## Abstract

This paper presents implementation techniques of fast Ate pairing of embedding degree 12. In this case, we have no trouble in finding a prime order pairing friendly curve E such as the Barreto-Naehrig curve y^{2} = x^{3} + a, a ∈ F_{p}. For the curve, an isomorphic substitution from G_{2} C E(F_{p}^{12}) into G _{2} in subfield-twisted elliptic curve E(Fp2 ) speeds up scalar multiplications over G_{2} and wipes out denominator calculations in Miller's algorithm. This paper mainly provides about 30% improvement of the Miller's algorithm calculation using proper subfield arithmetic operations. Moreover, we also provide the efficient parameter settings of the BN curves. When p is a 254-bit prime, the embedding degree is 12, and the processor is Pentium4 (3.6 GHz), it is shown that the proposed algorithm computes Ate pairing in 13.3 milli-seconds including final exponentiation.

Original language | English |
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Pages (from-to) | 508-516 |

Number of pages | 9 |

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

Volume | E92-A |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2009 |

## Keywords

- Ate pairing
- Fast computing
- Subfield arithmetic operation
- Twist

## ASJC Scopus subject areas

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