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Title of the thesis: Algebraic number theory and lattice based cryptography: equivalence and cryptanalysis of RLWE and PLWE

Thesis defender: Rahinatou Njah Epouse Nchiwo 
Opponent: Assistant Professor Laura Luzzi, ENSEA, France
Custos: Professor Camilla Hollanti, Aalto University School of Science

Modern public-key cryptography relies on computationally hard problems, most notably integer factorization and the discrete logarithm problem, which form the basis of security in our communication networks. However, rapid progress in quantum computing poses a fundamental threat to these classical assumptions. As a consequence, cryptographic protocols that rely on these problems are considered compromised. 

These concerns have motivated the development of cryptographic constructions that remain secure against quantum adversaries. The most notable post-quantum cryptographic paradigms are based on solving noisy linear systems, referred to as learning with errors (LWE) problems. They are based on some well-known lattice problems providing strong security guarantees. Namely, lattices are known to be a rich source of computationally intractable problems. 

In this thesis, we study the equivalence between different variants of the LWE problem that offer trade-offs between efficiency and presumed security. This line of research draws on algebraic number theory and enables more efficient cryptographic protocols without affecting the security assumptions. 

Another question we investigate is whether there are instances where these computationally hard lattice problems can become easier to solve. We answer this question in the affirmative by exploring the underlying structure of these schemes. 

The results contribute to a better understanding of the security and efficiency trade-offs in lattice-based cryptography. They can be used to design more efficient cryptographic systems and to avoid weak parameter choices in practice. Overall, the thesis shows that while lattice problems form a strong foundation for post-quantum cryptography, their security depends on subtle structural details that must be carefully analyzed.

Keywords: Lattices, RLWE

Thesis available for public display 7 days prior to the defence at . 

Contact Information: 
rahinatou.njah@aalto.fi