Forskning
Erik Mårtensson, Postdoktor
TL; DR
For excellent introductions to my research areas, watch these Youtube videos.
- This one on post-quantum cryptology by Veritasium.
- This one on efficient matrix multiplication by Quanta Magazine.
Research Area 1 - Post-quantum Cryptology
Asymmetric cryptography is based on the difficulty of solving the mathematical problems of factoring integers or finding discrete logarithms. After decades of intense research there are still no efficient algorithms for solving these problems in the general case on a classical computer.
However, having access to a large-scale quantum computer, both of these problems can be solved in polynomial time using Shor's algorithm. Post-quantum cryptography deals with the (potential) future threat of quantum computers by basing asymmetric cryptography on other mathemtical problems.
In lattice-based cryptography one promising mathematical problem to base cryptosystems on is the Learning with Errors problem (LWE).
In code-based cryptography, the underlying mathematical problem is decoding general linear codes. One famous cryptosystem in code-based cryptography is the McEliece system.
My research is about developing better algorithms for solving some of the underlying mathemtical problems in lattice-based and code-based cryptography. For more details about my research see my Publications.
For a quick introduction to post-quantum cryptography in general see Wikipedia. For a longer introduction I recommend this book.
Research Area 2 - Efficient Matrix Multiplication
Matrix multiplication is a fundamental operation in applied mathematics, statistics, physics, economics, engineering and machine learning. Finding more efficient methods for doing so is thus enormously important.
The schoolbook way of performing matrix multiplication seems asymptotically obvious, but in 1969 Volker Strassen developed a faster algorithm. Attempts at improving Strassen's algorithm for a long time did not lead to much faster algorithms. A large limitation is that most efforts were made with pen and paper, while the search space for finding these algorithms is astronomical.
However, in November 2022 Deepmind - known for developing AI systems to do anything from playing video games and boardgames, to folding proteins - entered the picture. By gamifying the process and using neural networks they discovered improvements of Strassen's algorithm. The potential for finding further improvements of matrix multiplication by using AI is very promising.