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Isogenies on twisted Hessian curves



Dustin Moody, Thinh Dang, Fouazou Lontouo Perez, Emmanuel Fouotsa


Elliptic curves are typically defined by Weierstrass equations. Given a kernel, the well-known Vélu's formula shows how to explicitly write down an isogeny between Weierstrass curves. However, it is not clear how to do the same on other forms of elliptic curves without isomorphisms mapping to and from the Weierstrass form. Previous papers have shown some isogeny formulas for (twisted) Edwards, Huff, and Montgomery forms of elliptic curves. Continuing this line of work, this paper derives explicit formulas for isogenies between elliptic curves in (twisted) Hessian form. In addition, we examine the numbers of operations in the base field to compute the formulas. In comparison with other isogeny formulas, we note that our formulas for twisted Hessian curves have the lowest costs for processing the kernel and our X-affine formula has the lowest cost for processing an input point in affine coordinates.
Journal of Mathematical Cryptography


Elliptic curves, Isogeny, Hessian curves, Vélu’s formulas


Moody, D. , Dang, T. , Perez, F. and Fouotsa, E. (2021), Isogenies on twisted Hessian curves, Journal of Mathematical Cryptography, [online],, (Accessed July 17, 2024)


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Created March 16, 2021, Updated April 23, 2024