4

I'm referencing RFC-6090 for an attempt at implementing ECC in my spare-time project.

In the RFC, pseudo-code examples are given to illustrate how to handle points-at-infinity in point arithmetic, and this involved several special cases. This is because the point doubling and point add formula in affine and homogeneous coordinates cannot correctly handle point at infinity.

So I want to ask: is there a coordinates system where the point doubling and adding formula can handle point at infinity cases?

DannyNiu
  • 10,640
  • 2
  • 27
  • 64

1 Answers1

4

For the homogenous coordinate systems it is possible to use addition and doubling formulas which are complete and exception-less for all odd order elliptic curves. These were originally derived by Bosma and Lenstra and optimized more recently by Renes et al. in this paper.

The main issue is that they are slower than incomplete formulas. The limitation of being valid only on odd order curves is not significant as most standards constains only prime-order short weierstass.

Here's a table, from the paper, describing the required field operation and comparing it with standard Jacobian: Table 3

Ruggero
  • 7,339
  • 33
  • 42