  This paper presents a new technique for disturbing the algebraic structure of
linear codes in code-based cryptography. This is a new attempt to exploit
Gabidulin codes in the McEliece setting and almost all the previous
cryptosystems of this type have been completely or partially broken. To be
specific, we introduce the so-called semilinear transformation in coding
theory, which is defined through an $\mathbb{F}_q$-linear automorphism of
$\mathbb{F}_{q^m}$, then apply them to construct a public key encryption
scheme. Our analysis shows that this scheme can resist all the existing
distinguisher attacks, such as Overbeck's attack and Coggia-Couvreur attack.
Meanwhile, we endow the underlying Gabidulin code with the so-called partial
cyclic structure to reduce the public key size. Compared with some other
code-based cryptosystems, our proposal has a much more compact representation
of public keys. For instance, 2592 bytes are enough for our proposal to achieve
the security of 256 bits, almost 403 times smaller than that of Classic
McEliece entering the third round of the NIST PQC project.
