<h1 align="center">ark-ff</h1>
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This crate defines Finite Field traits and useful abstraction models that follow these traits.
Implementations of concrete finite fields for some popular elliptic curves can be found in [`arkworks-rs/curves`](https://github.com/arkworks-rs/algebra/blob/master/curves/README.md) under `arkworks-rs/curves/<your favourite curve>/src/fields/`.
This crate contains two types of traits:
- `Field` traits: These define interfaces for manipulating field elements, such as addition, multiplication, inverses, square roots, and more.
- Field `Config`s: specifies the parameters defining the field in question. For extension fields, it also provides additional functionality required for the field, such as operations involving a (cubic or quadratic) non-residue used for constructing the field (`NONRESIDUE`).
The available field traits are:
- [`AdditiveGroup`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/mod.rs#L46) - Interface for additive groups that have a "scalar multiplication" operation with respect to the `Scalar` associated type. This applies to prime-order fields, field extensions, and elliptic-curve groups used in cryptography.
- [`Field`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/mod.rs#L161) - Interface for a generic finite field.
- [`FftField`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/fft_friendly.rs#L2) - Exposes methods that allow for performing efficient FFTs on field elements.
- [`PrimeField`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/prime.rs#L27) - Field with a prime `p` number of elements, also referred to as `Fp`.
The models implemented are:
- [`Quadratic Extension`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/quadratic_extension.rs)
- [`QuadExtField`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/quadratic_extension.rs#L113) - Struct representing a quadratic extension field, in this case holding two base field elements
- [`QuadExtConfig`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/quadratic_extension.rs#L29) - Trait defining the necessary parameters needed to instantiate a Quadratic Extension Field
- [`Cubic Extension`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/cubic_extension.rs)
- [`CubicExtField`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/cubic_extension.rs#L97) - Struct representing a cubic extension field, holds three base field elements
- [`CubicExtConfig`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/cubic_extension.rs#L28) - Trait defining the necessary parameters needed to instantiate a Cubic Extension Field
The above two models serve as abstractions for constructing the extension fields `Fp^m` directly (i.e. `m` equal 2 or 3) or for creating extension towers to arrive at higher `m`. The latter is done by applying the extensions iteratively, e.g. cubic extension over a quadratic extension field.
- [`Fp2`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/fp2.rs#L99) - Quadratic extension directly on the prime field, i.e. `BaseField == BasePrimeField`
- [`Fp3`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/fp3.rs#L72) - Cubic extension directly on the prime field, i.e. `BaseField == BasePrimeField`
- [`Fp6_2over3`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/fp6_2over3.rs#L59) - Extension tower: quadratic extension on a cubic extension field, i.e. `BaseField = Fp3`, but `BasePrimeField = Fp`.
- [`Fp6_3over2`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/fp6_3over2.rs#L64) - Extension tower, similar to the above except that the towering order is reversed: it's a cubic extension on a quadratic extension field, i.e. `BaseField = Fp2`, but `BasePrimeField = Fp`. Only this latter one is exported by default as `Fp6`.
- [`Fp12_2over3over2`](https://github.com/arkworks-rs/algebra/blob/master/ff/src/fields/models/fp12_2over3over2.rs#L66) - Extension tower: quadratic extension of `Fp6_3over2`, i.e. `BaseField = Fp6`.
## Usage
There are two important traits when working with finite fields: [`Field`],
and [`PrimeField`]. Let's explore these via examples.
### [`AdditiveGroup`][additive_group]
The [`AdditiveGroup`][additive_group] trait provides a generic interface for additive groups that have an associated scalar multiplication operations. Types implementing this trait support common group operations such as addition, subtraction, negation, as well as scalar multiplication by the [`Scalar`][group_scalar_type] associated type.
```rust
use ark_ff::AdditiveGroup;
// We'll use a field associated with the BLS12-381 pairing-friendly
// group for this example.
use ark_test_curves::bls12_381::Fq2 as F;
// `ark-std` is a utility crate that enables `arkworks` libraries
// to easily support `std` and `no_std` workloads, and also re-exports
// useful crates that should be common across the entire ecosystem, such as `rand`.
use ark_std::{One, UniformRand};
let mut rng = ark_std::test_rng();
// Let's sample uniformly random field elements:
let a = F::rand(&mut rng);
let b = F::rand(&mut rng);
let c = <F as AdditiveGroup>::Scalar::rand(&mut rng);
// We can add...
let c = a + b;
// ... subtract ...
let d = a - b;
// ... double elements ...
assert_eq!(c + d, a.double());
// ... negate them ...
assert_ne!(d, -d);
// ... and multiply them by scalars:
let e = d * c;
```
### [`Field`][field]
The [`Field`][field] trait provides a generic interface for any finite field.
Types implementing [`Field`][field] support common field operations
such as addition, subtraction, multiplication, and inverses, and are required
to be [`AdditiveGroup`][additive_group]s too.
```rust
use ark_ff::{AdditiveGroup, Field};
// We'll use a field associated with the BLS12-381 pairing-friendly
// group for this example.
use ark_test_curves::bls12_381::Fq2 as F;
// `ark-std` is a utility crate that enables `arkworks` libraries
// to easily support `std` and `no_std` workloads, and also re-exports
// useful crates that should be common across the entire ecosystem, such as `rand`.
use ark_std::{One, UniformRand};
let mut rng = ark_std::test_rng();
// Let's sample uniformly random field elements:
let a = F::rand(&mut rng);
let b = F::rand(&mut rng);
// We can perform all the operations from the `AdditiveGroup` trait:
// We can add...
let c = a + b;
// ... subtract ...
let d = a - b;
// ... double elements ...
assert_eq!(c + d, a.double());
// ... multiply ...
let e = c * d;
// ... square elements ...
assert_eq!(e, a.square() - b.square());
// ... and compute inverses ...
assert_eq!(a.inverse().unwrap() * a, F::one()); // have to unwrap, as `a` could be zero.
```
In some cases, it is useful to be able to compute square roots of field elements
(e.g.: for point compression of elliptic curve elements).
To support this, users can implement the `sqrt`-related methods for their field type. This method
is already implemented for prime fields (see below), and also for quadratic extension fields.
The `sqrt`-related methods can be used as follows:
```rust
use ark_ff::Field;
// As before, we'll use a field associated with the BLS12-381 pairing-friendly
// group for this example.
use ark_test_curves::bls12_381::Fq2 as F;
use ark_std::{One, UniformRand};
let mut rng = ark_std::test_rng();
let a = F::rand(&mut rng);
// We can check if a field element is a square by computing its Legendre symbol...
if a.legendre().is_qr() {
// ... and if it is, we can compute its square root.
let b = a.sqrt().unwrap();
assert_eq!(b.square(), a);
} else {
// Otherwise, we can check that the square root is `None`.
assert_eq!(a.sqrt(), None);
}
```
### [`PrimeField`][prime_field]
If the field is of prime order, then users can choose
to implement the [`PrimeField`][prime_field] trait for it. This provides access to the following
additional APIs:
```rust
use ark_ff::{Field, PrimeField, FpConfig, BigInteger, Zero};
// Now we'll use the prime field underlying the BLS12-381 G1 curve.
use ark_test_curves::bls12_381::Fq as F;
use ark_std::{One, UniformRand};
let mut rng = ark_std::test_rng();
let a = F::rand(&mut rng);
// We can access the prime modulus associated with `F`:
let modulus = <F as PrimeField>::MODULUS;
assert_eq!(a.pow(&modulus), a);
// We can convert field elements to integers in the range [0, MODULUS - 1]:
let one: num_bigint::BigUint = F::one().into();
assert_eq!(one, num_bigint::BigUint::one());
// We can construct field elements from an arbitrary sequence of bytes:
let n = F::from_le_bytes_mod_order(&modulus.to_bytes_le());
assert_eq!(n, F::zero());
```
[additive_group]: https://docs.rs/ark-ff/latest/ark_ff/fields/trait.AdditiveGroup.html
[group_scalar_type]: https://docs.rs/ark-ff/latest/ark_ff/fields/trait.AdditiveGroup.html#associatedtype.Scalar
[field]: https://docs.rs/ark-ff/latest/ark_ff/fields/trait.Field.html
[prime_field]: https://docs.rs/ark-ff/latest/ark_ff/fields/trait.PrimeField.html