[Resource Topic] 2003/097: Low Cost Security: Explicit Formulae for Genus 4 Hyperelliptic Curves

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Low Cost Security: Explicit Formulae for Genus 4 Hyperelliptic Curves

Authors: Jan Pelzl, Thomas Wollinger, Christof Paar


It is widely believed that genus four hyperelliptic curve
cryptosystems (HECC) are not attractive for practical applications
because of their complexity compared to systems based on lower
genera, especially elliptic curves. Our contribution shows that
for low cost security applications genus-4 hyperelliptic curves
(HEC) can outperform genus-2 HEC and that we can achieve a
performance similar to genus-3 HEC. Furthermore our implementation
results show that a genus-4 HECC is an alternative cryptosystem to
systems based on elliptic curves.

In the work at hand we present for the first time explicit
formulae for genus-4 HEC, resulting in a 60% speed-up compared to
the best published results. In addition we implemented genus-4
HECC on a Pentium4 and an ARM microprocessor. Our implementations
on the ARM show that for genus four HECC are only a factor of 1.66
slower than genus-2 curves considering group order ~2^{190}.
For the same group order ECC and genus-3 HECC are about
a factor of 2 faster than genus-4 curves on the ARM. The two most
surprising results are: 1) for low cost security application,
namely considering an underlying group of order 2^{128}, HECC
with genus 4 outperform genus-2 curves by a factor of 1.46 and has
similar performance to genus-3 curves on the ARM and 2) when
compared to genus-2 and genus-3, genus-4 HECC are better suited to
embedded microprocessors than to general purpose processors.

ePrint: https://eprint.iacr.org/2003/097

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