### Abstract

We show that the elliptic curve cryptosystems based on the Montgomery-form E^{M}: BY^{2} = X^{3}+ AX^{2} +X are immune to the timing-attacks by using our technique of randomized projective coordinates, while Montgomery originally introduced this type of curves for speeding up the Pollard and Elliptic Curve Methods of integer factorization [Math. Comp. Vol.48, No.177, (1987) pp.243-264]. However, it should be noted that not all the elliptic curves have the Montgomery-form, because the order of any elliptic curve with the Montgomery-form is divisible by “4”. Whereas recent ECC-standards [NIST,SEC-1] recommend that the cofactor of elliptic curve should be no greater than 4 for cryptographic applications. Therefore, we present an efficient algorithm for generating Montgomery-form elliptic curve whose cofactor is exactly “4”. Finally, we give the exact consition on the elliptic curves whether they can be represented as a Montgomery-form or not. We consider divisibility by “8” for Montgomery-form elliptic curves. We implement the proposed algorithm and give some numerical examples obtained by this.

Original language | English |
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Title of host publication | Public Key Cryptography - 3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000, Proceedings |

Editors | Hideki Imai, Yuliang Zheng |

Publisher | Springer Verlag |

Pages | 238-257 |

Number of pages | 20 |

ISBN (Print) | 3540669671, 9783540669678 |

DOIs | |

Publication status | Published - Jan 1 2000 |

Event | 3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000 - Melbourne, Australia Duration: Jan 18 2000 → Jan 20 2000 |

### 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 | 1751 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000 |
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Country | Australia |

City | Melbourne |

Period | 1/18/00 → 1/20/00 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Public Key Cryptography - 3rd International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000, Proceedings*(pp. 238-257). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1751). Springer Verlag. https://doi.org/10.1007/978-3-540-46588-1_17