Over the past few years there has been an interest in pairing-based cryptography. A pairing is a function which is a map from a subgroup of E(Fq) to a multiplicative ring of extension field of a defined field Fq^k. The k is called an embedding degree. This value is very important to construct an efficient pairing-based.
Since we necessary to construct an elliptic curve with prescribed embedding degree k<20. But it is known that an embedding degree become large in proportion to the size of defined field. So it is desired to construct an elliptic curve with an embedding degree k<20 which can be computed easily and satisfy a security level.

The aim of the research is construct an elliptic curve with embedding degree k<20 and large subgroup.