Deciding the Most Significant Bit of Binary Multiplication I am interested in determining the complexity of the following decision problem: Given two integers $l_1$ and $l_2$ (each with at most m bits), decide whether the most significant bit of the multiplication $l_1 \cdot l_2$ is 1 (where the result is printed in 2m bits with possibly leading 0's)? 

Some background on the problem: Obviously, this problem is a special case of binary multiplication that asks whether the $i$-th bit of the multiplication $l_1 \cdot l_2$ is 1.  In their paper, Uniform constant-depth threshold circuits for division and iterated multiplication, Hesse, Allender and Barrington prove that iterated (and thus binary) multiplication is in $\mathsf{DLogTime}$-uniform $\mathsf{TC}^0$. Moreover, it seems to be well-known that binary multiplication is already $\mathsf{DLogTime}$-uniform $\mathsf{TC}^0$-hard. However, I was not able to find a particular source proving this hardness result. As a non-expert in circuit complexity, I would also appreciate a pointer to this general hardness result. Finally, assuming that binary multiplication is $\mathsf{DLogTime}$-uniform $\mathsf{TC}^0$-hard, my question can also be read as: Does it remain $\mathsf{DLogTime}$-uniform $\mathsf{TC}^0$-hard if we want to decide only the most significant bit of binary multiplication?

UPDATE: Kaveh's answer clarifies why binary multiplication is $\mathsf{TC}^0$-hard (reduction from COUNT). The precise complexity of deciding the most significant bit of binary multiplication remains open (and the bounty is for this question).
Multiplication is complete for $\mathsf{TC}^0$ and 
this is a well know result. 
The reduction is from Count 
(number of 1 bits in a binary number). 
Comparison of binary numbers is in $\mathsf{AC^0}$ 
so $\mathsf{Majority}$ is reducible to $\mathsf{Count}$. 

To reduce $\mathsf{Count}$ to $\mathsf{Mult}$ do as follows: 
consider input is $a_0a_1\ldots a_n$. 
Insert $k$ 0s between $a_i$s and call it $a$. 
Multiply it with $b$ which is like $a$ except that 
$a_i$s in it are replaced with 1s. 
Pick $k&gt;3n$. 
The number in the middle section of $ab$ is the answer. 
The reduction is in $\mathsf{FO}$ and shows that 
$\mathsf{Count} \in \mathsf{FO(Mult)}$.