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Cryptography has become one of the most important areas in information technology. With cryptography it is possible to construct many security protocols which are the basic infrastructure for secure communications like SSL, IPsec, SSH,... Two inevitable concepts for these attractive applications are public-key cryptosystem and digital signature. Interestingly, the security of these concepts are underpinned by the difficulty of mathematical problems, for examples, a factoring problem, a discrete logarithm problem over elliptic curves, and lattice theory. Once the underlying problem is broken, the whole system using the problem is no longer secure. One of the main research subjects is to investigate the security of these problems and its implication to cryptographic protocols.