### Abstract

It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log_{2} r/φ(k), where φ(·) is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from ⌊log_{2} r⌋ to ⌊log_{2}(t-1)⌋, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "χ." For such a curve, this paper gives integer variable χ-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve, it reduces the number of loops to ⌊log _{2}χ⌋. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.

Original language | English |
---|---|

Pages (from-to) | 1859-1867 |

Number of pages | 9 |

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

Volume | E92-A |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 2009 |

### Fingerprint

### Keywords

- Ate pairing
- Miller's algorithm

### 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*(8), 1859-1867. https://doi.org/10.1587/transfun.E92.A.1859

**Integer variable χ-based cross twisted Ate pairing and its optimization for Barreto-Naehrig curve.** / Nogami, Yasuyuki; Sakemi, Yumi; Kato, Hidehiro; Akane, Masataka; Morikawa, Yoshitaka.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E92-A, no. 8, pp. 1859-1867. https://doi.org/10.1587/transfun.E92.A.1859

}

TY - JOUR

T1 - Integer variable χ-based cross twisted Ate pairing and its optimization for Barreto-Naehrig curve

AU - Nogami, Yasuyuki

AU - Sakemi, Yumi

AU - Kato, Hidehiro

AU - Akane, Masataka

AU - Morikawa, Yoshitaka

PY - 2009/8

Y1 - 2009/8

N2 - It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log2 r/φ(k), where φ(·) is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from ⌊log2 r⌋ to ⌊log2(t-1)⌋, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "χ." For such a curve, this paper gives integer variable χ-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve, it reduces the number of loops to ⌊log 2χ⌋. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.

AB - It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log2 r/φ(k), where φ(·) is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from ⌊log2 r⌋ to ⌊log2(t-1)⌋, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "χ." For such a curve, this paper gives integer variable χ-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve, it reduces the number of loops to ⌊log 2χ⌋. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.

KW - Ate pairing

KW - Miller's algorithm

UR - http://www.scopus.com/inward/record.url?scp=84881008209&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84881008209&partnerID=8YFLogxK

U2 - 10.1587/transfun.E92.A.1859

DO - 10.1587/transfun.E92.A.1859

M3 - Article

AN - SCOPUS:84881008209

VL - E92-A

SP - 1859

EP - 1867

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 - 8

ER -