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Isogenies on twisted Hessian curves.
Fouazou Lontouo Perez1, Thinh Dang2,3, Emmanuel Fouotsa4
1Department of Mathematics and Computer Sciences, Faculty of Sciences, The University of Maroua P.O.Box 814 Maroua,Cameroon.
Journal of Mathematical Cryptology
|July 29, 2021
Summary
This study presents new explicit formulas for isogenies between twisted Hessian elliptic curves. These formulas offer computational advantages, particularly for processing kernels and points in affine coordinates, advancing elliptic curve cryptography.
Area of Science:
- Number Theory
- Cryptography
- Algebraic Geometry
Background:
- Elliptic curves are fundamental in modern cryptography.
- Vélu's formula provides isogeny calculations for Weierstrass curves.
- Isogeny formulas for other curve forms (Edwards, Montgomery) exist but are limited.
Purpose of the Study:
- To derive explicit isogeny formulas for (twisted) Hessian elliptic curves.
- To analyze the computational cost of these new formulas.
- To compare their efficiency against existing methods.
Main Methods:
- Derivation of isogeny formulas specific to the (twisted) Hessian coordinate system.
- Operation count analysis in the base field for formula computation.
- Comparative performance evaluation with existing isogeny formulas.
Main Results:
- Explicit isogeny formulas for (twisted) Hessian elliptic curves are successfully derived.
- The new formulas demonstrate lower computational costs for kernel processing.
- The X-affine formula shows the lowest cost for affine point processing.
Conclusions:
- The derived formulas provide efficient tools for computations involving (twisted) Hessian elliptic curves.
- These findings contribute to the optimization of elliptic curve-based cryptographic protocols.
- The study highlights the potential of Hessian forms for performance improvements in isogeny-based cryptography.
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