### Abstract

Recently, the radix-3 representation of integers is used for the efficient implementation of pairing based cryptosystems. In this paper, we propose non-adjacent form of radix-r representation (rNAF) and efficient algorithms for generating rNAF. The number of non-trivial digits is (r - 2)(r + l)/2 and its average density of non-zero digit is asymptotically (r - l)/(2r -1). For r = 3, the non-trivial digits are {±2, ±4} and the non-zero density is 0.4. We then investigate the width-w version of rNAF for the general radix-r representation, which is a natural extension of the width-w NAF. Finally we compare the proposed algorithms with the generalized NAF (gNAF) discussed by Joye and Yen. The proposed scheme requires a larger table but its non-zero density is smaller even for large radix. We explain that gNAF is a simple degeneration of rNAF - we can consider that rNAF is a canonical form for the radix-r representation. Therefore, rNAF is a good alternative to gNAF. Keywords: Non-adjacent form, radix-r representation, signed window method, elliptic curve cryptosystem, pairing based cryptosystem

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 | Kan Zhang, Yuliang Zheng |

Publisher | Springer Verlag |

Pages | 99-110 |

Number of pages | 12 |

ISBN (Print) | 3540232087, 9783540232087 |

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 | 3225 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### All Science Journal Classification (ASJC) codes

- 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. 99-110). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3225). Springer Verlag. https://doi.org/10.1007/978-3-540-30144-8_9