Theoretical and Practical Possibilities of Elliptic Curves : From Elliptic Curve Cryptosystems to Post-Quantum Cryptosystems

Abstract

In 1985, Miller and Koblitz independently introduced elliptic curve cryptosystems, a type of public key cryptosys-tem. Elliptic curve cryptosystems use the fact that elliptic curves become a group to realize an ID-based cryptosystem for the first time, by applying a bilinear map on an elliptic curve. Furthermore, in recent years, isogenies on elliptic curves have been used to realize post-quantum cryptosystems. Elliptic curves are indeed treasures for solving various cryptographic problems. Elliptic curves have been applied to resolve many theoretical problems and are merely a theoretical breakthrough. The charm of elliptic curves is that they are highly practical. To verify the correctness of a blockchain, the elliptic curve DSA signature (ECDSA) is used, since the signature size of ECDSA is very short. Furthermore, the elliptic curve realizes a post-quantum cryptosystem. In this paper, we discuss various breakthroughs achieved by using elliptic curves as well as international standardization related to elliptic curves.