Superregular matrices, product codes and cryptographic sequences

• Grant number: 24/05051-7
• Support Opportunities: Regular Research Grants
• Start date: August 01, 2024
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Sara Díaz Cardell
• Grantee: Sara Díaz Cardell
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil
• Associated researchers: Amparo Fuster-Sabater ; Carina Alves ; Cintya Wink de Oliveira Benedito ; Grasiele Cristiane Jorge ; Gustavo Terra Bastos ; Joan Josep Climent ; Miguel Beltrá Vidal ; Veronica Requena Arevalo

Abstract

In this project, we present three lines of work. In the first, we explore the construction of superregular matrices, block superregular matrices, and triangular superregular matrices. These matrices play a crucial role in the construction of error-correcting codes with suitable properties for applications in communication theory. In the second line of work, we focus on the problem of counting correctable (or uncorrectable) patterns for SPC (single parity-check) product codes, with the aim of studying the correction capacity of this family of codes. To address this issue, we propose to examine the parity-check matrix of the code and the weight enumerator polynomial, as it can be proven that the number of codewords is a lower bound for the number of uncorrectable patterns. The third line of work introduces a new characterization of binary sequences called Boolean representation. In fact, each binary sequence with a period that is a power of two can be uniquely represented by a Boolean function. Our goal is to analyze the cryptographic properties of sequences through this function. Additionally, we propose a study on the randomness of sequences generated by ASCON, a lightweight encryption algorithm chosen by NIST as the new standard for devices with limited computational resources. (AU)