Investigation of problems related to lattice-based cryptography using algebraic number theory

• Grant number: 25/01467-7
• Support Opportunities: Scholarships abroad - Research
• Start date: November 10, 2025
• End date: December 09, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Robson Ricardo de Araujo
• Grantee: Robson Ricardo de Araujo
• Host Investigator: Cong Ling
• Host Institution: Instituto Federal de Educação, Ciência e Tecnologia de São Paulo (IFSP). Campus Catanduva. Catanduva , SP, Brazil
• Institution abroad: Imperial College London, England

Abstract

Lattices, which are discrete additive subgroups of Rn, have been studied for centuries and play a crucial role in various fields, including optimization, telecommunications, and cryptography. These structures are renowned for their significance associated with important mathematical problems, such as sphere packing. For decades, lattice codes have been proposed and used for signal transmission in communication channels, including Gaussian channels and Rayleigh fading channels. Recently, lattices - particularly those derived from algebraic number theory structures, have gained prominence in the search for cryptosystems resistant to quantum attacks. Indeed, several computational problems which are known to be relevant in the search for post-quantum protocols, such as LWE and its algebraic variants, have their security based on the difficulty of solving classical lattice problems, including SVP and CVP. One of the prominent problems recently emerging from post-quantum cryptography is the Lattice Isomorphism Problem (LIP), which, in its algebraic form, ensures the security of the promising digital signature scheme known as HAWK. In recent years, significant advances in the understanding and the cryptoanalysis of the LIP have been made, but the overall understanding of the problem's complexity remains unsolved, particularly in the context of cyclotomic fields. In this project, we aim to study and explore potential solutions to the LIP, while also discussing and investigating other problems related to lattice-based cryptography through the lens of algebraic number theory.