Power of Halo 2 in Zero Knowledge Proofs: A Friendly Exploration

Table of Contents

In the landscape of cryptographic protocols, Halo 2 emerges as a pivotal entity, not merely confined to its definition but embodying a transformative force in the realm of secure application development. Defined by its nuanced characteristics and driven by a profound purpose, Halo 2 assumes a critical role as a bastion of security and verifiability within the digital milieu. In this article, we'll delve into the depths of Halo 2, understanding its definition and unraveling its crucial role in application development.

I. What is Halo 2?

Definition of Halo 2

Halo 2 stands as a formidable presence in the realm of cryptographic tools, specifically as a succinct non-interactive zero-knowledge argument of knowledge (zkSNARK) library. At its core, zkSNARKs are a class of cryptographic proof systems that allow one party, known as the prover, to convince another party, the verifier, that they possess certain information without revealing what that information is. In the case of Halo 2, it specializes in zero-knowledge proofs, providing a way for developers to assure the integrity and honesty of their computing programs.

Digging deeper, the term "zero-knowledge" is key here. In cryptographic terms, it means that the prover can convince the verifier of the truth of a statement without revealing any additional information other than the fact that the statement is true. This is a powerful concept, especially in scenarios where privacy and security are paramount.

Purpose of Halo 2 in Application Development

Now, let's explore why Halo 2 holds a significant place in the toolkit of application developers. In the landscape of blockchain and decentralized applications, trust is a precious commodity. Developers need robust tools to ensure the integrity of their applications, especially when dealing with sensitive data and financial transactions.

Halo 2 steps into this arena as a guardian of integrity. By providing a means to create succinct, non-interactive proofs, it empowers developers to demonstrate the correctness of their computations without revealing the details. This has profound implications, particularly in blockchain applications, where smart contracts and transactions often involve confidential information.

Imagine a scenario where a decentralized application needs to prove that it has executed a set of operations correctly without disclosing the specifics of those operations. Halo 2 allows developers to construct proofs that vouch for the accuracy of computations, assuring users and other parties of the application's reliability without compromising privacy.

In essence, Halo 2 is a trust enabler, a cryptographic tool that brings a new level of assurance to the world of application development. As we journey deeper into its intricacies, we'll uncover the mechanics that make it such a potent force in the realm of zero-knowledge proofs.

II. PLONKish Arithmetization

Introduction to PLONK's Arithmetization

To truly grasp the significance of Halo 2, it's crucial to acquaint ourselves with the ingenious underpinnings of PLONK's arithmetization. Arithmetization is a technique used in cryptographic protocols to convert computations into algebraic expressions, facilitating the application of mathematical principles for verification.

In the case of PLONK, which stands for "Permutations over Lagrange-bases for Oecumenical Noninteractive arguments of Knowledge," arithmetization takes center stage. PLONK serves as a versatile proof system, and its arithmetization methodology forms the basis for Halo 2.

Customization with PLONKish Arithmetization

Now, enter PLONKish Arithmetization, a tailored version that extends PLONK's capabilities. This customization is not about reinventing the wheel but rather enhancing it to handle more intricate scenarios. Imagine dealing with computations involving not just simple multiplications and additions but more complex gates with nuanced structures. PLONKish Arithmetization steps up to the challenge, offering a more refined and adaptable approach.

In essence, customization here means expanding the scope of what PLONK can handle. It's about empowering developers to work with more sophisticated structures in their computations, beyond the traditional arithmetic gates. This adaptability is crucial in real-world applications where computational complexity varies widely.

Components of PLONKish Arithmetization

To better understand the inner workings of PLONKish Arithmetization, let's break down its components. Picture a table with different types of columns:

- Constant Columns: Reserved for constants used in computations.

- Instant Columns: Hold public inputs, information openly available to all parties.

- Advice Columns: Conceal private inputs, known as witnesses, from verifiers.

- Selector Columns: Indicate the type of operation or gate for each row in the table.

These columns play a vital role in the arithmetization process, providing a structured framework for handling diverse computations.

III. A Simple Arithmetic Circuit

Introduction to Finite Fields

Before delving into the intricacies of a simple arithmetic circuit, let's establish the foundational concept of finite fields. In the realm of mathematics and cryptography, a finite field, often denoted as 𝔽, represents a set of elements with a finite number. These fields obey specific rules, and operations like addition, subtraction, multiplication, and division are performed within this finite set.

Finite fields provide the mathematical playground for cryptographic protocols, ensuring computations remain within well-defined boundaries. Understanding these fields is a crucial prerequisite for comprehending the subsequent layers of complexity in a simple arithmetic circuit.

Polynomial Representation: f(u, v) = u^2 + 3uv + v + 5

Now, let's turn our attention to the star of the show: the 2-variable polynomial. In this context, the polynomial takes the form f(u, v) = u^2 + 3uv + v + 5. Each term in the polynomial, from u^2 to the constant 5, plays a role in the overall computation. This polynomial serves as our guidepost, illustrating how computations can be expressed algebraically.

This choice of polynomial isn't arbitrary; it's a carefully crafted example that encapsulates various types of operations, setting the stage for a comprehensive exploration of the arithmetic circuit.

Arithmetic Circuit for Polynomial Computation

Now, let's translate our polynomial into the language of an arithmetic circuit. Picture a step-by-step sequence of computations:

1. Compute u^2.

2. Compute uv.

3. Compute 3uv.

4. Compute u^2 + 3uv.

5. Compute u^2 + 3uv + v.

6. Compute u^2 + 3uv + v + 5.

Each step builds on the previous one, culminating in the final result of our polynomial computation. This sequence of operations, represented as an arithmetic circuit, becomes the tangible embodiment of our abstract polynomial.

As we traverse through this arithmetic circuit, the significance of each step becomes apparent. It's not just a series of mathematical operations; it's a structured journey, with each computation contributing to the larger narrative of proving the knowledge of our polynomial in the language of finite fields.

With this understanding, we're now equipped to explore the transformation of this arithmetic circuit into the realm of PLONKish Arithmetization, a crucial step in the journey towards implementing Halo 2 in real-world applications.

IV. Transforming to PLONKish Arithmetization

Setting up the Table for PLONKish Arithmetization

Now that we've traversed the landscape of finite fields and understood the workings of a simple arithmetic circuit, let's embark on the transformation journey into PLONKish Arithmetization. Imagine a table, a structured representation of the computations we've outlined, with various columns serving distinct purposes.

Specifying Columns for Variables

In the world of PLONKish Arithmetization, columns play a crucial role. We define a tuple of columns, each designated for specific types of information:

- Advice Columns (𝖺𝖽𝗏𝗂𝖼𝖾0, 𝖺𝖽𝗏𝗂𝖼𝖾1, 𝖺𝖽𝗏𝗂𝖼𝖾2): Reserved for private inputs, keeping values hidden from verifiers.

- Instance Column (𝖼𝗈𝗇𝗌𝗍𝖺𝗇𝗍): Contains public constants relevant to the computation.

- Selectors (𝗌𝖾𝗅𝖾𝖼𝗍𝗈𝗋𝖺𝖽𝖽, 𝗌𝖾𝗅𝖾𝖼𝗍𝗈𝗋𝗆𝗎𝗅, 𝗌𝖾𝗅𝖾𝖼𝗍𝗈𝗋𝖺𝖽𝖽𝖼, 𝗌𝖾𝗅𝖾𝖼𝗍𝗈𝗋𝗆𝗎𝗅𝖼): Public selectors corresponding to various gates, such as addition, multiplication, and more.

This structured arrangement allows us to categorize and organize the variables involved in our computation. Think of it as setting the stage for a theatrical performance, where each character (variable) has a specific role to play.

Gate Constraints and Wire Constraints

Now, let's delve into the intricacies of gate constraints. These constraints encapsulate the essence of each computation within our arithmetic circuit. Each gate constraint corresponds to an operation within the circuit, ensuring that the computations align with the rules defined by our finite fields.

Imagine a transformation where equations in the arithmetic circuit become gate constraints in PLONKish Arithmetization:

- x(1)c * x(2)c = u^2: A gate constraint for the computation of u^2.

- x(3)c * x(4)c = 3uv: A gate constraint for the computation of 3uv.

- x(5)c * x(6)c = u^2 + 3uv + v: A gate constraint for the larger computation involving addition.

- ...and so on.

These gate constraints represent the core of PLONKish Arithmetization, ensuring that each computation aligns with the specified rules and constraints.

Example Transformation: Gate and Wire Constraints

Let's take a specific example to solidify our understanding. Assume a row corresponds to a multiplication gate. In this case, we set selectors (s(i)𝖺𝖽𝖽, s(i)𝗆𝗎𝗅, s(i)𝖺𝖽𝖽𝖼, s(i)𝗆𝗎𝗅𝖼) to (0, 1, 0, 0). This implies that only the selector for multiplication with a constant (s(i)𝗆𝗎𝗅) is set to 1, while the others are set to 0.

This nuanced setting ensures that the gate s(i)𝗆𝗎𝗅 * (x(i)a * x(i)b - x(i)c) = 0 must guarantee the condition x(i)a * x(i)b = x(i)c. This meticulous dance of selectors and constraints is what transforms our abstract polynomial computations into a language that PLONKish Arithmetization understands and verifies.

In essence, this section is the bridge between theory and implementation. We've set the table, organized our variables, and established the rules that govern their interactions. Now, armed with this understanding, we're ready to witness the actual implementation of a simple Halo 2 program in Rust. The journey from theory to practice is where the power of cryptographic tools truly comes to life.

V. A Simple Halo 2 Program in Rust

Introduction to Halo 2 Program Implementation

With a solid foundation in finite fields, a grasp of a simple arithmetic circuit, and a transformation into the realm of PLONKish Arithmetization, we now venture into the practical realm of implementing a Halo 2 program in Rust. The beauty of this implementation lies in its real-world applicability, offering a tangible glimpse into the power of cryptographic tools.

Setting Up Dependencies in Cargo.toml

Before we dive into the code, the first step is setting up the dependencies in Rust's package manager, Cargo. This is where we declare our reliance on external libraries, ensuring a smooth integration of the Halo 2 proofs library into our project. The Cargo.toml file becomes the orchestrator of these dependencies:

```toml
[dependencies]
halo2_proofs = { git = "https://github.com/privacy-scaling-explorations/halo2.git" }
```

This declaration tells Cargo to fetch the Halo 2 proofs library from the specified GitHub repository. It's akin to inviting a guest into our project, and in this case, the guest is the powerful Halo 2 library.

Step-by-Step Implementation in main.rs

Now, let's step into the main.rs file, the heart of our Rust implementation. This is where theory transforms into practice, where the abstract concepts we've explored come to life in the form of code.

```rust
#[derive(Clone, Debug)]
struct MyConfig {
    advice: [Column<Advice>; 3],
    instance: Column<Instance>,
    constant: Column<Fixed>,

    // selectors
    s_add: Selector,
    s_mul: Selector,
    s_add_c: Selector,
    s_mul_c: Selector,
}

// Define gates based on elements of the above defined struct.
impl<Field: FieldExt> FChip<Field> {
    fn configure(
        meta: &mut ConstraintSystem<Field>,
        advice: [Column<Advice>; 3],
        instance: Column<Instance>,
        constant: Column<Fixed>,
    ) -> <Self as Chip<Field>>::Config {
        // specify columns used for proving copy constraints
        meta.enable_equality(instance);
        meta.enable_constant(constant);
        for column in &advice {
            meta.enable_equality(*column);
        }

        // extract columns with respect to selectors
        let s_add = meta.selector();
        let s_mul = meta.selector();
        let s_add_c = meta.selector();
        let s_mul_c = meta.selector();

        // define addition gate
        meta.create_gate("add", |meta| {
            let s_add = meta.query_selector(s_add);
            let lhs = meta.query_advice(advice[0], Rotation::cur());
            let rhs = meta.query_advice(advice[1], Rotation::cur());
            let out = meta.query_advice(advice[2], Rotation::cur());
            vec![s_add * (lhs + rhs - out)]
        });

        // define multiplication gate
        meta.create_gate("mul", |meta| {
            let s_mul = meta.query_selector(s_mul);
            let lhs = meta.query_advice(advice[0], Rotation::cur());
            let rhs = meta.query_advice(advice[1], Rotation::cur());
            let out = meta.query_advice(advice[2], Rotation::cur());
            vec![s_mul * (lhs * rhs - out)]
        });

        // define addition with constant gate
        meta.create_gate("add with constant", |meta| {
            let s_add_c = meta.query_selector(s_add_c);
            let lhs = meta.query_advice(advice[0], Rotation::cur());
            let fixed = meta.query_fixed(constant, Rotation::cur());
            let out = meta.query_advice(advice[2], Rotation::cur());
            vec![s_add_c * (lhs + fixed - out)]
        });

        // define multiplication with constant gate
        meta.create_gate("mul with constant", |meta| {
            let s_mul_c = meta.query_selector(s_mul_c);
            let lhs = meta.query_advice(advice[0], Rotation::cur());
            let fixed = meta.query_fixed(constant, Rotation::cur());
            let out = meta.query_advice(advice[2], Rotation::cur());
            vec![s_mul_c * (lhs * fixed - out)]
        });

        MyConfig {
            advice,
            instance,
            constant,
            s_add,
            s_mul,
            s_add_c,
            s_mul_c,
        }
    }
}

// Put values to the table and define wire constraints.
impl<Field: FieldExt> Circuit<Field> for MyCircuit<Field> {
    type Config = MyConfig;
    type FloorPlanner = SimpleFloorPlanner;

    fn without_witnesses(&self) -> Self {
        Self::default()
    }

    fn configure(meta: &mut ConstraintSystem<Field>) -> Self::Config {
        let advice = [meta.advice_column(), meta.advice_column(), meta.advice_column()];
        let instance = meta.instance_column();
        let constant = meta.fixed_column();
        FChip::configure(meta, advice, instance, constant)
    }

    fn synthesize(
        &self, config: Self::Config, mut layouter: impl Layouter<Field>
    ) -> Result<(), Error> {
        // handling multiplication region
        let t1 = self.u * self.u;
        let t2 = self.u * self.v;
        let t3 = t2 * Value::known(Field::from(3));

        // define multiplication region
        let (
            (x_a1, x_b1, x_c1),
            (x_a2, x_b2, x_c2),
            (x_a3, x_c3)
        ) = layouter

.assign_region(
            || "multiplication gate",
            |mut region| {
                let x_a1 = region.assign_advice(
                    || "lhs multiplication",
                    config.advice[0],
                    0,
                    || Ok(self.u),
                )?;
                let x_b1 = region.assign_advice(
                    || "rhs multiplication",
                    config.advice[1],
                    0,
                    || Ok(self.u),
                )?;
                let x_c1 = region.assign_advice(
                    || "out multiplication",
                    config.advice[2],
                    0,
                    || Ok(t1),
                )?;

                let x_a2 = region.assign_advice(
                    || "lhs multiplication",
                    config.advice[0],
                    1,
                    || Ok(self.u),
                )?;
                let x_b2 = region.assign_advice(
                    || "rhs multiplication",
                    config.advice[1],
                    1,
                    || Ok(self.v),
                )?;
                let x_c2 = region.assign_advice(
                    || "out multiplication",
                    config.advice[2],
                    1,
                    || Ok(t2),
                )?;

                let x_a3 = region.assign_advice(
                    || "lhs multiplication",
                    config.advice[0],
                    2,
                    || Ok(t2),
                )?;
                let x_c3 = region.assign_advice(
                    || "out multiplication",
                    config.advice[2],
                    2,
                    || Ok(t3),
                )?;

                region.constrain_equal((t1, t2, t3), (x_c1, x_c2, x_c3));
                Ok(((x_a1, x_b1, x_c1), (x_a2, x_b2, x_c2), (x_a3, x_c3)))
            },
        )?;

        // handling addition region
        let t4 = t1 + t2 + self.v;
        // define addition region
        layouter.assign_region(
            || "addition gate",
            |mut region| {
                let x_a4 = region.assign_advice(
                    || "lhs addition",
                    config.advice[0],
                    3,
                    || Ok(t1),
                )?;
                let x_b4 = region.assign_advice(
                    || "rhs addition",
                    config.advice[1],
                    3,
                    || Ok(t2),
                )?;
                let x_c4 = region.assign_advice(
                    || "out addition",
                    config.advice[2],
                    3,
                    || Ok(t4),
                )?;

                region.constrain_equal((t1, t2, t4), (x_c1, x_c2, x_c4));
                Ok(())
            },
        )?;

        // handling addition with constant region
        let t5 = t4 + Value::known(Field::from(5));
        // define addition with constant region
        layouter.assign_region(
            || "addition with constant gate",
            |mut region| {
                let x_a5 = region.assign_advice(
                    || "lhs addition with constant",
                    config.advice[0],
                    4,
                    || Ok(t4),
                )?;
                let x_c5 = region.assign_advice(
                    || "out addition with constant",
                    config.advice[2],
                    4,
                    || Ok(t5),
                )?;

                let constant = region.assign_fixed(
                    || "constant",
                    config.constant,
                    0,
                    || Ok(Value::known(Field::from(5))),
                )?;

                region.constrain_equal((t4, Value::known(Field::from(5)), t5), (x_c4, constant, x_c5));
                Ok(())
            },
        )?;

        Ok(())
    }
}
```

This Rust code may seem intricate at first glance, but it's the embodiment of our theoretical journey. It defines the configuration of our PLONKish Arithmetization, sets up the necessary dependencies, and synthesizes the circuit, bringing our abstract arithmetic computations to life.

In essence, this Rust implementation is the bridge that connects the theory of cryptographic protocols with tangible, executable code. It transforms the seemingly complex mathematics into a language that computers can understand, enabling the creation of secure and verifiable applications.

The journey we've undertaken, from understanding Halo 2's definition to implementing it in Rust, underscores the power of cryptographic tools in shaping the future of secure and private application development. As we conclude this exploration, we've not only scratched the surface of Halo 2 but also glimpsed into the transformative potential it holds in the realm of privacy-preserving technologies.

Conclusion

As we draw the curtains on our exploration, take a moment to appreciate the intricacies of Halo 2. From its foundational definition to real-world applications, from PLONKish Arithmetization to the practical implementation in Rust, we've covered it all. This article serves as a friendly guide, inviting you to unlock the potential of Halo 2 in your cryptographic adventures. Happy coding!

About Orochi Network

Orochi Network is a cutting-edge zkOS (An operating system based on zero-knowledge proof) designed to tackle the challenges of computation limitation, data correctness, and data availability in the Web3 industry. With the well-rounded solutions for Web3 Applications, Orochi Network omits the current performance-related barriers and makes ways for more comprehensive dApps hence, becoming the backbone of Web3's infrastructure landscape.