Scalar multiplication using frobenius expansion over twisted elliptic curve for ate pairing based cryptography

Yasuyuki Nogami, Yumi Sakemi, Takumi Okimoto, Kenta Nekado, Masataka Akane, Yoshitaka Morikawa

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


For ID-based cryptography, not only pairing but also scalar multiplication must be efficiently computable. In this paper, we propose a scalar multiplication method on the circumstances that we work at Ate pairing with Barreto-Naehrig (BN) curve. Note that the parameters of BN curve are given by a certain integer, namely mother parameter. Adhering the authors' previous policy that we execute scalar multiplication on subfield-twisted curve E∼(F p2) instead of doing on the original curve E(F p12), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) φ∼ in E∼(Fp2). On BN curves, note φ∼ is identified with the scalar multiplication by p. However a scalar is always smaller than the order r of BN curve for Ate pairing, so ST-SFM does not directly applicable to the above circumstances. We then exploit the expressions of the curve order r and the characteristic p by the mother parameter to derive some radices such that they are expressed as a polynomial of p. Thus, a scalar multiplication [s] can be written by the series of ST-SFMs φ∼. In combination with the binary method or multi-exponentiation technique, this paper shows that the proposed method runs about twice or more faster than plain binary method.

Original languageEnglish
Pages (from-to)182-189
Number of pages8
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number1
Publication statusPublished - Jan 2009


  • Ate pairing
  • BN curve
  • Frobenius mapping
  • Scalar multiplication
  • Twisted subfield computation

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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