### Abstract

In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, Frobenius-and-add algorithm using the τ-adic non-adjacent form (τ-NAF). The τ-NAF has the three properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming weight. On the other hand, Günther et al. (Speeding up the arithmetic on koblitz curves of genus two. LNCS, vol. 2012, pp. 106–117. Springer, Heidelberg, 2001) have proposed two generalizations of τ-NAF for a family of hyperelliptic curves (hyperelliptic Koblitz curves) which have been proposed by Koblitz (J Cryptol 1(3):139–150, 1989). We call these generalizations τ-adic sparse expansion, and τ-NAF, respectively. To our knowledge, it is not known whether the three properties are true or not, especially, the existence must be satisfied for concrete cryptographic implementations. We provide an answer to the question. Our investigation shows that the τ-adic sparse expansion has only the existence and the τ-NAF has the existence and uniqueness. Our results guarantee the concrete cryptographic implementations of these generalizations.

Original language | English |
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Pages (from-to) | 367-388 |

Number of pages | 22 |

Journal | Journal of Applied Mathematics and Computing |

Volume | 58 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Oct 1 2018 |

### All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics

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## Cite this

*Journal of Applied Mathematics and Computing*,

*58*(1-2), 367-388. https://doi.org/10.1007/s12190-017-1149-5