Fast Hashing to ${\mathbb {G}}_{2}$ on Aurifeuillean Pairing-Friendly Elliptic Curves.

In pairing-based cryptography, the computation of asymmetric pairings $$e: {\mathbb {G}}_{1}\times {\mathbb {G}}_{2}\longrightarrow {\mathbb {G}}_{T}$$ requires input points of prime order r. The process of getting those r-torsion point is known as hashing into $${\mathbb {G}}_{1}$$ or $${\mathbb {G}}_{2}$$ and is in general costly. Few recent works have considered the Scott et al.’s method and Fuentes et al.’s method for hashing on specific families of pairing-friendly curves. In this work, we apply those two methods on the recently discovered Scott–Guillevic Aurifeuillean curves with embedding degree $$k=6,9,15,18,27$$ and 54. The results obtained show that the Fuentes et al.’s method is at least twice faster than the Scott et al.’s method in terms of group operations. In addition, the computational cost of hashing into $${\mathbb {G}}_{2}$$ studied in this work is higher compared to the previous work done with BN curves, KSS curves and BLS curves at comparable embedding degrees.