ISOGENY FORMULAS FOR JACOBI INTERSECTION AND TWISTED HESSIAN CURVES

The security of public-key systems is based on the difficulty of solving certain mathematical problems. With the possible emergence of large-scale quantum computers several of these problems, such as factoring and computing discrete logarithms, would be efficiently solved. Research on quantum-resistant public-key cryptography, also called post-quantum cryptography (PQC), has been productive in recent years. Public-key cryptosystems based on the problem of computing isogenies between supersingular elliptic curves appear to be good candidates for the next generation of public-key cryptography standards in the PQC scenario. In this work, motivated by a previous work by D. Moody and D. Shumow {[}17], we derived maps for elliptic curves represented in Jacobi Intersection and Twisted Hessian models. Our derivation follows a multiplicative strategy that contrasts with the additive idea presented in the Velu formula. Finally, we present a comparison of computational cost to generate maps for isogenies of degree l, where l = 2k + 1. In affine coordinates, our formulas require 46.8% less computation than the Huff model and 48% less computation than the formulas given for the Extended Jacobi Quartic model when computing isogenies of degree 3. Considering higher degree isogenies as 101, our formulas require 23.4% less computation than the Huff model and 24.7% less computation than the formula for the Extended Jacobi Quartic model. (AU)