A Systematic Review of Irreducible Polynomial Selection for Fault-Tolerant ECC: Methods, Architectures, and Future Research Directions

Abstract

Elliptic Curve Cryptography (ECC) has become a cornerstone of modern secure communication systems due to its ability to provide strong security with relatively small key sizes. A critical aspect of ECC implementations over binary fields GF(2^m) is the selection of irreducible polynomials, which define the field arithmetic and directly influence computational efficiency, hardware complexity, and system reliability. This is particularly important in resource-constrained and security-sensitive environments such as IoT, embedded systems, and aerospace applications, where robustness against faults and attacks is essential. Faults may arise from radiation effects, hardware failures, or intentional fault injection, potentially compromising system integrity. Recent research has focused on optimizing polynomial selection to enhance fault tolerance, reduce implementation complexity, and improve resistance to side-channel and fault-based attacks. Special polynomial classes such as trinomials and pentanomials are widely explored due to their efficient hardware implementation. This review analyzes polynomial selection techniques, including algebraic methods, hardware-efficient designs, fault detection and correction strategies, and hybrid architectures incorporating redundancy and verification mechanisms. Findings indicate that optimized polynomial selection significantly improves performance and reliability, although challenges in balancing efficiency, security, and fault tolerance persist, highlighting the need for adaptive and intelligent design approaches.