## Cryptology ePrint Archive: Report 2008/171

**Binary Edwards Curves**

*Daniel J. Bernstein and Tanja Lange and Reza Rezaeian Farashahi*

**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.

**Category / Keywords: **public-key cryptography / Elliptic curves, Edwards curves, binary fields, complete addition law, Montgomery ladder, countermeasures against side-channel attacks

**Date: **received 15 Apr 2008, last revised 11 Jun 2008

**Contact author: **tanja at hyperelliptic org

**Available format(s): **PDF | BibTeX Citation

**Note: **Improved explicit formulas. See also the Explicit-Formulas Database, http://hyperelliptic.org/EFD.

**Version: **20080611:104806 (All versions of this report)

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