### Abstract

In 1999 Silverman [21] introduced a family of binary finite fields which are composite extensions of F_{2} and on which arithmetic operations can be performed more quickly than on prime extensions of F_{2} of the same size. We present here a fast approach to elliptic curve cryptography using a distinguished subset of the set of Silverman fields F_{2}N = F_{h}n. This approach leads to a theoretical computation speedup over fields of the same size, using a standard point of view (cf. [7]). We also analyse their security against prime extension fields F_{2}p, where p is prime, following the method of Menezes and Qu [12]. We conclude that our fields do not present any significant weakness towards the solution of the elliptic curve discrete logarithm problem and that often the Weil descent of Galbraith-Gaudry-Hess-Smart (GGHS) does not offer a better attack on elliptic curves defined over F_{2}N than on those defined over F_{2}p, with a prime p of the same size as N. A noteworthy example is provided by F_{2226}: a generic elliptic curve Y^{2} + XY = X^{3} + αX^{2} + β defined over F_{2226} is as prone to the GGHS Weil descent attack as a generic curve defined on the NIST field F_{2233}.

Original language | English |
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Title of host publication | Progress in Cryptology - INDOCRYPT 2001 - 2nd International Conference on Cryptology in India, Proceedings |

Publisher | Springer Verlag |

Pages | 108-116 |

Number of pages | 9 |

Volume | 2247 |

ISBN (Print) | 9783540453116 |

Publication status | Published - 2001 |

Event | 2nd International Conference on Cryptology in India, INDOCRYPT 2001 - Chennai, India Duration: Dec 16 2001 → Dec 20 2001 |

### 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 | 2247 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 2nd International Conference on Cryptology in India, INDOCRYPT 2001 |
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Country | India |

City | Chennai |

Period | 12/16/01 → 12/20/01 |

### Fingerprint

### Keywords

- Elliptic curve cryptography
- Fast performance
- Finite fields
- Weil descent

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Progress in Cryptology - INDOCRYPT 2001 - 2nd International Conference on Cryptology in India, Proceedings*(Vol. 2247, pp. 108-116). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2247). Springer Verlag.