Pairing-Friendly Elliptic Curves with Various Discriminants

Abstract

This paper extends the Brezing-Weng method by parameterizing the discriminant D by a polynomial D(x). To date, the maximum of CM discriminant can be adequately addressed is about 14-digits. Thus the degree of the square free part of D(x) has to be sufficiently small. By making the square free part of D(x) a linear monomial, the degree of the square free part is small and by substituting x to some quadratic monomial, pairing-friendly curves with various discriminants can be constructed. In order that a square free part of D(x) is of the form ax, ax has to be a square element as a polynomial representation in a number field. Two methods are introduced to apply this construction. For k = 5, 8, 9, 15, 16, 20, 24 and 28, the proposed method gives smaller ρ value than those in previous studies.