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 $\ell = 2$.

翻译：我们提出了一种用于基于格的公钥加密（PKE）的最大环面距离（MTD）码。通过将加密编码问题表述为在离散的$\ell$维环面$\mathbb{Z}_q^\ell$中选择$2^\ell$个点，所提出的构造最大化最小$L_2$范数环面距离，以降低后量子方案（如NIST ML-KEM（Crystals-Kyber））中的解密失败率（DFR）。对于$\ell = 2$，我们证明MTD码本质上是最近在IACR CHES 2025上提出的Minal码的一个变体。对于$\ell = 4$，我们提出了一种基于$D_4$格的构造，实现了目前已知的最大环面距离；而对于$\ell = 8$，MTD码对应于$\mathbb{Z}_4^8$中的$2E_8$格点。在Kyber设置下的数值评估表明，对于$\ell > 2$，所提出的码在DFR方面优于Minal码和最大Lee距离（$L_1$范数）码，而在$\ell = 2$时与Minal码性能相当。