Asymmetric Encryption Algorithms: Primitives and Applications

This paper mainly focuses on reviewing the preliminaries of asymmetric encryption techniques and systems. Starting from the trapdoor function, we discussed the math theories behind three representative asymmetric encryption algorithms and systems, including the RSA algorithm, ElGamal cryptosystem, and the elliptical curve cryptographies. In RSA algorithm, we mathematically analyze its security from the process of encryption and decryption. Since there are some disadvantages in RSA algorithm, based on Diffie Hellman key exchange, we studied the basic principle for the operation of ElGamal cryptosystem, deeply analyzing the mathematical meaning of ElGamal cryptosystem in DLP (Discrete Logarithm Problem). Then, from the definitions and properties of elliptic curves in fields, we deduced an elliptic curve version of DLP (Discrete Logarithm Problem), proving how ECDLP (Elliptic Curve Discrete Logarithm Problem) can be utilized as a trapdoor function in cryptography.