Consider two points P, Q over a pairing friendly elliptic curve $E[F_q]$, e.g., BN254. Let Z = e(P, Q). It is known that $Z \in F_{q^k}$ where $k$ is the embedding degree. The norm map N(Z) is defined as $\prod_{0\leq i\leq k-1} Z^{q^i}$. We observed that for BN254, N(Z) is always the 1 in $F_p$.

Is that the case for all pairing friendly groups?



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