### 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 . 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.

Original language | English |
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Title of host publication | Cryptographic Hardware and Embedded Systems - CHES 2008 - 10th International Workshop, Proceedings |

Editors | Elisabeth Oswald, Pankaj Rohatgi |

Pages | 244-265 |

Number of pages | 22 |

Volume | 5154 LNCS |

DOIs | |

Publication status | Published - 2008 |

Event | 10th International Workshop on Cryptographic Hardware and Embedded Systems, CHES 2008 - Washington, D.C., United States Duration: 10 Aug 2008 → 13 Aug 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5154 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 10th International Workshop on Cryptographic Hardware and Embedded Systems, CHES 2008 |
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Country | United States |

City | Washington, D.C. |

Period | 10/08/08 → 13/08/08 |

### Keywords

- Binary fields
- Complete addition law
- Countermeasures against side-channel attacks
- Edwards curves
- Elliptic curves
- Montgomery ladder

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## Cite this

*Cryptographic Hardware and Embedded Systems - CHES 2008 - 10th International Workshop, Proceedings*(Vol. 5154 LNCS, pp. 244-265). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5154 LNCS). https://doi.org/10.1007/978-3-540-85053-3_16