Universal Construction of a 12th Degree Extension Field for Asymmetric Pairing
Summary :
It is necessary to perform arithmetic in Fp12 to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z)=36z4+36z3+24z2+6z+1 for some integer z. In many implementations of Ate pairings, Fp12 has been regarded as a 6th degree extension of Fp2, and it has been constructed by Fp12=Fp2[v]/(v6-ξ) for an element ξ ∈ Fp2 such that v6-ξ is irreducible in Fp2[v]. Such a ξ depends on the value of p, and we may use a mathematical software package to find ξ. In this paper it is shown that when z ≡ 7,11 (mod 12), we can universally construct Fp12 as Fp12=Fp2[v]/(v6-u-1), where Fp2=Fp[u]/(u2+1).
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E94-A No.1 pp.156-164
- Publication Date
- 2011/01/01
- Publicized
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.E94.A.156
- Type of Manuscript
- Special Section PAPER (Special Section on Cryptography and Information Security)
- Category
- Mathematics
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Masaaki SHIRASE, "Universal Construction of a 12th Degree Extension Field for Asymmetric Pairing" in IEICE TRANSACTIONS on Fundamentals,
vol. E94-A, no. 1, pp. 156-164, January 2011, doi: 10.1587/transfun.E94.A.156.
Abstract: It is necessary to perform arithmetic in Fp12 to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z)=36z4+36z3+24z2+6z+1 for some integer z. In many implementations of Ate pairings, Fp12 has been regarded as a 6th degree extension of Fp2, and it has been constructed by Fp12=Fp2[v]/(v6-ξ) for an element ξ ∈ Fp2 such that v6-ξ is irreducible in Fp2[v]. Such a ξ depends on the value of p, and we may use a mathematical software package to find ξ. In this paper it is shown that when z ≡ 7,11 (mod 12), we can universally construct Fp12 as Fp12=Fp2[v]/(v6-u-1), where Fp2=Fp[u]/(u2+1).
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E94.A.156/_p
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@ARTICLE{e94-a_1_156,
author={Masaaki SHIRASE, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Universal Construction of a 12th Degree Extension Field for Asymmetric Pairing},
year={2011},
volume={E94-A},
number={1},
pages={156-164},
abstract={It is necessary to perform arithmetic in Fp12 to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z)=36z4+36z3+24z2+6z+1 for some integer z. In many implementations of Ate pairings, Fp12 has been regarded as a 6th degree extension of Fp2, and it has been constructed by Fp12=Fp2[v]/(v6-ξ) for an element ξ ∈ Fp2 such that v6-ξ is irreducible in Fp2[v]. Such a ξ depends on the value of p, and we may use a mathematical software package to find ξ. In this paper it is shown that when z ≡ 7,11 (mod 12), we can universally construct Fp12 as Fp12=Fp2[v]/(v6-u-1), where Fp2=Fp[u]/(u2+1).},
keywords={},
doi={10.1587/transfun.E94.A.156},
ISSN={1745-1337},
month={January},}
Copy
TY - JOUR
TI - Universal Construction of a 12th Degree Extension Field for Asymmetric Pairing
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 156
EP - 164
AU - Masaaki SHIRASE
PY - 2011
DO - 10.1587/transfun.E94.A.156
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E94-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2011
AB - It is necessary to perform arithmetic in Fp12 to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z)=36z4+36z3+24z2+6z+1 for some integer z. In many implementations of Ate pairings, Fp12 has been regarded as a 6th degree extension of Fp2, and it has been constructed by Fp12=Fp2[v]/(v6-ξ) for an element ξ ∈ Fp2 such that v6-ξ is irreducible in Fp2[v]. Such a ξ depends on the value of p, and we may use a mathematical software package to find ξ. In this paper it is shown that when z ≡ 7,11 (mod 12), we can universally construct Fp12 as Fp12=Fp2[v]/(v6-u-1), where Fp2=Fp[u]/(u2+1).
ER -
English
