Applications of number theory to asymmetric cryptography

• Grant number: 24/10773-1
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: October 01, 2024
• End date: September 30, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Sara Díaz Cardell
• Grantee: Raul Vicente Maistro
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

Abstract

Cryptography is the science that studies the possible transformations of a message so that, when transmitted through an insecure channel, it can only be interpreted by the recipient. Cryptography, which has existed since there have been secrets, is no longer exclusively for use by rulers or the military. Today, cryptography is part of our daily lives. The use of computers and the birth of the first networks have driven the extraordinary expansion of cryptographic methods over the past 40 years. Every time we make online purchases or use our digital certificate to carry out an administrative procedure, we are using cryptographic methods. Our money or the confidentiality of our personal data depends on all the involved processes being sufficiently secure.The goal of the project is to approach the main cryptographic algorithms currently in use, focusing on asymmetric algorithms, specifically RSA and its weaknesses. We want to focus the study on the theorems and mathematical concepts behind the implementation of these public key cryptography algorithms.The project will consist of four main stages: an initial stage of introductory study on general cryptography and the history of cryptography, a second stage on the study of RSA and its weaknesses, a third stage of analysis of other algorithms such as Rabin or ElGamal, and finally, we will review the concept of digital signatures and some digital signature algorithms based on asymmetric cryptography.