Integer Modular Multiplication With Barrett Reduction and Its Variants for Homomorphic Encryption Applications: A Comprehensive Review and an Empirical Study

Modular arithmetic calculations, such as modular addition and multiplication, are fundamental building blocks to Post-Quantum Cryptography (PQC) and Homomorphic Encryption (HE) systems. While modular addition has straightforward hardware implementations, integer modular multiplication is more challenging to implement. This work focuses on integer modular multiplication with Barrett modular reduction, or in short, the Barrett modular multiplication (BMM) technique, presenting an overview of its original algorithm, variants, and subsequent optimizations. The study offers comparative examples and a comprehensive analysis of the theoretical complexity and both empirical and experimental results for each BMM variant.