Paper 2008/171 Original: eprint.iacr.org/2008/171 ↗
Binary Edwards Curves
Abstract
This paper presents a new shape for ordinary elliptic curves over fields of characteristic 2. Using the new shape, this paper presents the first complete addition formulas for binary elliptic curves, i.e., addition formulas that work for all pairs of input points, with no exceptional cases. If n >= 3 then the complete curves cover all isomorphism classes of ordinary elliptic curves over F_2^n. This paper also presents dedicated doubling formulas for these curves using 2M + 6S + 3D, where M is the cost of a field multiplication, S is the cost of a field squaring, and D is the cost of multiplying by a curve parameter. These doubling formulas are also the first complete doubling formulas in the literature, with no exceptions for the neutral element, points of order 2, etc. Finally, this paper presents complete formulas for differential addition, i.e., addition of points with known difference. A differential addition and doubling, the basic step in a Montgomery ladder, uses 5M + 4S + 2D when the known difference is given in affine form.
Metadata
- Available format(s)
PDF- Category
- Public-key cryptography
- Publication info
- Preprint.
- History
- 2008-06-11: last updated
- 2008-04-15: published
- Short URL
- https://ia.cr/2008/171
- License
- https://creativecommons.org/licenses/by/4.0/
BibTeX
@misc{cryptoeprint:2008/171,
author = {Daniel J. Bernstein and Tanja Lange and Reza Rezaeian Farashahi},
title = {Binary Edwards Curves},
howpublished = {Cryptology {ePrint} Archive, Paper 2008/171},
year = {2008},
url = {https://eprint.iacr.org/2008/171}
}