A Simple Arithmetic Circuit

Let \(\mathbb{F}\) be some finite field. We would like to show in this section the transformation from the program computing polynomial \(f(u, v) = u^2 + 3uv + v + 5 \) where \(u, v \in \mathbb{F}\). With inputs \(u, v \in \mathbb{F}\), the arithmetic circuit for this polynomial is equivalently represented by topologically following the sequence of computations below.

1. Compute \(u^2\).
2. Compute \(uv\).
3. Compute \(3uv\).
4. Compute \(u^2 + 3uv\).
5. Compute \(u^2 + 3uv + v\).
6. Compute \(u^2 + 3uv + v + 5\).