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Since its invention in the 1980's, elliptic curve cryptography has become a popular and standardized approach to instantiate public-key cryptography. In this paper, the authors 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 user to work with moduli of half the size. They 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.
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