> ## Documentation Index
> Fetch the complete documentation index at: https://mintlify.com/provablehq/snarkvm/llms.txt
> Use this file to discover all available pages before exploring further.

# Algorithms Crate

> Overview of the snarkvm-algorithms crate and its cryptographic primitives

## About

The `snarkvm-algorithms` crate contains cryptographic algorithms and primitives for zero-knowledge proof construction in SnarkVM. This crate implements the mathematical foundations required for the Aleo blockchain's proof system.

**Package:** `snarkvm-algorithms`\
**Version:** 4.4.0\
**License:** Apache-2.0

## Core Modules

<CardGroup cols={2}>
  <Card title="Cryptographic Hashing" icon="hash" href="/api/algorithms/crypto-hash">
    Poseidon and SHA-256 hash functions optimized for ZK circuits
  </Card>

  <Card title="SNARK Systems" icon="shield-check" href="/api/algorithms/snark">
    Varuna zkSNARK implementation with AHP for R1CS
  </Card>

  <Card title="Polynomial Commitments" icon="signature" href="/api/algorithms/polycommit">
    KZG10 and SonicKZG10 polynomial commitment schemes
  </Card>

  <Card title="FFT Operations" icon="wave-sine" href="/api/algorithms/fft">
    Fast Fourier Transform for polynomial arithmetic
  </Card>
</CardGroup>

## Key Features

### Zero-Knowledge Proofs

* **Varuna zkSNARK** - Universal preprocessing zkSNARK with batch proving
* **AHP for R1CS** - Algebraic Holographic Proof compilation
* **Fiat-Shamir transformation** - Non-interactive proof generation

### Polynomial Arithmetic

* **FFT/IFFT** - O(n log n) polynomial operations
* **Evaluation domains** - Powers-of-two roots of unity
* **Dense and sparse polynomials** - Memory-efficient representations

### Cryptographic Commitments

* **KZG polynomial commitments** - Constant-size commitments with pairing-based verification
* **Batched opening proofs** - Amortized verification cost
* **Degree bounds** - Enforced polynomial degree constraints

### Optimized Primitives

* **Multi-scalar multiplication** - Pippenger's algorithm with parallel execution
* **Poseidon hash** - Algebraic hash function for efficient ZK circuits
* **Batch operations** - Parallelized cryptographic operations

## Module Organization

```
snarkvm-algorithms/
├── crypto_hash/        # Cryptographic hash functions
│   ├── poseidon.rs     # Poseidon hash and sponge
│   └── sha256.rs       # SHA-256 implementation
├── fft/                # Fast Fourier Transform
│   ├── domain.rs       # Evaluation domains
│   ├── evaluations.rs  # Lagrange evaluations
│   └── polynomial/     # Polynomial types
├── polycommit/         # Polynomial commitments
│   ├── kzg10/          # KZG10 scheme
│   └── sonic_pc/       # SonicKZG10 with batching
├── snark/              # zkSNARK implementations
│   └── varuna/         # Varuna proof system
├── msm/                # Multi-scalar multiplication
├── r1cs/               # R1CS constraint system
├── srs/                # Structured reference string
└── traits/             # Core trait definitions
```

## Usage Example

```rust theme={null}
use snarkvm_algorithms::{
    crypto_hash::Poseidon,
    fft::{DensePolynomial, EvaluationDomain},
    polycommit::kzg10::KZG10,
    snark::varuna::VarunaSNARK,
};

// Hash with Poseidon
let poseidon = Poseidon::<Fr, 4>::setup();
let hash = poseidon.evaluate(&[field1, field2, field3]);

// Polynomial operations
let domain = EvaluationDomain::new(256)?;
let poly = DensePolynomial::from_coefficients_vec(coeffs);
let evals = domain.fft(&poly);

// Polynomial commitments
let srs = KZG10::load_srs(max_degree)?;
let (commitment, randomness) = KZG10::commit(&powers, &poly, None, None)?;

// zkSNARK proving
type Varuna = VarunaSNARK<Bls12_377, PoseidonSponge<Fq, 2, 1>, DefaultMode>;
let proof = Varuna::prove(&prover, &fs_params, &pk, version, &circuit, rng)?;
let valid = Varuna::verify(&verifier, &fs_params, &vk, version, &input, &proof)?;
```

## Performance

### Parallelization

Most operations are parallelized using Rayon:

* FFT operations split across available cores
* MSM uses parallel bucket accumulation
* Polynomial commitment batching
* Parallel constraint synthesis

### CUDA Acceleration

Enable the `cuda` feature for GPU acceleration of:

* Multi-scalar multiplication (MSM)
* FFT/IFFT operations
* Polynomial evaluations

```toml theme={null}
[dependencies]
snarkvm-algorithms = { version = "4.4.0", features = ["cuda"] }
```

## Related Crates

* `snarkvm-fields` - Finite field arithmetic implementations
* `snarkvm-curves` - Elliptic curve group operations
* `snarkvm-console` - High-level VM types
* `snarkvm-circuit` - Circuit-level constraint synthesis

## See Also

* [Cryptographic Hash Functions](/api/algorithms/crypto-hash)
* [SNARK Implementations](/api/algorithms/snark)
* [Polynomial Commitments](/api/algorithms/polycommit)
* [FFT Implementations](/api/algorithms/fft)