13 releases
0.5.0-alpha.0 | Jun 20, 2024 |
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0.4.2 | Mar 18, 2023 |
0.4.1 | Feb 21, 2023 |
0.4.0-alpha.7 | Dec 29, 2022 |
0.2.0 | Mar 24, 2021 |
#262 in Cryptography
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Used in 1,962 crates
(46 directly)
570KB
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ark-poly
This crate implements traits and implementations for polynomials, FFT-friendly subsets of a field (dubbed "domains"), and FFTs for these domains.
Polynomials
The polynomial
module provides the following traits for defining polynomials in coefficient form:
Polynomial
: Requires implementors to support common operations on polynomials, such asAdd
,Sub
,Zero
, evaluation at a point, degree, etc, and defines methods to serialize to and from the coefficient representation of the polynomial.DenseUVPolynomial
: Specifies that aPolynomial
is actually a univariate polynomial.DenseMVPolynomial
: Specifies that aPolynomial
is actually a multivariate polynomial.
This crate also provides the following data structures that implement these traits:
univariate/DensePolynomial
: Represents degreed
univariate polynomials via a list ofd + 1
coefficients. This struct implements theDenseUVPolynomial
trait.univariate/SparsePolynomial
: Represents degreed
univariate polynomials via a list containing all non-zero monomials. This should only be used when most coefficients of the polynomial are zero. This struct implements thePolynomial
trait (but not theDenseUVPolynomial
trait).multivariate/SparsePolynomial
: Represents multivariate polynomials via a list containing all non-zero monomials.
This crate also provides the univariate/DenseOrSparsePolynomial
enum, which allows the user to abstract over the type of underlying univariate polynomial (dense or sparse).
Example
use ark_poly::{
polynomial::multivariate::{SparsePolynomial, SparseTerm, Term},
DenseMVPolynomial, Polynomial,
};
use ark_test_curves::bls12_381::Fq;
// Create a multivariate polynomial in 3 variables, with 4 terms:
// f(x_0, x_1, x_2) = 2*x_0^3 + x_0*x_2 + x_1*x_2 + 5
let poly = SparsePolynomial::from_coefficients_vec(
3,
vec![
(Fq::from(2), SparseTerm::new(vec![(0, 3)])),
(Fq::from(1), SparseTerm::new(vec![(0, 1), (2, 1)])),
(Fq::from(1), SparseTerm::new(vec![(1, 1), (2, 1)])),
(Fq::from(5), SparseTerm::new(vec![])),
],
);
assert_eq!(poly.evaluate(&vec![Fq::from(2), Fq::from(3), Fq::from(6)]), Fq::from(51));
Evaluations
The evaluations
module provides data structures to represent univariate polynomials in lagrange form.
univariate/Evaluations
Represents a univariate polynomial in evaluation form, which can be used for FFT.
The evaluations
module also provides the following traits for defining multivariate polynomials in lagrange form:
multivariate/multilinear/MultilinearExtension
Specifies a multilinear polynomial evaluated over boolean hypercube.
This crate provides some data structures to implement these traits.
-
multivariate/multilinear/DenseMultilinearExtension
Represents multilinear extension via a list of evaluations over boolean hypercube. -
multivariate/multilinear/SparseMultilinearExtension
Represents multilinear extension via a list of non-zero evaluations over boolean hypercube.
Example
use ark_test_curves::bls12_381::Fq;
use ark_poly::{DenseMultilinearExtension, MultilinearExtension, SparseMultilinearExtension};
use ark_poly::{
polynomial::multivariate::{SparsePolynomial, SparseTerm, Term},
DenseMVPolynomial, Polynomial,
};
use ark_std::One;
// Create a multivariate polynomial in 3 variables:
// f(x_0, x_1, x_2) = 2*x_0^3 + x_0*x_2 + x_1*x_2
let f = SparsePolynomial::from_coefficients_vec(
3,
vec![
(Fq::from(2), SparseTerm::new(vec![(0, 3)])),
(Fq::from(1), SparseTerm::new(vec![(0, 1), (2, 1)])),
(Fq::from(1), SparseTerm::new(vec![(1, 1), (2, 1)])),
(Fq::from(0), SparseTerm::new(vec![])),
],
);
// g is the multilinear extension of f, defined by the evaluations of f on the Boolean hypercube:
// f(0, 0, 0) = 2*0^3 + 0*0 + 0*0 = 0
// f(1, 0, 0) = 2*1^3 + 0*0 + 0*0 = 2
// ...
// f(1, 1, 1) = 2*1^3 + 1*1 + 1*1 = 4
let g: DenseMultilinearExtension<Fq> = DenseMultilinearExtension::from_evaluations_vec(
3,
vec![
Fq::from(0),
Fq::from(2),
Fq::from(0),
Fq::from(2),
Fq::from(0),
Fq::from(3),
Fq::from(1),
Fq::from(4),
]
);
// when evaluated at any point within the Boolean hypercube, f and g should be equal
let point_within_hypercube = &vec![Fq::from(0), Fq::from(1), Fq::from(1)];
assert_eq!(f.evaluate(&point_within_hypercube), g.evaluate(&point_within_hypercube));
// We can also define a MLE g'(x_0, x_1, x_2) by providing the list of non-zero evaluations:
let g_prime: SparseMultilinearExtension<Fq> = SparseMultilinearExtension::from_evaluations(
3,
&vec![
(1, Fq::from(2)),
(3, Fq::from(2)),
(5, Fq::from(3)),
(6, Fq::from(1)),
(7, Fq::from(4)),
]
);
// at any random point (X0, X1, X2), g == g' with negligible probability, unless they are the same function
let random_point = &vec![Fq::from(123), Fq::from(456), Fq::from(789)];
assert_eq!(g_prime.evaluate(&random_point), g.evaluate(&random_point));
Domains
TODO
Dependencies
~3.5–4.5MB
~86K SLoC