Arithmetic circuits over $GF(p)$ for computing $x \bmod 2$ given input $x \in GF(p)$ Let $p$ be a prime.  Suppose we consider the function $f: GF\left(p \right) \rightarrow \left\{0, 1 \right\}$ where $f(x) = x \bmod 2$ for all.

The question is the following:  are there any known circuit bounds or results on computing $f$ using arithmetic circuits over $GF(p)$?  

In other words, we'd like to evaluate $f$ using only field additions and multiplications.  

It's simple to do this with arithmetic circuits of size $O \left(p \right)$ and also probably straightforward to show that it cannot be done with arithmetic circuits of size $o\left( \log p \right)$.  However, it has been difficult for us to say anything concrete other than these simple bounds.  Any literature or references on this problem (or similar types of problem) would be greatly appreciated!  Thanks!
It's not clear if this is directly useful, but lower bounds are known for depth-3 circuits computing an $n$-variate version of your function over $\mathrm{GF}(p)$ for $p \neq 2$. See Theorem 3.7 in the survey of Shpilka and Yehudayoff for details and references.

I don't believe lower bounds of the form $\omega(\log d)$ are known for general circuits computing an explicit univariate polynomial of degree $d$, so proving an $\omega(\log p)$ lower bound for your problem will likely be difficult.