On multiple base discrete logarithm problem in $\mathbb {G}_m^{n}$ and $E_d^{n}$

In this paper, we consider local to global principle in $n$th power of $\mathbb {G}_m$ and $n$th power of CM elliptic curve $E_d: y^2=x^3-d^2x$. In both cases, for $n \gt 1$, we give examples where local to global multiple base discrete logarithm problem fails.