Subcodes of Lambda-Gabidulin Codes for Compact-Ciphertext Cryptography

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.