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Efficient Arithmetic on Hyperelliptic Curves
Authors: Tanja LangeAbstract:
Using the Frobenius endomorphism the operation of computing
scalar-mulitples in the Jacobian of a hyperelliptic curve is sped-up
considerably. The kind of curves considered are Kobiltz i.e. subfield
curves, defined over a small finite field which are then considered
over a large extension field. We deal with computation of
the group order over various extension fields, algorithms to obtain
the mentioned speed-up, and experimental results concerning both
issues. Additionally an alternative set-up is treated which uses arihtmetic in the finite field only and allows shorter code for similar security.
Furthermore explicit formulae to perform the arithmetic in the ideal class group explicitely are derived and can thus be used for implementation in hardware; in software they are also faster than the generic Cantor algorithm. As a second group suitable for cryptographic applications the trace-zero-variety is considered. Here we investigate the group operation and deal with security issues.
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