Universally Constructing 12-th Degree Extension Field for Ate Pairing

Masaaki Shirase

Paper 2009/623

Universally Constructing 12-th Degree Extension Field for Ate Pairing

Masaaki Shirase

Abstract

We need to perform arithmetic in $\Fpt$ to use Ate pairing on a Barreto-Naehrig (BN) curve, where $p (z)$ is a prime given by $p (z) = 36 z^{4} + 36 z^{3} + 24 z^{2} + 6 z + 1$ with an integer $z$ . In many implementations of Ate pairing, $\Fpt$ has been regarded as the 6-th extension of $\Fpp$ , and it has been constructed as $\Fpt = \Fpp [v] / (v^{6} - ξ)$ for an element $ξ \in \Fpp$ such that $v^{6} - ξ$ is irreducible in $\Fpp [v]$ . Such $ξ$ depends on the value of $p (z)$ , and we may use mathematic software to find $ξ$ . This paper shows that when $z \equiv 7, 11 (\mod 12)$ we can universally construct $\Fpp$ as $\Fpt = \Fpp [v] / (v^{6} - u - 1)$ , where $\Fpp = \Fp [u] / (u^{2} + 1)$ .

Note: I found some typos on my eprint report. Then I corrected them.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Published elsewhere. Unknown where it was published
Keywords: pairing Barreto-Naehrig curve extension field quadratic residue cubic residue Euler's conjecture
Contact author(s): shirase @ fun ac jp
History: 2010-02-19: last of 3 revisions; 2009-12-26: received; See all versions
Short URL: https://ia.cr/2009/623
License: CC BY

BibTeX

@misc{cryptoeprint:2009/623,
      author = {Masaaki Shirase},
      title = {Universally Constructing 12-th Degree Extension Field for Ate Pairing},
      howpublished = {Cryptology {ePrint} Archive, Paper 2009/623},
      year = {2009},
      url = {https://eprint.iacr.org/2009/623}
}