Two is Greater than One

In this paper we highlight the benefits of using genus-2 curves in public-key cryptography. Compared to the standardized genus-1 curves, or elliptic curves, arithmetic on genus-2 curves is typically more involved but allows us to work with moduli of half the size. We give a taxonomy of the best known techniques to realize genus-2 based cryptography, which includes fast formulas on the Kummer surface and efficient 4-dimensional GLV decompositions. By studying different modular arithmetic approaches on these curves, we present a range of genus-2 implementations. Our implementation on the Kummer surface breaks the 120 thousand cycle barrier which sets a new software speed record at the 128-bit security level for side-channel resistant scalar multiplications compared to all previous genus-1 and genus-2 implementations.