The Implementations and Applications of Elliptic Curve Cryptography

This paper introduces two distinct new software implementations of ECC over the finite field GF(p) utilizing character arrays and bit sets. Our novel implementations operate on ECC curves of the form y2 = x3 + ax + b (mod p). We have optimized the point addition operation and scalar multiplication on a real SEC (Standards for Efficient Cryptography) ECC curve over a prime field. Furthermore, we have tested and validated the Elliptic Curve Diffie-Hellman key exchange on a real SEC ECC curve using two different implementations of big integer classes. We then proceeded to compare and analyze the performance of these two distinct implementations. Elliptic Curve Cryptography (ECC) represents a promising public-key cryptography system due to its ability to achieve the same level of security as RSA with a significantly smaller key size. ECC stands out for its time efficiency and optimal resource utilization.