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Suppose $P\in E(\mathbb{F}_q)$ and $R=dP$.

In the MOV attack, we compute $\alpha=e(P,T)$ and $\beta=e(R,T)$ and try to solve the discrete logarithm problem for $\alpha$ and $\beta$ in the finite field $\mathbb{F}_{q^k}$ where $k$ is the embedding degree.

But if $E(\mathbb{F}_q)$ is cyclic (which it can be even in the supersingular case), $\alpha,\beta$ are doomed to be $1$.

So what am I missing here?

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    $\begingroup$ Why are they doomed to be 1? You would have $\alpha=\beta=1_{T}$ iff. $P=R=G_{1}$ and $T=G_2$, where $G_1$ and $G_2$ are the generators of the first group and second group respectively, which I suppose, is not the case. In the target group you have that $\beta=d\alpha$. Thus we have reduced the discrete logarithm problem on the group of points on an elliptic curve to the discrete logarithm on finite fields, where subexponential attacks are known. $\endgroup$ May 9 '20 at 7:54
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For the purposes of the MOV attack you choose $T$ to be a point in $E(\mathbb{F}_{q^k})$.

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