# Circuit Specification

Let $$\mathbb{F}$$ be a finite field. In this section, we describe the arithmetic circuit whose operations are over $$\mathbb{F}$$.

Let $$\ell_{\mathsf{in}} \in \mathbb{N}$$ be the number of input wires of the circuit $$\mathcal{C}$$. Assume that $$\mathcal{C}$$ has exactly $$n$$ gates. Each gate takes at most $$2$$ wires as inputs and returns $$1$$ output wires. In particular,

• Addition and multiplications gates takes $$2$$ inputs and return $$1$$ output.
• Gates of additions and multiplications with constants take $$1$$ input and return $$1$$ output.

Let's take a look at the following example.

Assume that $$f(u, v) = u^2 + 3uv + v + 5$$. Then the sequence are arranged in the following constraints, wrapped as below. $$\begin{cases} z_1 = u \cdot u &\text{(multiplication)},\\ z_2 = u \cdot v &\text{(multiplication)},\\ z_3 = z_2 \cdot 3 &\text{(multiplication with constant)},\\ z_4 = z_1 + z_3 &\text{(addition)},\\ z_5 = z_4 + v &\text{(addition)},\\ z_6 = z_5 + 5 &\text{(addition with constant)}. \end{cases}$$ The input size is $$\ell_{\mathsf{in}} = 2$$ for variables $$u, v$$ and the output is $$z_6$$.