### Abstract

The joint sparse form (JSF) is a representation of a pair of integers, which is famous for accelerating a multi-scalar multiplication in elliptic curve cryptosystems. Solinas' original paper showed three unsolved problems on the enhancement of JSF. Whereas two of them have been solved, the other still remains to be done. The remaining unsolved problem is as follows: To design a representation of a pair of integers using a larger digit set such as a set involving ±3, while the original JSF utilizes the digit set that consists of 0,±1 for representing a pair of integers. This paper puts an end to the problem; width-3 JSF. The proposed enhancement satisfies properties that are similar to that of the original. For example, the enhanced representation is defined as a representation that satisfies some rules. Some other properties are the existence, the uniqueness of such a representation, and the optimality of the Hamming weight. The non-zero density of the width-3 JSF is 563/1574(= 0.3577) and this is ideal. The conversion algorithm to the enhanced representation takes O(log n) memory and O(n) computational cost, which is very efficient, where n stands for the bit length of the integers. Keywords: elliptic curve cryptosystem, multi-scalar multiplication, joint sparse form (JSF).

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 | 67-84 |

Number of pages | 18 |

Volume | 6047 LNCS |

DOIs | |

Publication status | Published - 2010 |

Event | 6th International Conference on Information Security Practice and Experience, ISPEC 2010 - Seoul, Korea, Republic of Duration: May 12 2010 → May 13 2010 |

### 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 | 6047 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 6th International Conference on Information Security Practice and Experience, ISPEC 2010 |
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Country | Korea, Republic of |

City | Seoul |

Period | 5/12/10 → 5/13/10 |

### Fingerprint

### Keywords

- Joint sparse form (JSF)
- Ulliptic curve cryptosystem
- Uulti-scalar multiplication

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 6047 LNCS, pp. 67-84). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6047 LNCS). https://doi.org/10.1007/978-3-642-12827-1_6

**Width-3 joint sparse form.** / Okeya, Katsuyuki; Kato, Hidehiro; Nogami, Yasuyuki.

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. 6047 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6047 LNCS, pp. 67-84, 6th International Conference on Information Security Practice and Experience, ISPEC 2010, Seoul, Korea, Republic of, 5/12/10. https://doi.org/10.1007/978-3-642-12827-1_6

}

TY - GEN

T1 - Width-3 joint sparse form

AU - Okeya, Katsuyuki

AU - Kato, Hidehiro

AU - Nogami, Yasuyuki

PY - 2010

Y1 - 2010

N2 - The joint sparse form (JSF) is a representation of a pair of integers, which is famous for accelerating a multi-scalar multiplication in elliptic curve cryptosystems. Solinas' original paper showed three unsolved problems on the enhancement of JSF. Whereas two of them have been solved, the other still remains to be done. The remaining unsolved problem is as follows: To design a representation of a pair of integers using a larger digit set such as a set involving ±3, while the original JSF utilizes the digit set that consists of 0,±1 for representing a pair of integers. This paper puts an end to the problem; width-3 JSF. The proposed enhancement satisfies properties that are similar to that of the original. For example, the enhanced representation is defined as a representation that satisfies some rules. Some other properties are the existence, the uniqueness of such a representation, and the optimality of the Hamming weight. The non-zero density of the width-3 JSF is 563/1574(= 0.3577) and this is ideal. The conversion algorithm to the enhanced representation takes O(log n) memory and O(n) computational cost, which is very efficient, where n stands for the bit length of the integers. Keywords: elliptic curve cryptosystem, multi-scalar multiplication, joint sparse form (JSF).

AB - The joint sparse form (JSF) is a representation of a pair of integers, which is famous for accelerating a multi-scalar multiplication in elliptic curve cryptosystems. Solinas' original paper showed three unsolved problems on the enhancement of JSF. Whereas two of them have been solved, the other still remains to be done. The remaining unsolved problem is as follows: To design a representation of a pair of integers using a larger digit set such as a set involving ±3, while the original JSF utilizes the digit set that consists of 0,±1 for representing a pair of integers. This paper puts an end to the problem; width-3 JSF. The proposed enhancement satisfies properties that are similar to that of the original. For example, the enhanced representation is defined as a representation that satisfies some rules. Some other properties are the existence, the uniqueness of such a representation, and the optimality of the Hamming weight. The non-zero density of the width-3 JSF is 563/1574(= 0.3577) and this is ideal. The conversion algorithm to the enhanced representation takes O(log n) memory and O(n) computational cost, which is very efficient, where n stands for the bit length of the integers. Keywords: elliptic curve cryptosystem, multi-scalar multiplication, joint sparse form (JSF).

KW - Joint sparse form (JSF)

KW - Ulliptic curve cryptosystem

KW - Uulti-scalar multiplication

UR - http://www.scopus.com/inward/record.url?scp=78650271070&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650271070&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-12827-1_6

DO - 10.1007/978-3-642-12827-1_6

M3 - Conference contribution

AN - SCOPUS:78650271070

SN - 3642128262

SN - 9783642128264

VL - 6047 LNCS

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

SP - 67

EP - 84

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

ER -