# Is discrete log as hard when given same challenge in both pairing groups, in the case of BN curve and its sextic twist

This is a special case of the question Generalization of the DL-assumption in bilinear group pair, that wasn't answered.

Suppose $G_1$ is a BN curve over $F_q$. That is the set of elements $(x,y)\in F_q^2$ such that $$y^2 =x^3+b$$ for appropriately chosen $b$. and $G_2$ is a sextic twist of $G_1$. That is, the set of elements $(x,y)\in F_{q^2}^2$ such that $y^2=x^3+b/\xi,$ where $\xi\in F_{q^2}$ is such that $W^6-\xi$ is irreducible over $F_{q^2}$.

Given $(P,s\cdot P)\in G_1^2$ and $(Q,s\cdot Q)\in G_2^2$ is it hard to find $s$? Is it as hard as when given only one of the two pairs?