Efficient characteristic 3 Galois field operations for elliptic curve cryptographic applications

Galois fields of characteristic 3, where the number of field elements is a power of 3, have a distinctive application in building high-security elliptic curve cryptosystems. However, they are not typically used because of their relative inefficiency in computing polynomial operations when compared to conventional prime or binary Galois fields. The purpose of this research was to design and implement characteristic 3 Galois field arithmetic algorithms with greater overall efficiency than those presented in current literature, and to evaluate their applicability to elliptic curve cryptography. The algorithms designed were tested in a C++ program and using a mapping of field element logarithms, were able to simplify the operations of polynomial multiplication, division, cubing, and modular reduction to that of basic integer operations. They thus significantly outperformed the best characteristic 3 algorithms presented in literature and showed a distinct applicability to elliptic curve cryptosystems. In conclusion, this research presents a novel method of optimizing the performance of characteristic 3 Galois fields and has major implications for the field of elliptic curve cryptography.