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Determining if a constraint is „necessary” in zero-knowledge proof circuits
As a beginner in cryptography and Circom, it’s not uncommon to come across complex concepts like completeness. In zero-knowledge proof (ZKP) circuits, one of the most important aspects is determining if a particular constraint or requirement is necessary for the circuit to be correct. But what exactly does that mean? Let’s dive into the world of Circom and explore how to determine if a constraint is „necessary” in an IsZero() circuit.
What is completeness in ZKP circuits?
Completeness in zero-knowledge proof circuits refers to the property that, given a valid input (i.e. one that satisfies all constraints), the output is deterministically computable. In other words, if you have access to any input and can check its validity against the constraints enforced by the circuit, you should always get a consistent and deterministic output.
The IsZero() circuit: A simple example
Let’s look at Circomlib’s IsZero() circuit:
template IsZero() {
// Define inputs and outputs
Input x;
Output y;
// Define constraints
Constraint 1: {y = x > 0;};
Constraint 2: {y = y - 1;};
// Use constraints to generate the output
if (x > 0 && y >= 2) {
y = x * y;
} else {
return NoCommit;
}
}
In this example, we have two constraints:
Constraint 1: If $y$ is greater than 0, then $y – 1$ must be equal to a number.
Constraint 2: If $y$ is not greater than 0 (i.e. less than or equal to 0), then $y$ must be equal to 1.
Determining if a constraint is „necessary”
To determine if a constraint is necessary, we need to consider:
- Completeness: Can we get a consistent and deterministic output by checking arbitrary inputs? (We already checked this in
IsZero(), which returnsyfor valid inputs.)
- Correctness: Does the circuit produce the correct output given all constraints?
For Constraint 1, we can ensure that $y – 1 = x(y-1)$ if we check that $x > 0$ and $y \geq 2$. However, this constraint is not necessary because there are many possible values of $y$ that satisfy the output.
In particular, consider an input where $x = 0$. Then $y = 0(0-1) \equiv 1 \pmod{2}$. The circuit would still produce a valid output y, but it is not deterministically computable because there are infinitely many possible values of $y$ that satisfy the constraint.
Conclusion
In summary, to determine whether a constraint is „necessary” in an IsZero() circuit, both completeness and correctness must be considered. If a constraint does not guarantee deterministic output for all valid inputs, it may be necessary to include additional constraints or modify existing ones to achieve this goal.
By analyzing the IsZero() circuit and applying these principles, you can gain insights into when constraints are necessary in zero-knowledge proof circuits like IsZero(). Remember that completeness and correctness are key properties of a ZKP circuit, so check your inputs thoroughly before relying on them. Happy coding!
