Semilinear Transformations in Coding Theory: A New Technique in Code-Based Cryptography

Abstract: 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.