Multivariate Public Key Cryptosystem from Sidon Spaces

A Sidon space is a subspace of an extension field over a base field in which the product of any two elements can be factored uniquely, up to constants. This paper proposes a new public-key cryptosystem of the multivariate type which is based on Sidon spaces, and has the potential to remain secure even if quantum supremacy is attained. This system, whose security relies on the hardness of the well-known MinRank problem, is shown to be resilient to several straightforward algebraic attacks. In particular, it is proved that the two popular attacks on the MinRank problem, the kernel attack, and the minor attack, succeed only with exponentially small probability. The system is implemented in software, and its hardness is demonstrated experimentally.Comment: Appeared in Public-Key Cryptography - PKC 2021, 24th IACR International Conference on Practice and Theory of Public Key Cryptograph

Multi-Sidon spaces over finite fields

Sidon spaces have been introduced by Bachoc, Serra and Z\'emor in 2017 in connection with the linear analogue of Vosper's Theorem. In this paper, we propose a generalization of this notion to sets of subspaces, which we call multi-Sidon space. We analyze their structures, provide examples and introduce a notion of equivalnce among them. Making use of these results, we study a class of linear sets in PG$(r-1,q^n)$ determined by $r$ points and we investigate multi-orbit cyclic subspace codes

Tables of subspace codes

One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least $d$ over $\mathbb{F}_q^n$, where the dimensions of the codewords, which are vector spaces, are contained in $K\subseteq\{0,1,\dots,n\}$. In the special case of $K=\{k\}$ one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot

Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes

A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid

Weight Distributions, Automorphisms, and Isometries of Cyclic Orbit Codes

Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup Fqn* on an Fq-subspace U of Fqn. The weight distribution of a code is the vector whose ith entry is the number of codewords with distance i to a fixed reference space in the code. My dissertation investigates the structure of the weight distribution for cyclic orbit codes. We show that for full-length orbit codes with maximal possible distance the weight distribution depends only on q,n and the dimension of U. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the weight distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. We also briefly address the weight distribution of a union of full-length orbit codes with maximum distance. A related problem is to find the automorphism group of a cyclic orbit code, which plays a role in determining the isometry classes of the set of all cyclic orbit codes. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of Fqn. We then generalize to orbits under the normalizer of the Singer subgroup, although in this setup there is a remaining exceptional case. Finally, we can characterize linear isometries between such codes

On 4-general sets in finite projective spaces

A $4$-general set in ${\rm PG}(n,q)$ is a set of points of ${\rm PG}(n,q)$ spanning the whole ${\rm PG}(n,q)$ and such that no four of them are on a plane. Such a pointset is said to be complete if it is not contained in a larger $4$-general set of ${\rm PG}(n, q)$. In this paper upper and lower bounds for the size of the largest and the smallest complete $4$-general set in ${\rm PG}(n,q)$, respectively, are investigated. Complete $4$-general sets in ${\rm PG}(n,q)$, $q \in \{3,4\}$, whose size is close to the theoretical upper bound are provided. Further results are also presented, including a description of the complete $4$-general sets in projective spaces of small dimension over small fields and the construction of a transitive $4$-general set of size $3(q + 1)$ in ${\rm PG}(5, q)$, $q \equiv 1 \pmod{3}$