Cryptology ePrint Archive: Report 2002/107

Efficient Arithmetic on Hyperelliptic Curves

Tanja Lange

Abstract: 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.

Category / Keywords: public-key cryptography / hyperelliptic curves, scalar multiplication, arithmetic, implementation, class number

Publication Info: PhD Thesis, Preprint 4-2002, Universit\"at-Gesamthochschule Essen

Date: received 2 Aug 2002, last revised 15 Dec 2003

Contact author: lange at itsc ruhr-uni-bochum de

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Version: 20031215:222949 (All versions of this report)

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