Struct halo2_proofs::poly::EvaluationDomain

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pub struct EvaluationDomain<F: Field> { /* private fields */ }

Expand description

This structure contains precomputed constants and other details needed for performing operations on an evaluation domain of size $2^k$ and an extended domain of size $2^{k} * j$ with $j \neq 0$.

Implementations§

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impl<F: WithSmallOrderMulGroup<3>> EvaluationDomain<F>

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pub fn new(j: u32, k: u32) -> Self

This constructs a new evaluation domain object based on the provided values $j, k$.

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pub fn lagrange_from_vec(&self, values: Vec<F>) -> Polynomial<F, LagrangeCoeff>

Obtains a polynomial in Lagrange form when given a vector of Lagrange coefficients of size n; panics if the provided vector is the wrong length.

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pub fn coeff_from_vec(&self, values: Vec<F>) -> Polynomial<F, Coeff>

Obtains a polynomial in coefficient form when given a vector of coefficients of size n; panics if the provided vector is the wrong length.

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pub fn empty_coeff(&self) -> Polynomial<F, Coeff>

Returns an empty (zero) polynomial in the coefficient basis

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pub fn empty_lagrange(&self) -> Polynomial<F, LagrangeCoeff>

Returns an empty (zero) polynomial in the Lagrange coefficient basis

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pub fn constant_lagrange(&self, scalar: F) -> Polynomial<F, LagrangeCoeff>

Returns a constant polynomial in the Lagrange coefficient basis

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pub fn empty_extended(&self) -> Polynomial<F, ExtendedLagrangeCoeff>

Returns an empty (zero) polynomial in the extended Lagrange coefficient basis

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pub fn constant_extended( &self, scalar: F ) -> Polynomial<F, ExtendedLagrangeCoeff>

Returns a constant polynomial in the extended Lagrange coefficient basis

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pub fn lagrange_to_coeff( &self, a: Polynomial<F, LagrangeCoeff> ) -> Polynomial<F, Coeff>

This takes us from an n-length vector into the coefficient form.

This function will panic if the provided vector is not the correct length.

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pub fn coeff_to_extended( &self, a: Polynomial<F, Coeff> ) -> Polynomial<F, ExtendedLagrangeCoeff>

This takes us from an n-length coefficient vector into a coset of the extended evaluation domain, rotating by rotation if desired.

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pub fn rotate_extended( &self, poly: &Polynomial<F, ExtendedLagrangeCoeff>, rotation: Rotation ) -> Polynomial<F, ExtendedLagrangeCoeff>

Rotate the extended domain polynomial over the original domain.

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pub fn extended_to_coeff( &self, a: Polynomial<F, ExtendedLagrangeCoeff> ) -> Vec<F> ⓘ

This takes us from the extended evaluation domain and gets us the quotient polynomial coefficients.

This function will panic if the provided vector is not the correct length.

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pub fn divide_by_vanishing_poly( &self, a: Polynomial<F, ExtendedLagrangeCoeff> ) -> Polynomial<F, ExtendedLagrangeCoeff>

This divides the polynomial (in the extended domain) by the vanishing polynomial of the $2^k$ size domain.

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pub fn extended_len(&self) -> usize

Get the size of the extended domain

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pub fn get_omega(&self) -> F

Get $\omega$, the generator of the $2^k$ order multiplicative subgroup.

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pub fn get_omega_inv(&self) -> F

Get $\omega^{-1}$, the inverse of the generator of the $2^k$ order multiplicative subgroup.

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pub fn get_extended_omega(&self) -> F

Get the generator of the extended domain’s multiplicative subgroup.

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pub fn rotate_omega(&self, value: F, rotation: Rotation) -> F

Multiplies a value by some power of $\omega$, essentially rotating over the domain.

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pub fn l_i_range<I: IntoIterator<Item = i32> + Clone>( &self, x: F, xn: F, rotations: I ) -> Vec<F> ⓘ

Computes evaluations (at the point x, where xn = x^n) of Lagrange basis polynomials l_i(X) defined such that l_i(omega^i) = 1 and l_i(omega^j) = 0 for all j != i at each provided rotation i.

Implementation

The polynomial $$\prod_{j=0,j \neq i}^{n - 1} (X - \omega^j)$$ has a root at all points in the domain except $\omega^i$, where it evaluates to $$\prod_{j=0,j \neq i}^{n - 1} (\omega^i - \omega^j)$$ and so we divide that polynomial by this value to obtain $l_i(X)$. Since $$\prod_{j=0,j \neq i}^{n - 1} (X - \omega^j) = \frac{X^n - 1}{X - \omega^i}$$ then $l_i(x)$ for some $x$ is evaluated as $$\left(\frac{x^n - 1}{x - \omega^i}\right) \cdot \left(\frac{1}{\prod_{j=0,j \neq i}^{n - 1} (\omega^i - \omega^j)}\right).$$ We refer to $$1 \over \prod_{j=0,j \neq i}^{n - 1} (\omega^i - \omega^j)$$ as the barycentric weight of $\omega^i$.

We know that for $i = 0$ $$\frac{1}{\prod_{j=0,j \neq i}^{n - 1} (\omega^i - \omega^j)} = \frac{1}{n}.$$

If we multiply $(1 / n)$ by $\omega^i$ then we obtain $$\frac{1}{\prod_{j=0,j \neq 0}^{n - 1} (\omega^i - \omega^j)} = \frac{1}{\prod_{j=0,j \neq i}^{n - 1} (\omega^i - \omega^j)}$$ which is the barycentric weight of $\omega^i$.

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