Programming ECC - Compressed Pairings

]> Programming ECC - Compressed Pairings

Compressed Pairings

Suppose $𝔽_{q} [α]$ is a quadratic extension of $F_{q}$ where $α^{2} = δ$ for some $δ \in 𝔽_{q}$ . Then let $x = a + α b$ be some $r$ th root of unity in $𝔽_{q}$ for some $r$ dividing $q + 1$ . (Note $q + 1 = Φ_{2} (q)$ .)

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

1 = a^{q + 1} + α^{q + 1} b^{q + 1} = a^{2} + (α^{q}) α b^{2} = a^{2} - δ b^{2}

since $α^{q} = - α$ .