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In this thesis, the student will study the mathematics of codes in diffent metrics such as the Hamming metric, (sum-)rank metric, column/row-cover metric, etc.

The focus can lie on similar mathematical ideas shared across different metrics, such as bounds on codes, good code constructions, decoding algorithms, applications.  

In this thesis, the student will study the mathematics of linear codes and how they can be used to design cryptosystems. 

In this thesis, we want to investigate the list decoding complexity of random (linear) codes in the sum-rank metric.

List decoding is a technique to decode beyond the unique decoding radius of a code at the cost of obtaining a list of candidate solutions. The sum-rank metric [1] is a relatively novel metric where the weight of a vector is given by the sum of the ranks of its component blocks.

As a starting point, the student should familiarize themselves with the concept of the sum-rank metric. Then, the list decoding behavior of a random SR code should be investigated, perhaps along the lines of these papers [2,3] that have some similar results on random rank metric codes. It would also be nice to investigate if this other technique [4] can be applied to the sum-rank metric.

Resources:

[1] https://arxiv.org/pdf/2102.02244 (this is not the paper where this metric was first studied, but it has a very nice overview of existing results)

[2] https://arxiv.org/abs/1401.2693

[3] https://arxiv.org/abs/1710.11516

[4] https://arxiv.org/abs/1704.02420

Channel coding lecture or similar (i.e., basics of linear codes and their decoding)

strong background in linear algebra

An interest in combinatorics is beneficial, it is at the core of many of the related papers

Homomorphic encryption (HE) schemes are increasingly attracting attention in the era of large scale computing. While lattice-based approaches have been well-studied, recently first progress has been made towards establishing code-based alternatives. Preliminary results show that such alterative approaches might enable undiscovered functionalities not present in current lattice-based schemes. In this project, we particularily study novel code-based Partial/Somewhat HE schemes tailored to applications in artificial intelligence and federated learning.

After familiarizing with SotA methods in relevant fields (such as [1]), the student should analyze the requirements for use-cases at hand and explore suitable modifications to current schemes and novel approaches.

Please take note that this Master Thesis is further designed to open up the possibility for a subsequent PhD position in homomorphic encrpytion with Prof. Dr.-Ing. Antonia Wachter-Zeh.

[1] Aguilar-Melchor, Carlos, Victor Dyseryn, and Philippe Gaborit, "Somewhat Homomorphic Encryption based on Random Codes," Cryptology ePrint Archive (2023).

- Strong foundation in linear algebra
- Channel Coding
- Security in Communications and Storage
- Basic understanding of Machine Learning concepts

Multivariate polynomials have been attracting increasing interest in constructing codes with repair capabilities by accessing only a small amount of available symbols, which is required to build failure-resistant distributed storage systems.

List decoding is a technique to decode beyond half the minimum distance and has shown its advantages for codes based on univariate polynomials.

In this thesis, the student is expected to learn from the literature about the list decoding algorithm for multivariate polynomial codes and develop new decoding techniques upon it.

For our Advanced Technology team in Munich/Martinsried, we are looking for a motivated master thesis student at the intersection of optical (quantum) communications, electrical engineering, and cyber security. With the advancement of the continuous variable QKD (CV-QKD), information can be encoded on the quadratures of the incident electromagnetic field, like in commercial optical communication systems. This allows the information on coherent states of light to be captured at the receiver. Similar to coherent optical communications, Inphase & Quadrature (IQ) modulation is a crucial step for the generation of the QKD transmit signal . For this step the modulator, usually a Mach-Zehnder modulator (MZM), imprints the electrical baseband DAC output onto the continuous wave laser light.

Throughout this thesis work, the student’s main task will be to research, analyze and implement a practical IQ modulator for QKD applications.

Space-division multiplexing (SDM), which consists in exploiting multimode fibers (MMFs) or multicore fibers (MCFs) instead of single mode ones, is one of the future optical communications architectures to increase data rates and network planning flexibility. The nonlinear properties of MCFs are of primary interest in assessing the usefulness of SDM against the current network. With this thesis, the student has the chance to work on a state-of-the-art topic in the field of optical communication systems, and progress quickly thanks to a tight (if desired) supervision. Would you be curious to know more about it? If so, just get in touch with me at paolo.carniello@tum.de (personal page https://www.ce.cit.tum.de/lnt/mitarbeiter/doktoranden/carniello/).

-some knowledge on optical communications systems (e.g., Optical Communication Systems or Simulation of Optical Communication Systems Lab)

-some knowledge about communications engineering topics

Asymmetric cryptography is a core component of modern communication infrastructure. The existence of a sufficiently large quantum computer threatens all algorithms currently in use and recent developments in this field motivate the field of post-quantum cryptography (PQC), with the primary objective of developing secure and futureproof alternatives. To consolidate these efforts, the National Institute of Standards and Technology (NIST) is conducting a competition with the goal of selecting and standardizing the best available candidates.

In July 2022 four algorithms were selected, one Public-key Encryption and Key-establishment Algorithms (KEM) and three Digital Signature Algorithms (DSA). Of these four algorithms, three (including the two recommended for general purpose applications) are from the class of lattice-based schemes, i.e., they rely on difficult problems over lattices for their security.

The lattices in these schemes are represented by elements from a polynomial ring and arithmetic over this ring therefore plays a crucial role in their execution. To accelerate this arithmetic and, specifically, the multiplication of polynomials, the number-theoretic transform (NTT) is used. This approach is a generalization of the multiplication algorithms based on the fast Fourier transform (FFT), that have been long established in fields like signal processing. Since a considerable part of the computational complexity of these algorithms lies in this NTT, it is a prime candidate for HW acceleration and many works in literature have proposed such accelerators.

While considerable efforts have been made to offer fast and lean NTT accelerators, the topic of fault resilience has received little attention so far. In safety critical applications, as common in automotive or industrial fields, this resilience is an important feature and needs to be provided by the HW. On the other hand, these fields are traditionally price sensitive, so the additional chip area required for these features should be minimal. However, current approaches, such as those introduced by Sarker et al. [1], impose a large area (or latency) overhead.

The goal of this thesis is to address this shortcoming. First, the approaches published in literature shall be evaluated with regard to the relevant performance figures (area overhead, latency, …). Then, new approaches for error resilience in NTT calculation shall be developed, either based on improving upon existing methods or by adapting methods for error resilience in the computation of FFTs.

References

[1] Sarker, Ausmita, Alvaro Cintas Canto, Mehran Mozaffari Kermani, and Reza Azarderakhsh. “Error Detection Architectures for Hardware/Software Co-Design Approaches of Number-Theoretic Transform.” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2022, 1–1. https://doi.org/10.1109/TCAD.2022.3218614.

Basic understanding of HW design

Good knowledge of algebra

Cryptocurrencies like Bitcoin and Ethereum use a decentralized ledger called Blockchain to track transactions. Whenever a new block is added to the Blockchain, the change is spread through the network using a gossip-like protocol. This process is known as block propagation.

To increase scalability, the efficiency of block propagation is crucial. This thesis aims to explore the information theoretic limits of block propagation, derive realistic models based on real data, and investigate innovative and efficient techniques for block propagation.

The thesis will be conducted at the Institute of Communications and Navigation at DLR (German Aerospace Center) in Oberpfaffenhofen.

Required qualifications are

• basic knowledge of information theory
• programming experience in Matlab, C, or python.
• Interest in cryptocurrencies.

Homomorphic encryption (HE) schemes are increasingly attracting attention in the era of large scale computing. While lattice-based approaches have been well-studied, recently first progress has been made towards establishing code-based alternatives. Preliminary results show that such alterative approaches might enable undiscovered functionalities not present in current lattice-based schemes. In this project, we particularily study novel code-based Partial/Somewhat HE schemes tailored to applications in artificial intelligence and federated learning.

After familiarizing with SotA methods in relevant fields (such as [1]), the student should analyze the requirements for use-cases at hand and explore suitable modifications to current schemes and novel approaches.

Please take note that this Master Thesis is further designed to open up the possibility for a subsequent PhD position in homomorphic encrpytion with Prof. Dr.-Ing. Antonia Wachter-Zeh.

[1] Aguilar-Melchor, Carlos, Victor Dyseryn, and Philippe Gaborit, "Somewhat Homomorphic Encryption based on Random Codes," Cryptology ePrint Archive (2023).

- Strong foundation in linear algebra
- Channel Coding
- Security in Communications and Storage
- Basic understanding of Machine Learning concepts

Space-division multiplexing (SDM), which consists in exploiting multimode fibers (MMFs) or multicore fibers (MCFs) instead of single mode ones, is one of the future optical communications architectures to increase data rates and network planning flexibility. The nonlinear properties of MCFs are of primary interest in assessing the usefulness of SDM against the current network. With this thesis, the student has the chance to work on a state-of-the-art topic in the field of optical communication systems, and progress quickly thanks to a tight (if desired) supervision. Would you be curious to know more about it? If so, just get in touch with me at paolo.carniello@tum.de (personal page https://www.ce.cit.tum.de/lnt/mitarbeiter/doktoranden/carniello/).

-some knowledge on optical communications systems (e.g., Optical Communication Systems or Simulation of Optical Communication Systems Lab)

-some knowledge about communications engineering topics