Compressed Pairings

Suppose $\mathbb{F}_q[\alpha]$ is a quadratic extension of $F_q$ where $\alpha^2 = \delta$ for some $\delta \in \mathbb{F}_q$. Then let $x = a + \alpha b$ be some $r$th root of unity in $\mathbb{F}_q$ for some $r$ dividing $q+1$. (Note $q+1 = \Phi_2(q)$.)

Then $x^{q+1} = 1$, thus

\[ 1 = a^{q+1} + \alpha^{q+1} b^{q+1} = a^2 + (\alpha^q) \alpha b^2 = a^2 - \delta b^2 \]

since $\alpha^q = -\alpha$.