Abstract:

 From the beginning of the nineteenth century, mathematicians have put the elliptic curves as a research objective of algebra, geometry and number theory to study in-depth. So far, the theory of elliptic curves applied not only in the field of mathematics, and also widely in computing science, information security, physics and other fields. In this paper, we reviewed the application of elliptic curves in cryptography, from the primality testing, integer factorization to elliptic curve cryptosystem, bilinear pairing based cryptosystem and the quantum-resistant cryptosystems from elliptic curve isogenies. We introduced the basic principles and the status of these applications. Finally, we briefly discussed some open questions and possible future progress in this area.

Key words:  elliptic curve; cryptography; discrete logarithm problem (DLP); bilinear pairing; isogenies