Rank 1 Constrain Systems

Now, after compiling your circuit, you obtain a file.r1cs file. This indicates that the part where matrix operations come into play has begun.

This is the definition of R1CS. Az * Bz - Cz = 0 or Az * Bz = Cz

where A, B, and C are matrices representing the circuit.

and w is how to compute witness in term of

• (1 || x || w)

and then perform an element-wise product (multiplying 'w' with A, B, and C index by index)

Here are some code examples of how to convert arithmetic circuits to R1CS.

out = x * y + 2

Optimize to prevent the R1CS file from becoming larger than necessary.

out - 2 = x * y

import numpy as np
import random

# Define the matrices
A = np.array([[0,0,1,0]])
B = np.array([[0,0,0,1]])
C = np.array([[-2,1,0,0]])

# pick random values to test the equation
x = random.randint(1,1000)
y = random.randint(1,1000)
out = x * y + 2# witness vector
w = np.array([1, out, x, y])

# check the equality
result = C.dot(w) == np.multiply(A.dot(w),B.dot(w))
assert result.all(), "result contains an inequality"

The size of the matrix depends on w.

However, this is not yet complete, as it lacks security. Please proceed to the next part to learn how to enhance its security.