Discrete logs on elliptic curve with embedding degree 3 with the 'MOV' attack
Let's take your latter example. We will use the Weil pairing here, since that was the original MOV approach. Let's pick some arbitrary points in your curve:

$$
\begin{eqnarray}
P &amp;=&amp; (6116 : 2715) \\
Q &amp;=&amp; (3034 : 462)
\end{eqnarray}
$$

From now on, we'll actually work in an extension field of $\mathbb{F}_{8111}$, namely $\mathbb{F}_{8111^3}$. For concreteness, we choose the polynomial $x^3 + 4x - 11$ to define the extension field.

Now we need to find some arbitrary point that has order $8269$ but is not a multiple of $P$. We know that the group structure of the points of order $n$ in a pairing-friendly curve is $E(F_q)[n] = \mathbb{Z}_n \oplus \mathbb{Z}_n$, so there are definitely such points out there. An example point would be 

$$
R = (3515x^2 + 7098x + 2717 : 3745x^2 + 7258x + 4072).
$$

Now all it's left is to apply the Weil pairing to both points:

$$
\begin{eqnarray}
w_1 &amp;=&amp; e(P, R) \in \mathbb{F}_{8111^3} \\
w_2 &amp;=&amp; e(Q, R) = e(uP, R) = e(P, R)^u \in \mathbb{F}_{8111^3}
\end{eqnarray}
$$

Now we can solve the log of $w_1 = 7474x^2 + 74x + 6953$ and $w_2 = 2625x^2 + 6673x + 243$ over $\mathbb{F}_{8111^3}$. It's $2634$, and it's easy to see that $2634\cdot P = Q$. For large fields, it is considerably faster to find $\log_{w1} w2$ via the number or function field sieve ($L[1/3; c]$) than the elliptic curve discrete log $\log_P Q$ via Pollard's rho ($O(n^{1/2}))$.