General analysis on essential mathematical principles of elliptic curve cryptography

Prevalent is the practical application of Elliptic Curve Cryptography (ECC) in the modern public-key cryptosystem, especially the implementation of ECC algorithm in Bitcoin source code. With the thorough introduction of discrete logarithm and Diffie-Hellman key exchange, ECC has gradually progressed to be sophisticated and efficient simultaneously. Therefore, it currently has been widely regarded as the successor of RSA algorithm in terms of inheritance for its shorter lengths of keys, faster speed and higher safety under the same encryption strength. Due to the potential safety and complexity of Elliptic Curve Cryptosystem, it is apparently noticed that there is included a large volume of Maths principles related to the establishment of ECC algorithm. As a consequence, this paper will mainly focus on qualitative research and exemplary analysis to specifically elucidate the general knowledge on essential mathematical principles of ECC, including the Law of Addition, the Elliptic Curve Discrete Logarithm Problems (ECDLP) and the Elliptic Curve ElGamal (EC ElGamal), together with the corresponding applications combined with their deprivation processes.