This paper investigates subcodes of lambda-Gabidulin codes, viewed as rank-metric analogues of generalized Reed--Solomon codes, and their applications to compact-ciphertext cryptosystems. We first analyze subspace and generalized subspace subcodes of lambda-Gabidulin codes and relate them to corresponding subcodes of classical Gabidulin codes through coordinate-wise scaling. This relation yields cardinality bounds and structural properties for these families. When the extension degree equals the code length, we further characterize Gabidulin subspace subcodes in terms of linearized polynomials, which gives an explicit description of their encoding and dimension. We also study the matrix images of these subcodes over the base field through their stabilizer and annihilator algebras, showing that subspace restrictions may preserve nontrivial algebraic invariants despite the loss of extension-field linearity. Motivated by these results, we propose a generator-matrix-based construction of random subcodes designed to avoid such invariants. This construction is then used to design McEliece-like and Niederreiter-like encryption schemes in the MinRank setting. Among the parameter sets considered in this work, the most compact ciphertexts are obtained from random subcodes of classical Gabidulin codes. At the 128-, 192-, and 256-bit security levels, the resulting $\mathsf{LGS}$-Niederreiter instances achieve the smallest ciphertext sizes among the compared schemes, while maintaining competitive public-key sizes.

翻译：本文研究λ-加比杜林码的子码，将其视为广义里德-所罗门码的秩度量模拟，并探讨其在紧致密文密码系统中的应用。我们首先分析λ-加比杜林码的子空间子码和广义子空间子码，通过坐标缩放将其与经典加比杜林码的对应子码相关联。该关联揭示了这些码族的基数界和结构性质。当扩张次数等于码长时，我们进一步利用线性化多项式刻画加比杜林子空间子码，从而给出其编码和维数的显式描述。我们还通过稳定子代数和零化子代数研究这些子码在基域上的矩阵像，表明子空间限制可能保留非平凡代数不变量，即使扩展域线性性质已丧失。基于这些结果，我们提出一种生成矩阵驱动的随机子码构造方法，旨在避免此类不变量。进而将该构造用于设计MinRank框架下的类McEliece和类Niederreiter加密方案。在所考虑的参量集中，最紧凑的密文来自经典加比杜林码的随机子码。在128位、192位和256位安全等级下，所提出的$\mathsf{LGS}$-Niederreiter实例在保持竞争性公钥尺寸的同时，实现了所比较方案中最小的密文长度。