Some properties of τ -adic expansions on hyperelliptic Koblitz curves

Keisuke Hakuta, Hisayoshi Sato, Tsuyoshi Takagi

研究成果: ジャーナルへの寄稿学術誌査読

2 被引用数 (Scopus)


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.

ジャーナルJournal of Applied Mathematics and Computing
出版ステータス出版済み - 10月 1 2018

!!!All Science Journal Classification (ASJC) codes

  • 計算数学
  • 応用数学


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