### Abstract

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 _{p}^{2}) instead of doing on the original curve E(F _{p}^{12}), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) φ∼ in E∼(F_{p}^{2}). 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 language | English |
---|---|

Pages (from-to) | 182-189 |

Number of pages | 8 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E92-A |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2009 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E92-A*(1), 182-189. https://doi.org/10.1587/transfun.E92.A.182

**Scalar multiplication using frobenius expansion over twisted elliptic curve for ate pairing based cryptography.** / Nogami, Yasuyuki; Sakemi, Yumi; Okimoto, Takumi; Nekado, Kenta; Akane, Masataka; Morikawa, Yoshitaka.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E92-A, no. 1, pp. 182-189. https://doi.org/10.1587/transfun.E92.A.182

}

TY - JOUR

T1 - Scalar multiplication using frobenius expansion over twisted elliptic curve for ate pairing based cryptography

AU - Nogami, Yasuyuki

AU - Sakemi, Yumi

AU - Okimoto, Takumi

AU - Nekado, Kenta

AU - Akane, Masataka

AU - Morikawa, Yoshitaka

PY - 2009/1

Y1 - 2009/1

N2 - 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.

AB - 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.

KW - Ate pairing

KW - BN curve

KW - Frobenius mapping

KW - Scalar multiplication

KW - Twisted subfield computation

UR - http://www.scopus.com/inward/record.url?scp=77749294674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77749294674&partnerID=8YFLogxK

U2 - 10.1587/transfun.E92.A.182

DO - 10.1587/transfun.E92.A.182

M3 - Article

VL - E92-A

SP - 182

EP - 189

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 1

ER -