Isogenies on twisted Hessian curves

Abstract

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.

Enhanced Performance of Isogenies Over Huff Curve for Post Quantum Cryptography

“…We characterize the isogeny of degree 2, which maps 𝑃 4 to (0, 0). Φ: 𝑀 𝐴,𝐵 → 𝐹 (𝑅: 𝑆: 𝑇) → (𝑅(𝑅 − 𝑇) 2 : 𝑆(𝑅 2 − 𝑇 2 ): 𝑅 2 𝑇) (24) where 𝑟 = 𝑅 𝑇 𝑎𝑛𝑑 𝑠 = 𝑆 𝑇 for (𝑟, 𝑠) ∈ 𝑀 𝐴,𝐵 . The given corresponding image curve is…”

“…Recently, Broon et al proposed an optimized formula for isogenies between Hessian model (twisted) of elliptic curves in [24]. In the Montgomery type of elliptic curve, B-SIDH speed is determined by using a new technique to enumerate 𝑙isogeny curve developed by Huang et al [25].…”