### Production SRS Setup Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/plonk/src/dsl/README.md

An example demonstrating the setup and usage of the SRSManager, Prover, and Verifier for PLONK circuits.

```APIDOC
## Production SRS Setup (example)

### Description
This example shows how to load and check the SRS (Structured Reference String), create a KZG commitment scheme, and then use the Prover and Verifier to generate and verify a PLONK proof.

### Method
N/A (Example Code)

### Endpoint
N/A (Example Code)

### Parameters
N/A (Example Code)

### Request Example
N/A (Example Code)

### Response
N/A (Example Code)

### Code Example
```rust
use lambdaworks_plonk::srs::SRSManager;
use lambdaworks_plonk::prover::Prover;
use lambdaworks_plonk::test_utils::utils::{SecureRandomFieldGenerator, KZG};
use lambdaworks_plonk::verifier::Verifier;
use lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::{
    curve::BLS12381Curve,
    twist::BLS12381TwistCurve,
};
use lambdaworks_math::elliptic_curve::traits::IsEllipticCurve;

type G1Point = <BLS12381Curve as IsEllipticCurve>::PointRepresentation;
type G2Point = <BLS12381TwistCurve as IsEllipticCurve>::PointRepresentation;

// Load SRS and ensure it is large enough for the circuit size `cpi.n`
let srs = SRSManager::load::<G1Point, G2Point, _>("ceremony.srs")?;
SRSManager::check_size(&srs, cpi.n)?;

// Assumes `cpi`, `witness`, `public_inputs`, and `vk` are already built
let kzg = KZG::new(srs);
let prover = Prover::new(kzg.clone(), SecureRandomFieldGenerator);
let proof = prover.prove(&witness, &public_inputs, &cpi, &vk)?;

let verifier = Verifier::new(kzg);
verifier.verify_with_result(&proof, &public_inputs, &cpi, &vk)?;
```
```

--------------------------------

### Verification Key Setup in Rust

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/plonk/implementation.md

Performs the setup phase for the KZG commitment scheme, generating a verification key from the common preprocessed input.

```rust
let verifying_key = setup(&common_preprocessed_input, &kzg);
```

--------------------------------

### Install cargo-criterion

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/math/src/field/README.md

Install the cargo-criterion tool for running benchmarks locally.

```bash
cargo install cargo-criterion
```

--------------------------------

### Groth16 Setup, Prove, and Verify with Lambdaworks

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/groth16/arkworks-adapter/README.md

After converting to a Lambdaworks QAP, perform standard Groth16 operations: setup, prove, and verify. Ensure the public inputs are correctly sliced from the witness.

```rust
let (pk, vk) = setup(&qap);

let proof = Prover::prove(&w, &qap, &pk).expect("proving failed");

let public_inputs = &w[..qap.num_of_public_inputs];
let accept = verify(&vk, &proof, public_inputs);

assert!(accept);
```

--------------------------------

### Full Example: Linear Exponentiation Circuit

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/groth16/arkworks-adapter/README.md

An example demonstrating the creation of a linear exponentiation circuit using Arkworks, conversion to Lambdaworks format, and subsequent Groth16 proving and verification on BLS12-381.

```rust
use crate::to_lambda;
use ark_bls12_381::Fr;
use ark_relations::{lc, r1cs::ConstraintSystem, r1cs::Variable};
use lambdaworks_groth16::{setup, verify, Prover};
use rand::Rng;

// ...
// ...

let mut rng = rand::thread_rng();
let x = rng.gen::<u64>();
let exp = rng.gen::<u8>();

// Define the circuit using Arkworks

let cs = ConstraintSystem::<Fr>::new_ref();

let x = Fr::from(x);
let mut _x = cs.new_witness_variable(|| Ok(x)).unwrap();

let mut acc = Fr::from(x);
let mut _acc = cs.new_witness_variable(|| Ok(x)).unwrap();

for _ in 0..exp - 1 {
	acc *= x;
	let _new_acc = cs.new_witness_variable(|| Ok(acc)).unwrap();
	cs.enforce_constraint(lc!() + _acc, lc!() + _x, lc!() + _new_acc)
		.unwrap();
	_acc = _new_acc;
}

let _out = cs.new_input_variable(|| Ok(acc)).unwrap();
cs.enforce_constraint(lc!() + _out, lc!() + Variable::One, lc!() + _acc)
	.unwrap();

// Make Lambdaworks-compatible
let (qap, w) = to_lambda(&cs);

// Use Lambdaworks Groth16 backend

let (pk, vk) = setup(&qap);

let proof = Prover::prove(&w, &qap, &pk).expect("proving failed");

let public_inputs = &w[..qap.num_of_public_inputs];
let accept = verify(&vk, &proof, public_inputs);
assert!(accept);
```

--------------------------------

### Run Univariate Logup GKR Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/gkr-logup/README.md

Demonstrates univariate commitments on a cyclic domain using three examples: grand product, LogUp singles, and LogUp multiplicities.

```bash
cargo run -p lambdaworks-gkr-logup --example univariate_logup_gkr
```

--------------------------------

### Build Documentation Locally

Source: https://github.com/lambdaclass/lambdaworks/blob/main/README.md

Serve the project's documentation locally after installing mdbook and the Katex preprocessor. This command builds and serves the documentation.

```shell
make docs
```

--------------------------------

### Setup Evaluation and Verification Keys

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/pinocchio/Readme.md

Generates the EvaluationKey and VerificationKey required for the prover and verifier, respectively, using a QAP and toxic waste.

```rust
let (evaluation_key, verification_key) = setup(&test_qap, toxic_waste);
```

--------------------------------

### Run FROST Example with Cargo

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/frost-signature/README.md

Use this command to compile and run the FROST signature example using Cargo, the Rust build system and package manager.

```bash
cargo run -p frost-signature
```

--------------------------------

### Consolidated Setup Commands

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/groth16/circom-adapter/README.md

A consolidated bash snippet to perform circuit compilation, witness generation, and JSON export in one go. Replace 'test' with your circuit's name.

```bash
circom test.circom --r1cs --wasm -p bls12381;
node test_js/generate_witness.js test_js/test.wasm input.json witness.wtns;
snarkjs wtns export json witness.wtns;
snarkjs r1cs export json test.r1cs test.r1cs.json;
```

--------------------------------

### Groth16 Setup Parameters

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/groth16/README.md

Defines the initial random field elements required for Groth16 setup. These 'toxic waste' values are crucial for key generation and must be securely handled.

```plaintext
t, \u03b1, \u03b2, \u03b3, \u03b4
```

--------------------------------

### Verify Merkle Proof Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/merkle-tree-cli/README.md

This command demonstrates verifying a Merkle Proof using the generated root, proof, and leaf files for the sample tree.

```bash
cargo run --release verify-proof sample_tree_root.txt 0 sample_tree_proof_0.json sample_tree_leaf_0.txt
```

--------------------------------

### Virtual Column Trace Evaluation Example 2

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/starks/virtual_cols.md

Demonstrates row evaluation for a virtual column setup with a dummy flag 'D', where constraints are applied with a larger stride. This configuration is useful for aligning evaluations to powers of two.

```diff
+ A0 |
+ B0 |
+ C0 |
+ D0 |
+ A1 |
+ B1 |
+ C1 |
+ D1 |
  A2 |
  B2 |
  C2 |
  D2 |
  A3 |
  B3 |
  C3 |
  D3 |
```

```diff
  A0 |
  B0 |
  C0 |
  D0 |
+ A1 |
+ B1 |
+ C1 |
+ D1 |
+ A2 |
+ B2 |
+ C2 |
+ D2 |
  A3 |
  B3 |
  C3 |
  D3 |
```

```diff
  A0 |
  B0 |
  C0 |
  D0 |
  A1 |
  B1 |
  C1 |
  D1 |
+ A2 |
+ B2 |
+ C2 |
+ D2 |
+ A3 |
+ B3 |
+ C3 |
+ D3 |
```

--------------------------------

### Install nightly Rust toolchain

Source: https://github.com/lambdaclass/lambdaworks/blob/main/fuzz/README.md

Installs the nightly Rust toolchain, which is a prerequisite for using cargo-fuzz.

```bash
rustup toolchain install nightly
```

--------------------------------

### Merkle Tree Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/getting-started.md

Demonstrates building and verifying a Merkle tree using field elements and Keccak256 hashing. Requires `lambdaworks-crypto` and `lambdaworks-math`.

```rust
use lambdaworks_crypto::merkle_tree::merkle::MerkleTree;
use lambdaworks_crypto::merkle_tree::backends::field_element::FieldElementBackend;
use lambdaworks_math::field::element::FieldElement;
use lambdaworks_math::field::fields::fft_friendly::stark_252_prime_field::Stark252PrimeField;
use sha3::Keccak256;

type F = Stark252PrimeField;
type FE = FieldElement<F>;

fn main() {
    // Create some data
    let values: Vec<FE> = (1..9).map(FE::from).collect();

    // Build the Merkle tree
    let tree = MerkleTree::<FieldElementBackend<F, Keccak256, 32>>::build(&values).unwrap();

    // Generate a proof for element at index 3
    let proof = tree.get_proof_by_pos(3).unwrap();

    // Verify the proof
    let is_valid = proof.verify::<FieldElementBackend<F, Keccak256, 32>>(
        &tree.root,
        3,
        &values[3]
    );

    assert!(is_valid);
    println!("Merkle proof verified successfully!");
}
```

--------------------------------

### Schnorr Signature Example in Rust

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/schnorr-signature/README.md

This example demonstrates the basic usage of the Schnorr signature implementation. It covers key generation, message signing, and signature verification. Ensure the necessary elliptic curve and field elements are correctly set up.

```rust
use ark_bn254::Fr as FE;
use lambdaworks_gadgets::schnorr_signature::{get_public_key, sign_message, verify_signature};

// Choose a private key.
let private_key = FE::from(5);

// Compute the public key.
let public_key = get_public_key(&private_key);

// Write the message
let message = "hello world";

// Sign the message
let message_signed = sign_message(&private_key, message);

// Verify the signature
let is_the_signature_correct = verify_signature(
    &public_key,
    &message_signed
);

```

--------------------------------

### Run Reed-Solomon Codes Demo

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/reed-solomon-codes/README.md

Navigate to the examples directory and run the cargo command to see an interactive demonstration of Reed-Solomon encoding, decoding, and list decoding.

```bash
cd examples/reed-solomon-codes
cargo run
```

--------------------------------

### Simple STARK Proof (Fibonacci) Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/getting-started.md

Generates and verifies a STARK proof for the Fibonacci sequence. Requires `stark-platinum-prover`, `lambdaworks-math`, and `sha3` dependencies.

```rust
use lambdaworks_math::field::element::FieldElement;
use lambdaworks_math::field::fields::fft_friendly::stark_252_prime_field::Stark252PrimeField;
use stark_platinum_prover::proof::options::ProofOptions;
use stark_platinum_prover::prover::{IsStarkProver, Prover};
use stark_platinum_prover::verifier::{IsStarkVerifier, Verifier};
use stark_platinum_prover::examples::simple_fibonacci::{
    FibonacciAIR, FibonacciPublicInputs, fibonacci_trace
};
use stark_platinum_prover::transcript::StoneProverTranscript;

type Felt = FieldElement<Stark252PrimeField>;

fn main() {
    // Generate the Fibonacci trace (first 8 numbers)
    let mut trace = fibonacci_trace([Felt::from(1), Felt::from(1)], 8);

    // Set proof options
    let proof_options = ProofOptions::default_test_options();

    // Define public inputs (initial values)
    let pub_inputs = FibonacciPublicInputs {
        a0: Felt::one(),
        a1: Felt::one(),
    };

    // Generate the proof
    let proof = Prover::<FibonacciAIR<Stark252PrimeField>>::prove(
        &mut trace,
        &pub_inputs,
        &proof_options,
        StoneProverTranscript::new(&[]),
    ).unwrap();

    // Verify the proof
    let is_valid = Verifier::<FibonacciAIR<Stark252PrimeField>>::verify(
        &proof,
        &pub_inputs,
        &proof_options,
        StoneProverTranscript::new(&[]),
    );

    assert!(is_valid);
    println!("STARK proof verified successfully!");
}
```

--------------------------------

### GKR Protocol Example Usage

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/gkr/README.md

Demonstrates how to generate and verify a GKR proof for a predefined circuit. Ensure the necessary field and circuit components are defined before use.

```rust
use lambdaworks_math::field::fields::u64_prime_field::U64PrimeField;
use lambdaworks_math::field::element::FieldElement;
use lambdaworks_gkr::{gkr_prove, gkr_verify, circuit_from_lambda};

// Define the field (We use modulus 23 as in the example of our blog post).
const MODULUS23: u64 = 23;
type F23 = U64PrimeField<MODULUS23>;
type F23E = FieldElement<F23>;

// Create the circuit of our blog post.
// This creates a 2-layer circuit plus the input layer.
let circuit = lambda_post_circuit.unwrap();

// Define input values (from the post example).
let input = [F23E::from(3), F23E::from(1)];

// Generate proof.
let proof_result = gkr_prove(&circuit, &input);
assert!(proof_result.is_ok());
let proof = proof_result.unwrap();

// Verify proof.
let verification_result = gkr_verify(&proof, &circuit, &input);
assert!(verification_result.is_ok() && verification_result.unwrap());
println!("GKR verification successful!");

// You can also check the actual output.
let evaluation = circuit.evaluate(&input);
println!("Circuit output: {:?}", evaluation.layers[0]);
```

--------------------------------

### Build and Prove Plonk Circuits

Source: https://context7.com/lambdaclass/lambdaworks/llms.txt

Demonstrates building a constraint system for a simple multiplication circuit and generating a proof. Requires setup and verification components not shown.

```rust
use lambdaworks_plonk::constraint_system::ConstraintSystem;
use lambdaworks_plonk::setup::{setup, CommonPreprocessedInput};
use lambdaworks_plonk::prover::Prover;
use lambdaworks_plonk::verifier::Verifier;
use lambdaworks_crypto::commitments::kzg::KateZaveruchaGoldberg;
use lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::default_types::FrField;
use lambdaworks_math::field::element::FieldElement;
use std::collections::HashMap;

type FE = FieldElement<FrField>;

// Build constraint system: prove x * e = y
let system = &mut ConstraintSystem::<FrField>::new();

// Define variables
let x = system.new_public_input();   // Public input
let y = system.new_public_input();   // Public input
let e = system.new_variable();       // Private witness

// Add constraint: x * e = y
let z = system.mul(&x, &e);
system.assert_eq(&y, &z);

// Setup phase
// let common = CommonPreprocessedInput::from_constraint_system(&system, &ORDER_R_MINUS_1_ROOT_UNITY);
// let srs = test_srs(common.n);
// let kzg = KZG::new(srs);
// let verifying_key = setup(&common, &kzg);

// Prove with concrete values: x=4, e=3, y=12
let inputs = HashMap::from([
    (x, FE::from(4u64)),
    (e, FE::from(3u64)),
]);
let assignments = system.solve(inputs).unwrap();

// Generate and verify proof
// let witness = Witness::new(assignments, &system);
// let public_inputs = system.public_input_values(&assignments);
// let prover = Prover::new(kzg.clone(), TestRandomFieldGenerator {});
// let proof = prover.prove(&witness, &public_inputs, &common, &verifying_key);

// Verify
// let verifier = Verifier::new(kzg);
// assert!(verifier.verify(&proof, &public_inputs, &common, &verifying_key));
```

--------------------------------

### Setup Proving and Verification Keys in Rust

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/baby-snark/README.md

Generates the proving (pk) and verification (vk) keys from the constructed Square Span Program (ssp).

```rust
    let (pk, vk) = setup(&ssp);
```

--------------------------------

### Generate Merkle Proof Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/merkle-tree-cli/README.md

This command demonstrates generating a Merkle Proof for the leaf at position 0 in the sample tree. It will create a JSON file for the proof and a TXT file for the leaf's value.

```bash
cargo run --release generate-proof sample_tree.csv 0
```

--------------------------------

### Production SRS Setup and Verification

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/plonk/src/dsl/README.md

Loads a Secure Random Sampling (SRS) and performs prover and verifier operations. Ensure the SRS is sufficiently large for the circuit size.

```rust
use lambdaworks_plonk::srs::SRSManager;
use lambdaworks_plonk::prover::Prover;
use lambdaworks_plonk::test_utils::utils::{SecureRandomFieldGenerator, KZG};
use lambdaworks_plonk::verifier::Verifier;
use lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::
    curve::BLS12381Curve,
    twist::BLS12381TwistCurve,
;
use lambdaworks_math::elliptic_curve::traits::IsEllipticCurve;

type G1Point = <BLS12381Curve as IsEllipticCurve>::PointRepresentation;
type G2Point = <BLS12381TwistCurve as IsEllipticCurve>::PointRepresentation;

// Load SRS and ensure it is large enough for the circuit size `cpi.n`
let srs = SRSManager::load::<G1Point, G2Point, _>("ceremony.srs")?;
SRSManager::check_size(&srs, cpi.n)?;

// Assumes `cpi`, `witness`, `public_inputs`, and `vk` are already built
let kzg = KZG::new(srs);
let prover = Prover::new(kzg.clone(), SecureRandomFieldGenerator);
let proof = prover.prove(&witness, &public_inputs, &cpi, &vk)?;

let verifier = Verifier::new(kzg);
verifier.verify_with_result(&proof, &public_inputs, &cpi, &vk)?;
```

--------------------------------

### Run Logup GKR Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/gkr-logup/README.md

Executes the LogUpSingles and LogUpMultiplicities batch instances to verify memory accesses against a ROM table. This is used for read-only memory checks.

```bash
cargo run -p lambdaworks-gkr-logup --example logup_gkr
```

--------------------------------

### Finite Field Arithmetic Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/getting-started.md

Demonstrates basic arithmetic operations on field elements using Stark252PrimeField. Ensure `lambdaworks-math` is added to your dependencies.

```rust
use lambdaworks_math::field::element::FieldElement;
use lambdaworks_math::field::fields::fft_friendly::stark_252_prime_field::Stark252PrimeField;

// Create a type alias for convenience
type FE = FieldElement<Stark252PrimeField>;

fn main() {
    // Create field elements
    let a = FE::from(42u64);
    let b = FE::from(7u64);

    // Basic operations
    let sum = &a + &b;          // Addition
    let product = &a * &b;      // Multiplication
    let quotient = &a / &b;     // Division
    let squared = a.square();   // Squaring
    let inverse = b.inv().unwrap(); // Multiplicative inverse

    println!("a + b = {:?}", sum);
    println!("a * b = {:?}", product);
    println!("a / b = {:?}", quotient);
    println!("a^2 = {:?}", squared);
    println!("b^(-1) = {:?}", inverse);

    // Verify: b * b^(-1) = 1
    assert_eq!(&b * &inverse, FE::one());
}
```

--------------------------------

### Creating a Custom Cube Gadget

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/plonk/src/dsl/README.md

Illustrates how to create a reusable custom gadget by implementing the `Gadget` trait. This example defines a `Cube` gadget that computes `x^3` using two multiplication gates.

```rust
use lambdaworks_plonk::dsl::gadgets::{Gadget, GadgetError};
use lambdaworks_plonk::dsl::{CircuitBuilder, FieldVar};

/// Computes x^3
pub struct Cube;

impl<F: IsField> Gadget<F> for Cube {
    type Input = FieldVar;
    type Output = FieldVar;

    fn synthesize(
        builder: &mut CircuitBuilder<F>,
        x: Self::Input,
    ) -> Result<Self::Output, GadgetError> {
        let x_squared = builder.mul(&x, &x);
        let x_cubed = builder.mul(&x_squared, &x);
        Ok(x_cubed)
    }

    fn constraint_count() -> usize {
        2  // Two multiplication gates
    }

    fn name() -> &'static str {
        "Cube"
    }
}
```

--------------------------------

### Create Simple Main Boundary Constraint

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/stark/README.md

Illustrates the creation of a single boundary constraint. This specific example applies a constraint to the second row (index 1) using a public input value.

```rust
let a1 = BoundaryConstraint::new_simple_main(1, self.pub_inputs.a1.clone());
```

--------------------------------

### KZG Polynomial Commitment Placeholder

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/crypto/README.md

A placeholder indicating the usage of the Kate-Zaverucha-Goldberg (KZG) polynomial commitment scheme. A full example requires a trusted setup (SRS).

```rust
use lambdaworks_crypto::commitments::kzg::KateZaveruchaGoldberg;
use lambdaworks_math::polynomial::Polynomial;

// KZG requires a trusted setup (SRS)
// See the full example in the crate documentation
```

--------------------------------

### Virtual Column Trace Evaluation Example 1

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/starks/virtual_cols.md

Illustrates the row evaluation for a virtual column setup where constraints are applied over consecutive rows. This is used when constraints do not involve other trace cells.

```diff
+ A0 | B0 | C0
+ A1 | B1 | C1
  A2 | B2 | C2
  A3 | B3 | C3
```

```diff
  A0 | B0 | C0
+ A1 | B1 | C1
+ A2 | B2 | C2
  A3 | B3 | C3
```

--------------------------------

### Install honggfuzz

Source: https://github.com/lambdaclass/lambdaworks/blob/main/fuzz/README.md

Installs honggfuzz, the fuzzer used for CUDA targets. This also requires installing build essentials and development libraries.

```bash
cargo install honggfuzz
```

```bash
apt install build-essential
```

```bash
apt-get install binutils-dev
```

```bash
sudo apt-get install libunwind-dev
```

```bash
sudo apt-get install lldb
```

--------------------------------

### Install cargo-fuzzer

Source: https://github.com/lambdaclass/lambdaworks/blob/main/README.md

Use this command to install the cargo-fuzzer tool, which is required for running CPU fuzzers.

```bash
cargo install cargo-fuzz
```

--------------------------------

### Run Prover with Default Settings

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/stark/PROFILING.md

Execute the profiling binary with default trace length (2^16 rows).

```bash
./target/release/deps/prover_profile-*[!.d] --trace-length 16
```

--------------------------------

### Example dhat Memory Output

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/stark/PROFILING.md

Example output from dhat memory profiling, showing total allocations, peak usage, and final memory state.

```text
dhat: Total:     200,078,545 bytes in 952,106 blocks
dhat: At t-gmax: 33,292,816 bytes in 98,327 blocks
dhat: At t-end:  64 bytes in 1 blocks
```

--------------------------------

### Setup and SRS for PLONK

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/plonk/constraint_system.md

Generates the common preprocessed input and the structured reference string (SRS) required for PLONK setup. The SRS is used by the commitment scheme.

```rust
let common = CommonPreprocessedInput::from_constraint_system(&system, &ORDER_R_MINUS_1_ROOT_UNITY);
let srs = test_srs(common.n);
let kzg = KZG::new(srs); // The commitment scheme for plonk.
let vk = setup(&common, &kzg);
```

--------------------------------

### KZG Initialization

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/crypto/src/commitments/README.md

Demonstrates how to load a Structured Reference String (SRS) and initialize the KZG commitment scheme instance.

```APIDOC
## KZG Initialization

### Description
Initializes the KZG commitment scheme by loading a Structured Reference String (SRS) and creating a KZG instance.

### Method
Initialization

### Endpoint
N/A (Code-based initialization)

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Request Example
```rust
use lambdaworks_crypto::commitments::kzg::{KateZaveruchaGoldberg, StructuredReferenceString};
use lambdaworks_crypto::commitments::traits::IsCommitmentScheme;
use lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::{
    curve::BLS12381Curve,
    default_types::{FrElement, FrField},
    pairing::BLS12381AtePairing,
    twist::BLS12381TwistCurve,
};
use lambdaworks_math::elliptic_curve::short_weierstrass::point::ShortWeierstrassProjectivePoint;

// Load SRS from a file
let srs_file = "path/to/srs.bin";
let srs = StructuredReferenceString::from_file(srs_file).unwrap();

// Create a KZG instance
let kzg = KateZaveruchaGoldberg::<FrField, BLS12381AtePairing>::new(srs);
```

### Response
#### Success Response (200)
N/A (Initialization is a code operation)

#### Response Example
N/A
```

--------------------------------

### Create IPA Instance in Rust

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/crypto/src/commitments/README.md

Demonstrates how to create an IPA instance with deterministic generators. Ensure the number of generators is a power of two.

```rust
use lambdaworks_crypto::commitments::ipa::{Ipa, IpaSetup};
use lambdaworks_math::{
    elliptic_curve::short_weierstrass::curves::pallas::curve::PallasCurve,
    elliptic_curve::traits::IsEllipticCurve,
    field::element::FieldElement,
    field::fields::vesta_field::Vesta255PrimeField,
    polynomial::Polynomial,
};

type F = Vesta255PrimeField;
type FE = FieldElement<F>;
type G = <PallasCurve as IsEllipticCurve>::PointRepresentation;

// Create generators deterministically
let g = PallasCurve::generator();
let n = 8; // must be a power of two
let generators: Vec<G> = (1..=n as u64)
    .map(|i| g.operate_with_self(i))
    .collect();
let u = g.operate_with_self(n as u64 + 1337);

let setup = IpaSetup { generators, u };
let ipa = Ipa::<4, F, G>::new(setup);
```

--------------------------------

### Full Happy Path Proof Generation and Verification in Rust

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/plonk/implementation.md

This snippet illustrates the complete process of generating a proof for a circuit and then verifying it. It requires setting up the common preprocessed input, witness, SRS, and KZG commitment scheme.

```rust
// This is the common preprocessed input for
// the test circuit ( ASSERT(x * e + 5 == y) )
let common_preprocessed_input = test_common_preprocessed_input_2();

// Input
let x = FieldElement::from(2_u64);

// Private input
let e = FieldElement::from(3_u64);

let y, witness = test_witness_2(x, e);

let srs = test_srs(common_preprocessed_input.n);
let kzg = KZG::new(srs);

let verifying_key = setup(&common_preprocessed_input, &kzg);

let random_generator = TestRandomFieldGenerator {};
let prover = Prover::new(kzg.clone(), random_generator);

let public_input = vec![x.clone(), y];

let proof = prover.prove(
    &witness,
    &public_input,
    &common_preprocessed_input,
    &verifying_key,
);

let verifier = Verifier::new(kzg);
assert!(verifier.verify(
    &proof,
    &public_input,
    &common_preprocessed_input,
    &verifying_key
));
```

--------------------------------

### Build and Verify Merkle Tree with BatchPoseidonTree

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/crypto/src/merkle_tree/README.md

Shows how to build a Merkle tree using BatchPoseidonTree for efficient batch hashing with Poseidon, and then generate and verify an inclusion proof. This backend is optimized for ZK systems.

```rust
use lambdaworks_crypto::merkle_tree::{
    merkle::MerkleTree,
    backends::field_element_vector::BatchPoseidonTree,
};
use lambdaworks_crypto::hash::poseidon::starknet::PoseidonCairoStark252;
use lambdaworks_math::field::{
    element::FieldElement,
    fields::fft_friendly::stark_252_prime_field::Stark252PrimeField,
};

// Define the types we'll use
type F = Stark252PrimeField;
type FE = FieldElement<F>;

// Create some data (vectors of field elements)
let values: Vec<Vec<FE>> = vec![
    vec![FE::from(1), FE::from(2)],
    vec![FE::from(3), FE::from(4)],
    vec![FE::from(5), FE::from(6)],
    vec![FE::from(7), FE::from(8)],
];

// Build the Merkle tree using Poseidon hash function
let merkle_tree = MerkleTree::<BatchPoseidonTree<PoseidonCairoStark252>>::build(&values).unwrap();

// Generate a proof for a specific element
let proof = merkle_tree.get_proof_by_pos(1).unwrap();

// Verify the proof
let is_valid = proof.verify::<BatchPoseidonTree<PoseidonCairoStark252>>(
    &merkle_tree.root,
    1,
    &values[1]
);

assert!(is_valid, "Proof verification failed");
```

--------------------------------

### Build Profiling Binary

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/stark/PROFILING.md

Build the release binary for the STARK prover with profiling capabilities enabled.

```bash
cargo build --release -p stark-platinum-prover --bench prover_profile
```

--------------------------------

### Check Rust Version

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/getting-started.md

Verify that your Rust installation meets the minimum version requirement.

```bash
rustc --version
```

--------------------------------

### Serialization and Deserialization of SRS

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/crypto/src/commitments/README.md

Provides examples for serializing the SRS to bytes and deserializing it back.

```APIDOC
## Serialization and Deserialization of SRS

### Description
Allows for the serialization of the Structured Reference String (SRS) into a byte format for storage or transmission, and deserialization back into an SRS object.

### Method
Serialization and Deserialization

### Endpoint
N/A (Code-based operations)

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Request Example
```rust
// Assuming 'srs' is an initialized StructuredReferenceString instance

// Serialize the SRS
let bytes = srs.as_bytes();

// Deserialize the SRS
let deserialized_srs = StructuredReferenceString::<
    ShortWeierstrassProjectivePoint<BLS12381Curve>,
    ShortWeierstrassProjectivePoint<BLS12381TwistCurve>,
>::deserialize(&bytes).unwrap();
```

### Response
#### Success Response (200)
- **bytes** (Vec<u8>) - The serialized SRS as a byte vector.
- **deserialized_srs** (StructuredReferenceString) - The deserialized SRS object.

#### Response Example
N/A (Serialization/Deserialization are data transformations within code)
```

--------------------------------

### Sample Merkle Tree CSV Content

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/merkle-tree-cli/README.md

An example of the content within a sample Merkle Tree CSV file.

```csv
0x12345;0x6789A;0xBCDEF
```

--------------------------------

### Build and Verify Merkle Tree with Field Element Vectors

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/crypto/src/merkle_tree/README.md

Demonstrates building a Merkle tree using vectors of field elements as leaves and then generating and verifying a proof. This example requires importing necessary traits and types for field elements and the Merkle tree backend.

```rust
use lambdaworks_crypto::merkle_tree::{
    merkle::MerkleTree,
    backends::field_element_vector::FieldElementVectorBackend,
};
use lambdaworks_math::field::{
    element::FieldElement,
    fields::fft_friendly::stark_252_prime_field::Stark252PrimeField,
};
use sha3::Keccak256;

// Define the types we'll use
type F = Stark252PrimeField;
type FE = FieldElement<F>;

// Create some data (vectors of field elements)
let values: Vec<Vec<FE>> = vec![
    vec![FE::from(1), FE::from(2)],
    vec![FE::from(3), FE::from(4)],
    vec![FE::from(5), FE::from(6)],
];

// Build the Merkle tree
let merkle_tree = MerkleTree::<FieldElementVectorBackend<F, Keccak256, 32>>::build(&values).unwrap();

// Generate a proof
let proof = merkle_tree.get_proof_by_pos(0).unwrap();

// Verify the proof
let is_valid = proof.verify::<FieldElementVectorBackend<F, Keccak256, 32>>(
    &merkle_tree.root,
    0,
    &values[0]
);

assert!(is_valid, "Proof verification failed");
```

--------------------------------

### Run CPU Fuzzer

Source: https://github.com/lambdaclass/lambdaworks/blob/main/README.md

Execute a specific CPU fuzzer by providing its name to the make command. Ensure cargo-fuzzer is installed.

```bash
make run-fuzzer FUZZER=stark252
```

--------------------------------

### Construct a Test R1CS QAP

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/pinocchio/Readme.md

Initializes a test Quadratic Arithmetic Program (QAP) for use in the Pinocchio protocol setup.

```rust
let test_qap = new_test_r1cs().into();
```

--------------------------------

### Elliptic Curve Operations in Rust

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/math/README.md

Shows how to get the generator of the BLS12-381 curve and perform scalar multiplication and addition operations.

```rust
use lambdaworks_math::elliptic_curve::
    short_weierstrass::curves::bls12_381::curve::BLS12381Curve,
    traits::IsEllipticCurve,
;

let generator = BLS12381Curve::generator();
let doubled = generator.operate_with_self(2u64);
let sum = generator.operate_with(&doubled);
```

--------------------------------

### Run All Benchmarks

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/math/src/field/README.md

Execute the complete benchmark suite for the lambdaworks project.

```bash
make benchmarks
```

--------------------------------

### Rust XMSS Usage Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/xmss-signature/README.md

Demonstrates how to use the XMSS library in Rust, including creating an instance, generating a key pair, signing a message, and verifying the signature. Ensure secure random number generation for production environments.

```rust
use xmss_signature::{Xmss, Sha256Hasher};

// Create XMSS instance
let xmss = Xmss::new(Sha256Hasher::new());

// Generate key pair (use secure random in production!)
let mut seed = [0u8; 96];
rand::thread_rng().fill_bytes(&mut seed);
let (public_key, mut secret_key) = xmss.keygen(&seed);

// Sign a message
let message = b"Hello, post-quantum world!";
let signature = xmss.sign(message, &mut secret_key)?;

// Verify
assert!(xmss.verify(message, &signature, &public_key));

// Check remaining signatures
println!("Remaining signatures: {}", secret_key.remaining_signatures());
```

--------------------------------

### Merkle Leaf Value Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/merkle-tree-cli/README.md

The content of the TXT file storing the value of a specific leaf, generated after running the generate-proof command.

```plaintext
0x12345
```

--------------------------------

### Elliptic Curve Operations Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/docs/src/getting-started.md

Shows scalar multiplication and point addition on the BLS12-381 elliptic curve. Requires `lambdaworks-math` dependency.

```rust
use lambdaworks_math::elliptic_curve::short_weierstrass::curves::bls12_381::curve::BLS12381Curve;
use lambdaworks_math::elliptic_curve::traits::IsEllipticCurve;
use lambdaworks_math::cyclic_group::IsGroup;

fn main() {
    // Get the generator point
    let g = BLS12381Curve::generator();

    // Scalar multiplication: compute 5 * G
    let five_g = g.operate_with_self(5u64);

    // Point addition: compute G + 5G = 6G
    let six_g = g.operate_with(&five_g);

    // Verify: 6G = 6 * G
    let six_g_direct = g.operate_with_self(6u64);
    assert_eq!(six_g.to_affine(), six_g_direct.to_affine());

    println!("6G computed two ways - they match!");
}
```

--------------------------------

### Run Univariate to Multilinear Transform Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/provers/gkr-logup/README.md

Converts polynomial evaluations from a cyclic domain to a multilinear extension layout using IFFT and bit-reversal, then applies the standard multilinear GKR prover. This is an alternative to the univariate IOP.

```bash
cargo run -p lambdaworks-gkr-logup --example univariate_to_multilinear
```

--------------------------------

### Generating and Verifying Proofs

Source: https://github.com/lambdaclass/lambdaworks/blob/main/crates/crypto/src/commitments/README.md

Explains how to generate an opening proof for a polynomial evaluation and verify its validity.

```APIDOC
## Generating and Verifying Proofs

### Description
Generates a proof for a polynomial's evaluation at a specific point and verifies the proof against the commitment.

### Method
Proof Generation and Verification

### Endpoint
N/A (Code-based operations)

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Request Example
```rust
// Assuming 'kzg' is an initialized KateZaveruchaGoldberg instance and 'p' is a Polynomial

// Choose a point to evaluate the polynomial
let x = -FieldElement::one();

// Compute the evaluation
let y = p.evaluate(&x);  // Should be 0 for p(x) = x + 1 when x = -1

// Generate a proof for this evaluation
let proof = kzg.open(&x, &y, &p);

// Verify the proof
let is_valid = kzg.verify(&x, &y, &commitment, &proof);
assert!(is_valid, "Proof verification failed");
```

### Response
#### Success Response (200)
- **is_valid** (bool) - True if the proof is valid, false otherwise.

#### Response Example
N/A (Verification is a boolean result within code)
```

--------------------------------

### Merkle Tree Root Output Example

Source: https://github.com/lambdaclass/lambdaworks/blob/main/examples/merkle-tree-cli/README.md

The content of the TXT file that stores the Merkle Tree root, generated after running the generate-tree command.

```plaintext
0xa3bbbb9eac9f79d18862b802ea79f87e75efc37d4f4af4464976784c14a851b69c09aa04b1e8a8d1eb9825b713dc6ca
```