tripartite diffie hellman with Weil pairing I try to understand how the tripartite Diffie-Hellman key exchange works.
I read Joux's paper for this: https://www.semanticscholar.org/paper/A-One-Round-Protocol-for-Tripartite-Diffie-Hellman-Joux/845e96c20e5a5ff3b03f4caf72c3cb817a7fa542/pdf

However on page 388 he states the following equality which I don't see why it is true:
$F_W(a,P_b,Q_c) = F_W(a,Q_b, P_c)$

where $F_W(x,y,z)= e_m(y,z)^x$.
For me the first value of the equality is the inverse of the second one?
This does indeed appear to be a typo. Using that $e_m$ is bilinear and alternating, one calculates
$$\begin{align*}
F_W(a,P_b,Q_c)
\;&amp;=\; e_m([b]P,[c]Q)^a
\\&amp;=\; e_m(P,Q)^{abc}
\\&amp;=\; e_m(Q,P)^{-abc}
\\&amp;=\; e_m([b]Q,[c]P)^{-a}
\\&amp;=\; F_W(a,Q_b,P_c)^{-1}
\text.
\end{align*}$$

(However, note that these are only different if the order of $e_m(P,Q))$ does not divide $abc$; in particular, we need $e_m(P,Q)\neq1$.)