3- and 5-Isogenies of Supersingular Edwards Curves
This work addresses a specific bottleneck in isogeny-based cryptography for quantum-resistant security, but it is incremental as it builds on known methods for odd-degree isogenies.
The paper tackles the problem of singular points in 2-isogenies for key encapsulation in the SIDH algorithm by proposing the use of 3- and 5-isogenies on supersingular Edwards curves, resulting in algorithms with complexities of 6M + 4S and 12M + 5S, respectively, and conditions for curves of specific orders.
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.