A New Method for Constructing Pairing-Friendly Abelian Surfaces

Abstract

We present a new method for constructing simple ordinary abelian surfaces with a small embedding degree. To a quartic CM field K, we associate a quadric surface H ⊂ ℙ3(ℚ) and use its parametrization to determine Weil numbers in K corresponding in the sense of Honda-Tate theory to such surfaces. In general, the resulting surfaces have parameter ρ ≈ 8. However, if there exist rational lines on H, they can be used to achieve ρ ≈ 4. We give examples of non-primitive quartic CM fields such that H has rulings by rational lines. Furthermore, we show how our method can be used to construct parametric families of pairing-friendly surfaces.