Design

Notation

We work on a cycle of curves composed by $E_1$ and $E_2$, where $E_1.F_r=E_2.F_q$ and $E_1.F_q=E_2.F_r$. We will use $F_r$ to refer to $E_1.F_r=E_2.F_q$, and $F_q$ to refer to $E_1.F_q=E_2.F_r$. The main circuit constraint field is $F_r$, and $C_{EC}$ circuit constraint field is $F_q$ (in some cases we use also $C^{\ast }$, both notations come from papers).