### Abstract

Let E be an elliptic curve defined over F_{2n}. The inverse operation of point doubling, called point halving, can be done up to three times as fast as doubling. Some authors have therefore proposed to perform a scalar multiplication by an "halve-and-add" algorithm, which is faster than the classical double-and-add method. If the coefficients of the equation defining the curve lie in a small subfield of F_{2n}, one can use the Frobenius endomorphism τ of the field extension to replace doublings. Since the cost of τ is negligible if normal bases are used, the scalar multiplication is written in "base τ" and the resulting "τ-and-add" algorithm gives very good performance. For elliptic Koblitz curves, this work combines the two ideas for the first time to achieve a novel decomposition of the scalar. This gives a new scalar multiplication algorithm which is up to 14.29% faster than the Frobenius method, without any additional precomputation.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Editors | Feng Bao, Robert Deng, Jianying Zhou |

Publisher | Springer Verlag |

Pages | 28-40 |

Number of pages | 13 |

ISBN (Print) | 3540210180, 9783540210184 |

DOIs | |

Publication status | Published - 2004 |

### 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 | 2947 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Keywords

- Integer decomposition
- Koblitz curves
- Point halving
- Scalar multiplication
- τ-adic expansion

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(pp. 28-40). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2947). Springer Verlag. https://doi.org/10.1007/978-3-540-24632-9_3