3- and 5-Isogenies of Supersingular Edwards Curves

TL;DR

This paper analyzes 3- and 5-isogenies of supersingular Edwards curves, proposing their use in SIDH cryptography to avoid issues with 2-isogenies, and provides algorithms and complexity estimates for their computation.

Contribution

It introduces minimal odd degree isogenies for Edwards curves in SIDH, with explicit equations, algorithms, and complexity analysis, enhancing quantum-resistant cryptographic methods.

Findings

Derived equations for 3- and 5-isogenies of Edwards curves.

Provided algorithms with specific complexity estimates for isogeny computation.

Identified parameters for cryptosystem implementation at 128-bit quantum security.

Abstract

An analysis is made of the properties and conditions for the existence of 3- and 5-isogenies of complete and quadratic supersingular Edwards curves. For the encapsulation of keys based on the SIDH algorithm, it is proposed to use isogeny of minimal odd degrees 3 and 5, which allows bypassing the problem of singular points of the 2nd and 4th orders, characteristic of 2-isogenies. A review of the main properties of the classes of complete, quadratic, and twisted Edwards curves over a simple field is given. Equations for the isogeny of odd degrees are reduced to a form adapted to curves in the form of Weierstrass. To do this, use the modified law of addition of curve points in the generalized Edwards form, which preserves the horizontal symmetry of the curve return points. Examples of the calculation of 3- and 5-isogenies of complete Edwards supersingular curves over small simple fields are given, and the properties of the isogeny composition for their calculation with large-order kernels are discussed. Equations are obtained for upper complexity estimates for computing isogeny of odd degrees 3 and 5 in the classes of complete and quadratic Edwards curves in projective coordinates; algorithms are constructed for calculating 3- and 5-isogenies of Edwards curves with complexity 6M + 4S and 12M + 5S, respectively. The conditions for the existence of supersingular complete and quadratic Edwards curves of order 4x3mx5n and 8x3mx5n are found. Some parameters of the cryptosystem are determined when implementing the SIDH algorithm at the level of quantum security of 128 bits.