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

Due to the recent advances in quantum computers, the search for cryptosystems that survive quantum attacks is of great interest. Code-based cryptography is a promising candidate, since it is build on the NP-hard problem of decoding a random code [1].

The McEliece cryptosystem is a promising candidate for asymmetric encryption.
However, many attempts at constructing a code-based signature scheme have resulted in impractical parameters or security problems.

NIST's announcement of a competetion dedicated to standardizing post-quantum signatures has lead to the publication of several new code-based schemes

In this work we consider LESS [2] a signature scheme based on the hardness of the code equivalence problem [3]. State-of-the-art solvers of the problem [4] are analysed and modifications are made to improve their performance.

References:

[1] Weger, V., Gassner, N., & Rosenthal, J. (2022). A Survey on Code-Based Cryptography. arXiv preprint arXiv:2201.07119.

[2] Barenghi, A., Biasse, J. F., Persichetti, E., & Santini, P. (2021). LESS-FM: fine-tuning signatures from the code equivalence problem. In Post-Quantum Cryptography: 12th International Workshop, PQCrypto 2021, Daejeon, South Korea, July 20–22, 2021, Proceedings 12 (pp. 23-43). Springer International Publishing.

[3] Barenghi, A., Biasse, J. F., Persichetti, E., & Santini, P. (2022). On the computational hardness of the code equivalence problem in cryptography. Cryptology ePrint Archive.

[4] Beullens, W. (2021, July). Not enough LESS: An improved algorithm for solving code equivalence problems over F q. In Selected Areas in Cryptography: 27th International Conference, Halifax, NS, Canada (Virtual Event), October 21-23, 2020, Revised Selected Papers (pp. 387-403). Cham: Springer International Publishing.

Due to the recent advances in quantum computers, searching for cryptosystems that survive quantum attacks is of great interest. Code-based cryptography is a promising candidate since it is built on the NP-hard problem of decoding a random code [5].

Random-looking codes replace random codes for most cryptosystems to enable a trapdoor. McEliece originally proposed to use binary Goppa codes. Later, MDPC codes were successfully introduced.

More recently, it was suggested that generalized (U|U+V) codes be used, which belong to the class of generalized concatenated codes.
Concatenated codes play an important role in classical channel coding, but further research is required to evaluate which variants also form a promising basis for code-based cryptography.

This topic aims to analyze constructions of generalized concatenated codes proposed for cryptographic applications [2-4], in particular, the WAVE signature scheme [1].

The research internship addresses the following questions:
- how do proposed constructions work?
- which properties do they have?
- is the secret structure well hidden?

Main Papers:

[1] Debris-Alazard, T., Sendrier, N., & Tillich, J. P. (2018). Wave: A new code-based signature scheme.

[2] Márquez-Corbella, Irene, and Jean-Pierre Tillich. "Using Reed-Solomon codes in the (U| U+ V) construction and an application to cryptography." 2016 IEEE International Symposium on Information Theory (ISIT). IEEE, 2016.

[3] Puchinger, S., Müelich, S., Ishak, K., & Bossert, M. (2017). Code-based cryptosystems using generalized concatenated codes. In Applications of Computer Algebra: Kalamata, Greece, July 20–23 2015 (pp. 397-423). Springer International Publishing.

[4] Cho, J., No, J. S., Lee, Y., Koo, Z., & Kim, Y. S. (2022). Enhanced pqsigRM: code-based digital signature scheme with short signature and fast verification for post-quantum cryptography. Cryptology ePrint Archive.

Further References:

[5] Weger, V., Gassner, N., & Rosenthal, J. (2022). A Survey on Code-Based Cryptography. arXiv preprint arXiv:2201.07119.

Security in Communications and Storage
Channel Coding