http://dcherukhin.i2p/research/research.htm
For every fixed depth $d$, we prove a lower bound $Omega(n \lambda_{d-1}(n))$ for the size (i.e. the number of
wires)
of any circuit for computing the cyclic convolution over the field $K$.
In particular, for $d=2,3,4$ , our bounds are $\Omega(n^{1.5})$, $\Omega(n \log n)$ and $Omega(n \log \log n)$ respectively;
for $d \ge 5$, the function $\lambda_{d-1}(n)$ is slowly
growing.