improved DFR performance in Kyber settings, outperforming Minal and MLD for and matching Minal for . The work also demonstrates practical decoding considerations, achieving lower ciphertext expansion rates and enabling efficient encoding/decoding at small dimensions, with a fast lattice CVP decoder available for larger via the structure. An open question remains to design fast-decodable MTD codes for dimensions beyond .

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 points in the discrete -dimensional torus , the proposed construction maximizes the minimum -norm toroidal distance to reduce the decryption failure rate (DFR) in post-quantum schemes such as the NIST ML-KEM (Crystals-Kyber). For , we show that the MTD code is essentially a variant of the Minal code recently introduced at IACR CHES 2025. For , we present a construction based on the lattice that achieves the largest known toroidal distance, while for , the MTD code corresponds to lattice points in . Numerical evaluations under the Kyber setting show that the proposed codes outperform both Minal and maximum Lee-distance (-norm) codes in DFR for , while matching Minal code performance for .
Paper Structure (12 sections, 9 theorems, 37 equations, 2 tables, 3 algorithms)

This paper contains 12 sections, 9 theorems, 37 equations, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. Notation
  4. Kyber public key encryption (PKE)
  5. Decryption noise, toroidal distance, and Minal codes
  6. Maximum Toroidal Distance Code
  7. MTD code for $\ell=2$
  8. GTD code construction via lattices for any $\ell$
  9. GTD code for $\ell=4$
  10. MTD and GTD codes for $\ell=8$
  11. Comparison and decoding complexity
  12. Conclusions

Key Result

Proposition 1

Let $\ell$, $\nu$, and $n$ be positive integers with $n$ and $\ell$ powers of $2$, $1 < \ell < n$, and $\nu = n/\ell$. Let $a,b \in R_q = \mathbb{Z}_q[x]/(x^n + 1)$, with coefficients sampled from distributions $\mathcal{D}_1$ and $\mathcal{D}_2$, where $\mathcal{D}_2$ is symmetric. Let $c = ab\in R

Theorems & Definitions (16)

  • Proposition 1: polynomial splitting Minal_Code_Kyber2024
  • Corollary 1: Minal_Code_Kyber2024
  • Definition 1: $L_2$-norm toroidal distance
  • Definition 2: Minal code Minal_Code_Kyber2024
  • Definition 3: MTD code
  • Example 1
  • Definition 4: Good Toroidal Distance Code (GTD)
  • Lemma 1
  • Proposition 2: Optimal $\gamma$
  • Theorem 1: MTD code for $\ell=2$
  • ...and 6 more