Trait halo2_backend::arithmetic::Field

pub trait Field: Sized + Eq + Copy + Clone + Default + Send + Sync + Debug + 'static + ConditionallySelectable + ConstantTimeEq + Neg<Output = Self> + Add<Output = Self, Output = Self> + Sub<Output = Self, Output = Self> + Mul<Output = Self, Output = Self> + Sum + Product + for<'a> Add<&'a Self> + for<'a> Sub<&'a Self> + for<'a> Mul<&'a Self> + for<'a> Sum<&'a Self> + for<'a> Product<&'a Self> + AddAssign + SubAssign + MulAssign + for<'a> AddAssign<&'a Self> + for<'a> SubAssign<&'a Self> + for<'a> MulAssign<&'a Self> {
    const ZERO: Self;
    const ONE: Self;

    // Required methods
    fn random(rng: impl RngCore) -> Self;
    fn square(&self) -> Self;
    fn double(&self) -> Self;
    fn invert(&self) -> CtOption<Self>;
    fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self);

    // Provided methods
    fn is_zero(&self) -> Choice { ... }
    fn is_zero_vartime(&self) -> bool { ... }
    fn cube(&self) -> Self { ... }
    fn sqrt_alt(&self) -> (Choice, Self) { ... }
    fn sqrt(&self) -> CtOption<Self> { ... }
    fn pow<S>(&self, exp: S) -> Self
       where S: AsRef<[u64]> { ... }
    fn pow_vartime<S>(&self, exp: S) -> Self
       where S: AsRef<[u64]> { ... }
}

Expand description

This trait represents an element of a field.

Required Associated Constants§

const ZERO: Self

The zero element of the field, the additive identity.

const ONE: Self

The one element of the field, the multiplicative identity.

Required Methods§

fn random(rng: impl RngCore) -> Self

Returns an element chosen uniformly at random using a user-provided RNG.

fn square(&self) -> Self

Squares this element.

fn double(&self) -> Self

Doubles this element.

fn invert(&self) -> CtOption<Self>

Computes the multiplicative inverse of this element, failing if the element is zero.

fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self)

Computes:

$(\textsf{true}, \sqrt{\textsf{num}/\textsf{div}})$, if $\textsf{num}$ and $\textsf{div}$ are nonzero and $\textsf{num}/\textsf{div}$ is a square in the field;
$(\textsf{true}, 0)$, if $\textsf{num}$ is zero;
$(\textsf{false}, 0)$, if $\textsf{num}$ is nonzero and $\textsf{div}$ is zero;
$(\textsf{false}, \sqrt{G_S \cdot \textsf{num}/\textsf{div}})$, if $\textsf{num}$ and $\textsf{div}$ are nonzero and $\textsf{num}/\textsf{div}$ is a nonsquare in the field;

where $G_S$ is a non-square.

§Warnings

The choice of root from sqrt is unspecified.
The value of $G_S$ is unspecified, and cannot be assumed to have any specific value in a generic context.

Provided Methods§

fn is_zero(&self) -> Choice

Returns true iff this element is zero.

fn is_zero_vartime(&self) -> bool

Returns true iff this element is zero.

§Security

This method provides no constant-time guarantees. Implementors of the Field trait may optimise this method using non-constant-time logic.

fn cube(&self) -> Self

Cubes this element.

fn sqrt_alt(&self) -> (Choice, Self)

Equivalent to Self::sqrt_ratio(self, one()).

The provided method is implemented in terms of Self::sqrt_ratio.

fn sqrt(&self) -> CtOption<Self>

Returns the square root of the field element, if it is quadratic residue.

The provided method is implemented in terms of Self::sqrt_ratio.

fn pow<S>(&self, exp: S) -> Self
where S: AsRef<[u64]>,

Exponentiates self by exp, where exp is a little-endian order integer exponent.

§Guarantees

This operation is constant time with respect to self, for all exponents with the same number of digits (exp.as_ref().len()). It is variable time with respect to the number of digits in the exponent.

fn pow_vartime<S>(&self, exp: S) -> Self
where S: AsRef<[u64]>,

Exponentiates self by exp, where exp is a little-endian order integer exponent.

§Guarantees

This operation is variable time with respect to self, for all exponent. If the exponent is fixed, this operation is effectively constant time. However, for stronger constant-time guarantees, Field::pow should be used.