2. Arithmetic Circuits, R1CS and Circom

From Code to Arithmetic Constraints

Computations are converted into arithmetic circuits consisting of addition and multiplication gates over finite field elements. Every statement must be expressed in Rank-1 Constraint System (R1CS) form: $(A \cdot s) \times (B \cdot s) = (C \cdot s)$, where $s$ is the witness vector containing inputs and intermediate variables.

Writing Circom Circuits

Circom is a domain-specific language for defining arithmetic constraints. Circuits define input signals (public or private) and generate constraints that output a witness file used by the prover.

// Circom arithmetic constraint example
pragma circom 2.1.6;

template Multiplier() {
    signal input a;
    signal input b;
    signal output c;

    // R1CS constraint: a * b === c
    c <== a * b;
}

component main = Multiplier();