Randomized communication complexity of indexing The function $\mathrm{INDEX}:\{0,1\}^n\times\{1,\dots,n\}\to \{0,1\}$ is defined as

$$\mathrm{INDEX}(x,i)=x_i,$$
where $x=x_1\dots x_n$. I am looking for the randomized communication complexity of $\mathrm{INDEX}$ for an arbitrary number of rounds.

Somehow, I do not manage to find it in the literature.

The deterministic communication complexity is known to be $\Theta(\log n)$, while its one-way randomized communication complexity is $\Theta(n)$.

I guess the randomized communication complexity is still $\Theta(\log n)$, I was not able to find a lower bound.

Since this function its well-studied, it should be stated somewhere.

Thanks for your help.
It is known that the randomized communication complexity of inner product on $m$ bits is $\Omega(m)$. You can compute inner product using a protocol for the indexing function on $\{0,1\}^{2^m} \times [2^m]$ as follows: denoting Alice's input by $x \in \{0,1\}^m$ and Bob's by $y \in \{0,1\}^m$, Alice computes a new vector $X$ by $X_y = \mathsf{IP}(x,y)$, and the parties run the indexing protocol on the inputs $X$ and $y$. This gives an $\Omega(m) = \Omega(\log n)$ lower bound for indexing.