**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 $p(z)=36z^4+36z^3+24z^2+6z+1$ 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 $\Fpt=\Fpp[v]/(v^6-\xi)$ for an element $\xi\in \Fpp$ such that $v^6-\xi$ is irreducible in $\Fpp[v]$. Such $\xi$ depends on the value of $p(z)$, and we may use mathematic software to find $\xi$. This paper shows that when $z \equiv 7,11 \pmod{12}$ we can universally construct $\Fpp$ as $\Fpt=\Fpp[v]/(v^6-u-1)$, where $\Fpp=\Fp[u]/(u^2+1)$.

**Category / Keywords: **public-key cryptography / pairing, Barreto-Naehrig curve, extension field, quadratic residue, cubic residue, Euler's conjecture

**Date: **received 17 Dec 2009, last revised 18 Feb 2010

**Contact author: **shirase at fun ac jp

**Available format(s): **PDF | BibTeX Citation

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

**Version: **20100219:040115 (All versions of this report)

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