### Abstract

This paper proposes an efficient implementation of Ate pairing on twisted elliptic curve. Suppose that a pairing-friendly elliptic curve E has a twisted elliptic curve E′ of degree d, and let Ψ_{d} be an isomorphic map from E′ (double struck F sign_{p}
^{e}) to the corresponding subgroup of E(double struck F sign_{p}
^{k}). Then, consider double struck G sign′_{1} = Ψ_{d} ^{-1}(double struck G sign_{1}) and double struck G sign′_{2} = Ψ_{d}
^{-1}(double struck G sign_{2}) for double struck G sign_{1}, double struck G sign_{2} at Ate pairing α. Let P ∈ double struck G sign _{1}, Q ∈ double struck G sign_{2}, P′ ∈ double struck G sign′_{1} and Q′ ∈ double struck G sign′_{2}, the authors have shown α(Q, P) = f _{t-1,Q},(P)(p^{k} - 1)/r = f_{t-1},Q′(P′) (p^{k} - 1)/^{r}. This paper shows that this new Ate pairing, namely cross twisted (Xt) Ate pairing, provides an quite efficient implementation.

Original language | English |
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Title of host publication | Proceedings - 3rd International Conference on Convergence and Hybrid Information Technology, ICCIT 2008 |

Pages | 430-439 |

Number of pages | 10 |

Volume | 2 |

DOIs | |

Publication status | Published - 2008 |

Event | 3rd International Conference on Convergence and Hybrid Information Technology, ICCIT 2008 - Busan, Korea, Republic of Duration: Nov 11 2008 → Nov 13 2008 |

### Other

Other | 3rd International Conference on Convergence and Hybrid Information Technology, ICCIT 2008 |
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Country | Korea, Republic of |

City | Busan |

Period | 11/11/08 → 11/13/08 |

### ASJC Scopus subject areas

- Information Systems
- Software

### Cite this

*Proceedings - 3rd International Conference on Convergence and Hybrid Information Technology, ICCIT 2008*(Vol. 2, pp. 430-439). [4682279] https://doi.org/10.1109/ICCIT.2008.172

**Efficient pairings on twisted elliptic curve.** / Nogami, Yasuyuki; Sakemi, Yumi; Akane, Masataka; Morikawa, Yoshitaka.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings - 3rd International Conference on Convergence and Hybrid Information Technology, ICCIT 2008.*vol. 2, 4682279, pp. 430-439, 3rd International Conference on Convergence and Hybrid Information Technology, ICCIT 2008, Busan, Korea, Republic of, 11/11/08. https://doi.org/10.1109/ICCIT.2008.172

}

TY - GEN

T1 - Efficient pairings on twisted elliptic curve

AU - Nogami, Yasuyuki

AU - Sakemi, Yumi

AU - Akane, Masataka

AU - Morikawa, Yoshitaka

PY - 2008

Y1 - 2008

N2 - This paper proposes an efficient implementation of Ate pairing on twisted elliptic curve. Suppose that a pairing-friendly elliptic curve E has a twisted elliptic curve E′ of degree d, and let Ψd be an isomorphic map from E′ (double struck F signp e) to the corresponding subgroup of E(double struck F signp k). Then, consider double struck G sign′1 = Ψd -1(double struck G sign1) and double struck G sign′2 = Ψd -1(double struck G sign2) for double struck G sign1, double struck G sign2 at Ate pairing α. Let P ∈ double struck G sign 1, Q ∈ double struck G sign2, P′ ∈ double struck G sign′1 and Q′ ∈ double struck G sign′2, the authors have shown α(Q, P) = f t-1,Q,(P)(pk - 1)/r = ft-1,Q′(P′) (pk - 1)/r. This paper shows that this new Ate pairing, namely cross twisted (Xt) Ate pairing, provides an quite efficient implementation.

AB - This paper proposes an efficient implementation of Ate pairing on twisted elliptic curve. Suppose that a pairing-friendly elliptic curve E has a twisted elliptic curve E′ of degree d, and let Ψd be an isomorphic map from E′ (double struck F signp e) to the corresponding subgroup of E(double struck F signp k). Then, consider double struck G sign′1 = Ψd -1(double struck G sign1) and double struck G sign′2 = Ψd -1(double struck G sign2) for double struck G sign1, double struck G sign2 at Ate pairing α. Let P ∈ double struck G sign 1, Q ∈ double struck G sign2, P′ ∈ double struck G sign′1 and Q′ ∈ double struck G sign′2, the authors have shown α(Q, P) = f t-1,Q,(P)(pk - 1)/r = ft-1,Q′(P′) (pk - 1)/r. This paper shows that this new Ate pairing, namely cross twisted (Xt) Ate pairing, provides an quite efficient implementation.

UR - http://www.scopus.com/inward/record.url?scp=57849134187&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57849134187&partnerID=8YFLogxK

U2 - 10.1109/ICCIT.2008.172

DO - 10.1109/ICCIT.2008.172

M3 - Conference contribution

AN - SCOPUS:57849134187

SN - 9780769534077

VL - 2

SP - 430

EP - 439

BT - Proceedings - 3rd International Conference on Convergence and Hybrid Information Technology, ICCIT 2008

ER -