Cryptography was once considered to be a means of maintaining secrecy of communications only in military affairs and diplomacy. However, today, modern cryptography is used for various purposes in familiar circumstances. Public-key cryptography is a key technology of modern society; it is used for personal authentication, electronic commerce on the Internet, copyright protection of DVDs, and so on. In particular, the RSA public-key cryptosystem, which was proposed more than 30 years ago, has become the de facto standard of cryptographic software since the spread of the Internet in the 1990s. Another technology, called elliptic curve cryptography, was proposed in 1985. It can perform arithmetic processing at high speed, and since the beginning of the 2000s, it has been implemented in devices such as DVD players and personal digital assistants. Pairing-based cryptography, first proposed in 2000, can be incorporated in security technologies that are not practical with the previous public-key cryptographies. It is actively studied by various organizations around the world. In this chapter, we explain the basic mathematics and security evaluations of public-key cryptography.