Periodic Reporting for period 1 - ENCODE (European Network in Coding Theory and Applications)
Berichtszeitraum: 2023-03-01 bis 2025-02-28
While modern society is heavily dependent on digital technologies, data can be compromised, damaged or disappear due to errors induced by noise, disc failures, untrustworthy servers and malicious cyber-attacks. Transmission and storage of data also incur energy costs. Coding theory plays a pivotal role in providing the theoretical basis of a secure, robust, and efficient digital infrastructure. Coding theory is an interdisciplinary field at the intersection of mathematics, computer science and electrical engineering. Its many applications include error-correction, network coding, distributed storage, cryptography, private information retrieval and distributed computation. The basic principle of coding theory is that encoding data makes it robust to corruption from noise and packet loss. It arises as a fundamental tool of every system of digital communications, including satellite communications, cloud storage systems, multi-party communications and code-based cryptosystems. The principal scientific aim of the ENCODE project is to develop fundamental coding theoretic tools to support modern digital communication and its evolving needs. The ENCODE doctoral network was designed with the goal to deliver a specialized and innovative training in coding theory and its applications to eight doctoral candidates.
The ENCODE project has two major strands of work. The focus of the first is on developing fundamental mathematical tools to preserve privacy in distributed information retrieval and computation. In this domain, doctoral candidates are pursuing projects in leakage resistance in secret sharing, Schur products, secret sharing in the rank metric and private information retrieval. Secret sharing refers to technique that enables a collection of participants to perform a computation or to identify some data only in conjunction with agreed coalitions of participants. As a principle, it underlies many of the questions relevant to the project. Information theoretic secret sharing may have vulnerabilities respect to leakage. One of the major goals is to find the minimal amount of leakage that can be tolerated while still preserving secrecy. Our first results in this direction use the structure of Gabidulin codes. Secret sharing in the rank metric is closely connected to the wire-tap problem in a random network. Our results in this direction have been the establishment of the concept of a q-polymatroid port and its relation to a generalized access structure. We also demonstrate a secret sharing protocol using a rank metric code. We developed the first code-based single-server private information retrieval protocol, whose security is based on the hardness of decoding a random linear code. We also have conducted a security analysis of some graph-based PIR schemes. Schur products are applied in secret sharing as well as for distinguisher attacks on code-based cryptosystems. In this direction, we have results on the cryptanalysis of a McEliece-like cryptosystem.
The focus of the second strand of research is on the development of the theory of error-correcting codes for in distributed settings. The main mathematical objects of study are linear matrix codes, tensor codes, codes for the rank metric, sum-rank metric, subspace codes, and submodule codes. A common theme is the notion of an anti-code in coding theory. We have studied a class of tensor codes for the tensor-rank distance. We also introduced new metrics that generalize the rank metric. We have devised new decoding algorithms for this family of codes with respect to the new metrics, which also yield decoders for the tensor rank. The structure of rank-metric codes is an important area of study both for the theory of rank-metrics codes itself, and also for analysing the robustness of rank-metric codes for code-based cryptosystems. We have defined new invariants of equivalence for linear codes for a range of metrics, including the rank and sum-rank metrics. Such invariants allows us to identify inequivalent codes. For matrix codes over a principal ideal ring, a distance function and a lifting construction were defined, in such a way that the lift of a length-metric code over a ring is a submodule code and the lifting is an isometry up to a constant factor. A structure theorem for optimal anticodes was obtained and a definition of generalized weights for matrix codes over principal ideal rings was proposed.
The focus of the second strand of research is on the development of the theory of error-correcting codes for in distributed settings. The main mathematical objects of study are linear matrix codes, tensor codes, codes for the rank metric, sum-rank metric, subspace codes, and submodule codes. A common theme is the notion of an anti-code in coding theory. We have studied a class of tensor codes for the tensor-rank distance. We also introduced new metrics that generalize the rank metric. We have devised new decoding algorithms for this family of codes with respect to the new metrics, which also yield decoders for the tensor rank. The structure of rank-metric codes is an important area of study both for the theory of rank-metrics codes itself, and also for analysing the robustness of rank-metric codes for code-based cryptosystems. We have defined new invariants of equivalence for linear codes for a range of metrics, including the rank and sum-rank metrics. Such invariants allows us to identify inequivalent codes. For matrix codes over a principal ideal ring, a distance function and a lifting construction were defined, in such a way that the lift of a length-metric code over a ring is a submodule code and the lifting is an isometry up to a constant factor. A structure theorem for optimal anticodes was obtained and a definition of generalized weights for matrix codes over principal ideal rings was proposed.
There are direct scientific impacts for the creation of new scientific knowledge and new mathematical theories related to coding theory. Several of our results may have impacts for code-based cryptography. In particular, the results on coding theoretic invariants and the Schur product may be useful for distinguisher attacks. Further research is necessary in some cases to determine how practical these results are.