We need to perform arithmetic in Fp(z)12 to use Ate pairing on a Barreto-Naehrig (BN) curve, where p(z) is a prime given by p(z) = 36z + 36z + 24z + 6z+ 1 with an integer z. In many implementations of Ate pairing, Fp(z)12 has been regarded as the 6-th extension of Fp(z)2 , and it has been constructed as Fp(z)12 = Fp(z)2 [v]/(v−ξ) for an element ξ ∈ Fp(z)2 such that v − ξ is irreducible in Fp(z)...

IN this paper we shall study the connection between local and global nilpotency for groups which satisfy certain chain conditions. In the first section, we shall prove that a group satisfying the chain condition on centralizers has a unique maximal normal nilpotent subgroup (the Fitting subgroup), i.e. the group generated by all normal nilpotent subgroups must again be nilpotent. This generaliz...