This thesis investigates the basic security properties of addition operations within prime fields and their importance for building countermeasures against side-channel attacks. The research focuses on two prominent side-channel information exploited by attacks: Bit Leakage and Hamming Weight Leakage. The challenge is to balance physical security and efficient implementation when designing these countermeasures. However, the current security analyses of these countermeasures do not describe how the vulnerability to side-channel attacks varies with different prime numbers and their respective sizes. The results of the work show a fundamental trend: while larger prime fields increase the risk of bit leakage, increasing the field size can provide exponential security improvements for Hamming weight leakage.

This observation represents a significant advance in developing future cryptographic schemes that work over large prime fields. It also highlights the need for further research to better understand and characterise side-channel attacks to ensure better protection in the long term.