Fuzzy k-Means is a popular generalization of the classical k-Means problem to soft clusterings. From an algorithmic perspective however, it is much more difficult to compute provably good solutions. Typically, these problems use the squared Euclidean distance to measure how far data points are apart. The squared Euclidean distance is part of the mu-similar Bregman divergences, a large class of dissimilarity measures sharing desirable characteristics.

Using sampling techniques to find good approximations of optimal centroids has proven to work for both, the k-Means problem using mu-similar Bregman divergences and also for the Fuzzy k-Means problem using the squared Euclidean distance. The goal of this thesis is to explore whether this can actually be combined to obtain a good approximation algorithm for the Fuzzy k-Means problem using mu-similar Bregman divergences.

Contact: Johannes Blömer

Threshold signature schemes enhance “normal” digital signature schemes in the sense that from n parties a fraction k<n is (at least) required to create a valid signature with respect to a jointly computed public key. This is a special use case for secure multiparty computation protocols where n parties jointly compute a given function (here, for example, the signing algorithm) without revealing their private data (here, their “shares” of a private key). Typically, this privacy of inputs is only guaranteed when a threshold k of all n parties honestly runs a given code.

This topic aims to give an overview on existing threshold signature solutions, underlying assumptions, their security properties, and further features we/you identify as meaningful. A very recent publication concerning threshold signatures and MPC is, for example, https://eprint.iacr.org/2022/374.pdf.

Supervisor: Henrik Bröcher (Mail)

In lattice cryptography there are many parameters. Many, if not all useful theorems require certain bounds on these parameters. Examples for these parameters are the length of some vectors under the L2 norm or the infinity norm, the spectral norm of matrices of certain distributions, or the so-called smoothing parameter.

When instantiating cryptographic schemes based on lattices, one has to decide on values for these parameters, such that security still holds. However, theoretical bounds can often be quite conservative, leading to not-so efficient schemes.

The question now is whether there are heuristics with which one could choose values for the parameters and how much these heuristics could improve the efficiency of the cryptographic schemes.

Your task is to identify interesting parameters, to create and implement heuristic tests for bounds and to compare the heuristics to the theoretical bounds. Afterwards you compare the efficiency of instantiations of some cryptographic schemes based on your heuristics and theoretical bounds.

A Decade of Lattice Cryptography: https://eprint.iacr.org/2015/939.pdf A good starting point to learn about lattices.

Supervisor: Laurens Porzenheim Mail