Construction of Pairing-Friendly Elliptic Curves

The aim of this paper is to construct pairing friendly elliptic curves. In this paper, we explain a method of finding the polynomials representing √−D and ζk over the field containing √−D and ζk and how to construct a pairing friendly elliptic curves over the cyclotomic fields containing Q(ζk, √−D) for arbitrary k and D by CP method. By using the factorization of the cyclotomic polynomial combined some polynomial, we extend the construction over cyclotomic fields to the construction over some extensions of the cyclotomic fields containing Q(ζk, √−D). We explain the limitation of finding more families of pairing friendly elliptic curves with embedding degree 10. For all computation, we use the PARI-GP [13].