Elliptic vs. hyperelliptic, part 1

Daniel J. Bernstein

The "Curve25519" speed records for Diffie-Hellman software were set with a Montgomery-form elliptic curve. Is it possible to achieve even better scalar-multiplication speeds with a Gaudry-form hyperelliptic curve? Exactly how fast is arithmetic modulo 2127-1, compared to arithmetic modulo 2255-19?

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