# Efficient Arithmetic on Hessian Curves

```@inproceedings{Farashahi2010EfficientAO,
title={Efficient Arithmetic on Hessian Curves},
author={Reza Rezaeian Farashahi and Marc Joye},
booktitle={Public Key Cryptography},
year={2010}
}```
• Published in Public Key Cryptography 26 May 2010
• Computer Science, Mathematics
This paper considers a generalized form for Hessian curves. The family of generalized Hessian curves covers more isomorphism classes of elliptic curves. Over a finite field \$\mathbb{F}_q\$, it is shown to be equivalent to the family of elliptic curves with a torsion subgroup isomorphic to ℤ/3ℤ. This paper provides efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen parameters. This paper also presents extremely fast…
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## References

SHOWING 1-10 OF 52 REFERENCES
Binary Edwards Curves
• Mathematics, Computer Science
CHES
• 2008
This paper presents the first complete addition formulas for binary elliptic curves, i.e., addition formulas that work for all pairs of input points, with no exceptional cases, in the literature.
The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines
• Mathematics, Computer Science
Finite Fields Their Appl.
• 2009
Fast Multiplication on Elliptic Curves over GF(2m) without Precomputation
• Computer Science, Mathematics
CHES
• 1999
The improved method possesses many desirable features for implementing elliptic curves in restricted environments and requires less memory than projective schemes and the amount of computation needed for a scalar multiplication is fixed for all multipliers of the same binary length.
The arithmetic of characteristic 2 Kummer surfaces
• Mathematics, Computer Science
IACR Cryptol. ePrint Arch.
• 2008
It is shown that applying the same strategy to elliptic curves gives Montgomery-like formulas in odd characteristic that are of some interest, and the formulas obtained are very efficient and may be useful in cryptographic applications.
A New Addition Formula for Elliptic Curves over GF(2n)
• Mathematics, Computer Science
IEEE Trans. Computers
• 2002
The complexity analysis shows that the new addition formula speeds up the addition in projective coordinates by about 10-2 percent, which leads to enhanced scalar multiplication methods for random and Koblitz curves.
Faster Addition and Doubling on Elliptic Curves
• Mathematics, Computer Science
ASIACRYPT
• 2007
An extensive comparison of different forms of elliptic curves and different coordinate systems for the basic group operations (doubling, mixed addition, non-mixed addition, and unified addition) as well as higher-level operations such as multi-scalar multiplication.
Lectures on elliptic curves
Introduction 1. Curves of genus: introduction 2. p-adic numbers 3. The local-global principle for conics 4. Geometry of numbers 5. Local-global principle: conclusion of proof 6. Cubic curves 7.
Improved Algorithms for Elliptic Curve Arithmetic in GF(2n)
• Computer Science, Mathematics
Selected Areas in Cryptography
• 1998
A new method for doubling an elliptic curve point, which is simpler to implement than the fastest known method, due to Schroeppel, and which favors sparse elliptic Curve coefficients, and a new kind of projective coordinates that provides the fastestknown arithmetic on elliptic curves.
Improved Algorithms for Elliptic Curve Arithmetic in Gf(2 N ) Improved Algorithms for Elliptic Curve Arithmetic in Gf (2 N )
A new method for doubling an elliptic Curve point, which is simpler to implement than the fastest known method, due to Schroeppel, and which favors sparse elliptic curve coeecients, and a new kind of projective coordinates that provides the fastestknown arithmetic on elliptic curves.
A New Method for Speeding Up Arithmetic on Elliptic Curves over Binary Fields
• Computer Science, Mathematics
IACR Cryptol. ePrint Arch.
• 2007
This paper reduces the costs of point doubling and addition on elliptic curves over binary fields to less than S M 5 8 + S M 3 3 + and , respectively, by using a new projective coordinates the authors call PL-coordinates and rewriting the point doubling formula.