DFG project Semirings: Fundamental research for new encryption methods

Jens Zumbrägel, Professor of Cryptography at the University of Passau, is leading a project on basic mathematical research. The project, which is funded by the German Research Foundation (DFG), investigates the structure theory of so-called semirings.

Semirings are general structures in mathematics that allow arithmetic operations such as addition and multiplication to be carried out according to certain rules. Examples of these are number ranges familiar from school, such as natural numbers or positive real numbers. However, semirings can also be used to describe completely different elements. These include objects that occur in the description of systems of equations, such as polynomials or matrices.

A particular feature of the concept of semirings is that no subtraction is required. This often allows to model structures of theoretical computer science that cannot be grasped using conventional terms. The theory of semirings is comparatively young and still has many open questions. For example, little is known about solving systems of linear equations over semirings.

One of the main aims of the DFG project is to decompose such structures into basic building blocks or “simple semirings”. These have already been investigated and classified recently with the participation of Professor Jens Zumbrägel. The current project is now focussing on the composition of general semirings with the aim of improving their understanding and shedding light on new fields of application.

“The versatility of semirings makes them suitable for a wide range of applications,” says Zumbrägel. “For example, they are relevant for analysing information flows in databases or for formal reasoning in the field of symbolic artificial intelligence.” This research is also interesting for encryption technology, especially in the current field of post-quantum cryptography. “It is important to find a good balance between efficiency at application level and security against potential attacks using quantum computers,” explains Zumbrägel. “Here, semirings offer a promising approach due to their generality, which makes it possible to thwart some common attacks.”

The project started in November 2024 and will be funded by the DFG for three years.

Symbolic picture: Colourbox