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

This paper presents a new technique for disturbing the algebraic structure of linear codes in code-based cryptography. Specifically, we introduce the so-called semilinear transformations in coding theory and then apply them to the construction of code-based cryptosystems. Note that Fqm can be viewed as an Fq -linear space of dimension m , a semilinear transformation φ is therefore defined as an Fq -linear automorphism of Fqm . Then we impose this transformation to a linear code C over Fqm . It is clear that φ (C) forms an Fq -linear space, but generally does not preserve the Fqm -linearity any longer. Inspired by this observation, a new technique for masking the structure of linear codes is developed in this paper. Meanwhile, we endow the underlying Gabidulin code with the so-called partial cyclic structure to reduce the public-key size. Compared to some other code-based cryptosystems, our proposal admits a much more compact representation of public keys. For instance, 2592 bytes are enough 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.

Encoding classical information in gauge subsystems of quantum codes

In this paper, we show how to construct hybrid quantum-classical codes from subsystem codes by encoding the classical information into the gauge qudits using gauge fixing. Unlike previous work on hybrid codes, we allow for two separate minimum distances, one for the quantum information and one for the classical information. We give an explicit construction of hybrid codes from two classical linear codes using Bacon–Casaccino subsystem codes, as well as several new examples of good hybrid code.

On cutting blocking sets and their codes

Let PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} be the r-dimensional projective space over the finite field GF ⁡ ( q ) {\operatorname{GF}(q)} . A set 𝒳 {\mathcal{X}} of points of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} the set Π ∩ 𝒳 {\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG ⁡ ( 3 , q 3 ) {\operatorname{PG}(3,q^{3})} of size 3 ⁢ ( q + 1 ) ⁢ ( q 2 + 1 ) {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} of size 7 ⁢ ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} . In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q-ary linear code having dimension 4 and 6.