Implementing Elliptic Curve Cryptography

摘要

The strength of public key cryptography utilizing Elliptic Curves relies on the difficulty of computing discrete logarithms in a finite field. Diffie-Hellman key exchange algorithm also relies on the same fact. There are two flavors of this algorithm, one using Elliptic Curves and another without using Elliptic Curves. Both flavors of the algorithm rely on the difficulty of computing discrete logarithms in a finite field. Other public key cryptographic algorithms, such as RSA, rely on the difficulty of integer factorization. Both flavors of Diffie-Hellman key exchange algorithm will be discussed in this paper, and we will show implementation details of both of them. Additionally, we will describe what Elliptic Curve Cryptography (ECC) is, and how we can implement different cryptographic algorithms in java, such as digital signatures, encryption/decryption and Key exchange. We will not utilize the java built-in implementations of ECC. Instead, we will use the java programming language as a platform to implement several cryptographic algorithms from the ground up, thus revealing the details of each algorithm and the proofs and reasons these algorithms work. We will describe the theory of ECC and show implementation details that would help students, practitioners, and researchers understand, implement and experiment with such algorithms.