### Abstract

In this paper we produce a practical and efficient algorithm to find a decomposition of type n = ∑_{i=1} ^{k}2^{si}3 ^{ti}, s^{i} t^{i} ∈ ℕ ∪ {0} with k ≤ (c +o(1)) log n/log log n . It is conjectured that one can take c = 2 above. Then this decomposition is refined into an effective scalar multiplication algorithm to compute nP on some supersingular elliptic curves of characteristic 3 with running time bounded by O (log n/log log n) and essentially no storage. To our knowledge, this is the first instance of a scalar multiplication algorithm that requires o(log n) curve operations on an elliptic curve over double struck F sign_{q}, with log q ≈ log n and uses comparable storage as in the standard double-and-add algorithm. This leads to an efficient algorithm very useful for cryptographic protocols based on supersingular curves. This is for example the case of the well-studied (in the past four years) identity based schemes. The method carries over to any supersingular curve of fixed characteristic.

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) |

Pages | 171-182 |

Number of pages | 12 |

Volume | 3715 LNCS |

DOIs | |

Publication status | Published - 2005 |

Externally published | Yes |

Event | 1st International Conference on Cryptology in Malaysia on Progress in Cryptology - Mycrypt 2005 - Kuala Lumpur, Malaysia Duration: Sep 28 2005 → Sep 30 2005 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 3715 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 1st International Conference on Cryptology in Malaysia on Progress in Cryptology - Mycrypt 2005 |
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Country | Malaysia |

City | Kuala Lumpur |

Period | 9/28/05 → 9/30/05 |

### Fingerprint

### Keywords

- Exponentiation algorithms
- Integer decomposition

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 3715 LNCS, pp. 171-182). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3715 LNCS). https://doi.org/10.1007/11554868_12

**An analysis of double base number systems and a sublinear scalar multiplication algorithm.** / Ciet, Mathieu; Sica, Francesco.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 3715 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 3715 LNCS, pp. 171-182, 1st International Conference on Cryptology in Malaysia on Progress in Cryptology - Mycrypt 2005, Kuala Lumpur, Malaysia, 9/28/05. https://doi.org/10.1007/11554868_12

}

TY - GEN

T1 - An analysis of double base number systems and a sublinear scalar multiplication algorithm

AU - Ciet, Mathieu

AU - Sica, Francesco

PY - 2005

Y1 - 2005

N2 - In this paper we produce a practical and efficient algorithm to find a decomposition of type n = ∑i=1 k2si3 ti, si ti ∈ ℕ ∪ {0} with k ≤ (c +o(1)) log n/log log n . It is conjectured that one can take c = 2 above. Then this decomposition is refined into an effective scalar multiplication algorithm to compute nP on some supersingular elliptic curves of characteristic 3 with running time bounded by O (log n/log log n) and essentially no storage. To our knowledge, this is the first instance of a scalar multiplication algorithm that requires o(log n) curve operations on an elliptic curve over double struck F signq, with log q ≈ log n and uses comparable storage as in the standard double-and-add algorithm. This leads to an efficient algorithm very useful for cryptographic protocols based on supersingular curves. This is for example the case of the well-studied (in the past four years) identity based schemes. The method carries over to any supersingular curve of fixed characteristic.

AB - In this paper we produce a practical and efficient algorithm to find a decomposition of type n = ∑i=1 k2si3 ti, si ti ∈ ℕ ∪ {0} with k ≤ (c +o(1)) log n/log log n . It is conjectured that one can take c = 2 above. Then this decomposition is refined into an effective scalar multiplication algorithm to compute nP on some supersingular elliptic curves of characteristic 3 with running time bounded by O (log n/log log n) and essentially no storage. To our knowledge, this is the first instance of a scalar multiplication algorithm that requires o(log n) curve operations on an elliptic curve over double struck F signq, with log q ≈ log n and uses comparable storage as in the standard double-and-add algorithm. This leads to an efficient algorithm very useful for cryptographic protocols based on supersingular curves. This is for example the case of the well-studied (in the past four years) identity based schemes. The method carries over to any supersingular curve of fixed characteristic.

KW - Exponentiation algorithms

KW - Integer decomposition

UR - http://www.scopus.com/inward/record.url?scp=33646198552&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646198552&partnerID=8YFLogxK

U2 - 10.1007/11554868_12

DO - 10.1007/11554868_12

M3 - Conference contribution

SN - 3540289380

SN - 9783540289388

VL - 3715 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 171

EP - 182

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -