Struct Projective 

pub struct Projective<P>
where
    P: DOCurveConfig,
{
    pub e: <P as CurveConfig>::BaseField,
    pub z: <P as CurveConfig>::BaseField,
    pub u: <P as CurveConfig>::BaseField,
    pub t: <P as CurveConfig>::BaseField,
}

Fractional coordinates as utilised in https://doubleodd.group/doubleodd-jq.pdf, but first defined in https://www.sciencedirect.com/science/article/pii/S0020019007001433. Affine point (e,u) is represented as (E:Z:U:T) such that:

• Z != 0
• e = E/Z
• u = U/Z
• u**2 / T/Z

Fields

e: <P as CurveConfig>::BaseField

E / Z projection of the affine e

z: <P as CurveConfig>::BaseField

Projective multiplicative inverse.

u: <P as CurveConfig>::BaseField

U / Z projection of the affine u

t: <P as CurveConfig>::BaseField

Additional formula for faster addition: T = U^2/Z

Implementations

impl<P> Projective<P>

where P: DOCurveConfig,

pub const fn new_unchecked( e: <P as CurveConfig>::BaseField, z: <P as CurveConfig>::BaseField, u: <P as CurveConfig>::BaseField, t: <P as CurveConfig>::BaseField, ) -> Projective<P>

Constructs a new group element without checking whether the coordinates specify a point in the subgroup.

pub fn new( e: <P as CurveConfig>::BaseField, z: <P as CurveConfig>::BaseField, u: <P as CurveConfig>::BaseField, t: <P as CurveConfig>::BaseField, ) -> Projective<P>

Constructs a new group element in a way while enforcing that points are in the prime-order subgroup.

Trait Implementations

impl<'a, 'b, P> Add<&'a Projective<P>> for &'b Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: &'a Projective<P>) -> Projective<P>

Performs the + operation.

impl<'a, P> Add<&'a Projective<P>> for Affine<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: &'a Projective<P>) -> Projective<P>

Performs the + operation.

impl<'a, P> Add<&'a Projective<P>> for Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: &'a Projective<P>) -> Projective<P>

Performs the + operation.

impl<'a, 'b, P> Add<&'a mut Projective<P>> for &'b Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: &'a mut Projective<P>) -> Projective<P>

Performs the + operation.

impl<'a, P> Add<&'a mut Projective<P>> for Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: &'a mut Projective<P>) -> Projective<P>

Performs the + operation.

impl<'b, P> Add<Projective<P>> for &'b Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: Projective<P>) -> Projective<P>

Performs the + operation.

impl<P> Add<Projective<P>> for Affine<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: Projective<P>) -> Projective<P>

Performs the + operation.

impl<P, T> Add<T> for Projective<P>

where P: DOCurveConfig, T: Borrow<Affine<P>>,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: T) -> Projective<P>

Performs the + operation.

impl<P> Add for Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the + operator.

fn add(self, other: Projective<P>) -> Projective<P>

Performs the + operation.

impl<'a, P> AddAssign<&'a Projective<P>> for Projective<P>

where P: DOCurveConfig,

fn add_assign(&mut self, other: &'a Projective<P>)

Using Algorithm 3 from https://doubleodd.group/doubleodd-jq.pdf.

impl<'a, P> AddAssign<&'a mut Projective<P>> for Projective<P>

where P: DOCurveConfig,

fn add_assign(&mut self, other: &'a mut Projective<P>)

Performs the += operation.

impl<P, T> AddAssign<T> for Projective<P>

where P: DOCurveConfig, T: Borrow<Affine<P>>,

fn add_assign(&mut self, other: T)

Using Algorithm 3 from https://doubleodd.group/doubleodd-jq.pdf, simplified because the second point is affine (n2 = Z1, n5 = Z1*T2 + T1).

impl<P> AddAssign for Projective<P>

where P: DOCurveConfig,

fn add_assign(&mut self, other: Projective<P>)

Performs the += operation.

impl<P> AdditiveGroup for Projective<P>

where P: DOCurveConfig,

const ZERO: Projective<P>

The additive identity of the field.

type Scalar = <P as CurveConfig>::ScalarField

fn double_in_place(&mut self) -> &mut Projective<P>

Doubles self in place.

fn double(&self) -> Self

Doubles self.

fn neg_in_place(&mut self) -> &mut Self

Negates self in place.

impl<P> CanonicalDeserialize for Projective<P>

where P: DOCurveConfig,

fn deserialize_with_mode<R>( reader: R, compress: Compress, validate: Validate, ) -> Result<Projective<P>, SerializationError>

where R: Read,

The general deserialize method that takes in customization flags.

fn deserialize_compressed<R>(reader: R) -> Result<Self, SerializationError>

where R: Read,

Reads Self from reader using the compressed form if applicable. Performs validation if applicable.

fn deserialize_compressed_unchecked<R>( reader: R, ) -> Result<Self, SerializationError>

where R: Read,

Reads Self from reader using the compressed form if applicable, without validating the deserialized value.

fn deserialize_uncompressed<R>(reader: R) -> Result<Self, SerializationError>

where R: Read,

Reads Self from reader using the uncompressed form. Performs validation if applicable.

fn deserialize_uncompressed_unchecked<R>( reader: R, ) -> Result<Self, SerializationError>

where R: Read,

Reads Self from reader using the uncompressed form, without validating the deserialized value.

impl<P> CanonicalSerialize for Projective<P>

where P: DOCurveConfig,

fn serialize_with_mode<W>( &self, writer: W, compress: Compress, ) -> Result<(), SerializationError>

where W: Write,

The general serialize method that takes in customization flags.

fn serialized_size(&self, compress: Compress) -> usize

Returns the size in bytes of the serialized version of self with the given compression mode.

fn serialize_compressed<W>(&self, writer: W) -> Result<(), SerializationError>

where W: Write,

Serializes self into writer using the compressed form if applicable.

fn compressed_size(&self) -> usize

Returns the size in bytes of the compressed serialized version of self.

fn serialize_uncompressed<W>(&self, writer: W) -> Result<(), SerializationError>

where W: Write,

Serializes self into writer using the uncompressed form.

fn uncompressed_size(&self) -> usize

Returns the size in bytes of the uncompressed serialized version of self.

impl<P> Clone for Projective<P>

where P: DOCurveConfig, <P as CurveConfig>::BaseField: Copy,

fn clone(&self) -> Projective<P>

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl<P> CurveGroup for Projective<P>

where P: DOCurveConfig,

type Config = P

type BaseField = <P as CurveConfig>::BaseField

The field over which this curve is defined.

type Affine = Affine<P>

The affine representation of this element.

type FullGroup = Affine<P>

Type representing an element of the full elliptic curve group, not just the prime order subgroup.

fn normalize_batch( v: &[Projective<P>], ) -> Vec<<Projective<P> as CurveGroup>::Affine>

Normalizes a slice of group elements into affine.

fn into_affine(self) -> Self::Affine

Converts self into the affine representation.

impl<P> Debug for Projective<P>

where P: DOCurveConfig,

fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), Error>

Formats the value using the given formatter.

impl<P> Default for Projective<P>

where P: DOCurveConfig,

fn default() -> Projective<P>

Returns the “default value” for a type.

impl<P> Display for Projective<P>

where P: DOCurveConfig,

fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), Error>

Formats the value using the given formatter.

impl<P> From<Affine<P>> for Projective<P>

where P: DOCurveConfig,

fn from(p: Affine<P>) -> Projective<P>

Converts to this type from the input type.

impl<P> From<Projective<P>> for Affine<P>

where P: DOCurveConfig,

fn from(p: Projective<P>) -> Affine<P>

Converts to this type from the input type.

impl<P> Hash for Projective<P>

where P: DOCurveConfig,

fn hash<H>(&self, state: &mut H)

where H: Hasher,

Feeds this value into the given Hasher.

fn hash_slice<H>(data: &[Self], state: &mut H)

where H: Hasher, Self: Sized,

Feeds a slice of this type into the given Hasher.

impl<P, T> Mul<T> for Projective<P>

where P: DOCurveConfig, T: Borrow<<P as CurveConfig>::ScalarField>,

type Output = Projective<P>

The resulting type after applying the * operator.

fn mul(self, other: T) -> Projective<P>

Performs the * operation.

impl<P, T> MulAssign<T> for Projective<P>

where P: DOCurveConfig, T: Borrow<<P as CurveConfig>::ScalarField>,

fn mul_assign(&mut self, other: T)

Performs the *= operation.

impl<P> Neg for Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn neg(self) -> Projective<P>

Performs the unary - operation.

impl<P> PartialEq<Affine<P>> for Projective<P>

where P: DOCurveConfig,

fn eq(&self, other: &Affine<P>) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl<P> PartialEq<Projective<P>> for Affine<P>

where P: DOCurveConfig,

fn eq(&self, other: &Projective<P>) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl<P> PartialEq for Projective<P>

where P: DOCurveConfig,

fn eq(&self, other: &Projective<P>) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl<P> PrimeGroup for Projective<P>

where P: DOCurveConfig,

type ScalarField = <P as CurveConfig>::ScalarField

The scalar field F_r, where r is the order of this group.

fn generator() -> Projective<P>

Returns a fixed generator of this group.

fn mul_bigint(&self, other: impl AsRef<[u64]>) -> Projective<P>

Performs scalar multiplication of this element.

fn mul_bits_be(&self, other: impl Iterator<Item = bool>) -> Self

Computes other * self, where other is a big-endian bit representation of some integer.

impl<P> ScalarMul for Projective<P>

where P: DOCurveConfig,

const NEGATION_IS_CHEAP: bool = true

type MulBase = Affine<P>

fn batch_convert_to_mul_base( bases: &[Projective<P>], ) -> Vec<<Projective<P> as ScalarMul>::MulBase>

fn batch_mul(self, v: &[Self::ScalarField]) -> Vec<Self::MulBase>

Compute the vector v[0].G, v[1].G, …, v[n-1].G, given:

fn batch_mul_with_preprocessing( table: &BatchMulPreprocessing<Self>, v: &[Self::ScalarField], ) -> Vec<Self::MulBase>

Compute the vector v[0].G, v[1].G, …, v[n-1].G, given:

impl<'a, 'b, P> Sub<&'a Projective<P>> for &'b Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: &'a Projective<P>) -> Projective<P>

Performs the - operation.

impl<'a, P> Sub<&'a Projective<P>> for Affine<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: &'a Projective<P>) -> Projective<P>

Performs the - operation.

impl<'a, P> Sub<&'a Projective<P>> for Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: &'a Projective<P>) -> Projective<P>

Performs the - operation.

impl<'a, 'b, P> Sub<&'a mut Projective<P>> for &'b Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: &'a mut Projective<P>) -> Projective<P>

Performs the - operation.

impl<'a, P> Sub<&'a mut Projective<P>> for Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: &'a mut Projective<P>) -> Projective<P>

Performs the - operation.

impl<'b, P> Sub<Projective<P>> for &'b Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: Projective<P>) -> Projective<P>

Performs the - operation.

impl<P> Sub<Projective<P>> for Affine<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: Projective<P>) -> Projective<P>

Performs the - operation.

impl<P, T> Sub<T> for Projective<P>

where P: DOCurveConfig, T: Borrow<Affine<P>>,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: T) -> Projective<P>

Performs the - operation.

impl<P> Sub for Projective<P>

where P: DOCurveConfig,

type Output = Projective<P>

The resulting type after applying the - operator.

fn sub(self, other: Projective<P>) -> Projective<P>

Performs the - operation.

impl<'a, P> SubAssign<&'a Projective<P>> for Projective<P>

where P: DOCurveConfig,

fn sub_assign(&mut self, other: &'a Projective<P>)

Performs the -= operation.

impl<'a, P> SubAssign<&'a mut Projective<P>> for Projective<P>

where P: DOCurveConfig,

fn sub_assign(&mut self, other: &'a mut Projective<P>)

Performs the -= operation.

impl<P, T> SubAssign<T> for Projective<P>

where P: DOCurveConfig, T: Borrow<Affine<P>>,

fn sub_assign(&mut self, other: T)

Performs the -= operation.

impl<P> SubAssign for Projective<P>

where P: DOCurveConfig,

fn sub_assign(&mut self, other: Projective<P>)

Performs the -= operation.

impl<'a, P> Sum<&'a Projective<P>> for Projective<P>

where P: DOCurveConfig,

fn sum<I>(iter: I) -> Projective<P>

where I: Iterator<Item = &'a Projective<P>>,

Takes an iterator and generates Self from the elements by “summing up” the items.

impl<P, T> Sum<T> for Projective<P>

where P: DOCurveConfig, T: Borrow<Affine<P>>,

fn sum<I>(iter: I) -> Projective<P>

where I: Iterator<Item = T>,

Takes an iterator and generates Self from the elements by “summing up” the items.

impl<P> Sum for Projective<P>

where P: DOCurveConfig,

fn sum<I>(iter: I) -> Projective<P>

where I: Iterator<Item = Projective<P>>,

Takes an iterator and generates Self from the elements by “summing up” the items.

impl<M, ConstraintF> ToConstraintField<ConstraintF> for Projective<M>

where M: DOCurveConfig, ConstraintF: Field, <M as CurveConfig>::BaseField: ToConstraintField<ConstraintF>,

fn to_field_elements(&self) -> Option<Vec<ConstraintF>>

impl<P> Valid for Projective<P>

where P: DOCurveConfig,

fn check(&self) -> Result<(), SerializationError>

Checks whether self is valid. If self is valid, returns Ok(()). Otherwise, returns an error describing the failure. This method is called by deserialize_with_mode if validate is Validate::Yes.

fn batch_check<'a>( batch: impl Iterator<Item = &'a Projective<P>> + Send, ) -> Result<(), SerializationError>

where Projective<P>: 'a,

Checks whether all items in batch are valid. If all items are valid, returns Ok(()). Otherwise, returns an error describing the first failure.

const TRIVIAL_CHECK: bool = false

Whether the check method is trivial (i.e. always returns Ok(())). If this is true, the batch_check method will skip all checks and return Ok(()). This should be set to true for types where check is trivial, e.g. integers, field elements, etc. This is false by default. This is primarily an optimization to skip unnecessary checks in batch_check.

impl<P> VariableBaseMSM for Projective<P>

where P: DOCurveConfig,

const ZERO_BUCKET: Projective<P> = Self::ZERO

type Bucket = Projective<P>

fn msm( bases: &[<Projective<P> as ScalarMul>::MulBase], bigints: &[<Projective<P> as PrimeGroup>::ScalarField], ) -> Result<Projective<P>, usize>

Performs multi-scalar multiplication.

fn msm_unchecked(bases: &[Self::MulBase], scalars: &[Self::ScalarField]) -> Self

Computes an inner product between the PrimeField elements in scalars and the corresponding group elements in bases.

fn msm_bigint( bases: &[Self::MulBase], bigints: &[<Self::ScalarField as PrimeField>::BigInt], ) -> Self

Optimized implementation of multi-scalar multiplication.

fn msm_u1(bases: &[Self::MulBase], scalars: &[bool]) -> Self

Performs multi-scalar multiplication when the scalars are known to be boolean. The default implementation is faster than Self::msm_bigint.

fn msm_u8(bases: &[Self::MulBase], scalars: &[u8]) -> Self

Performs multi-scalar multiplication when the scalars are known to be u8-sized. The default implementation is faster than Self::msm_bigint.

fn msm_u16(bases: &[Self::MulBase], scalars: &[u16]) -> Self

Performs multi-scalar multiplication when the scalars are known to be u16-sized. The default implementation is faster than Self::msm_bigint.

fn msm_u32(bases: &[Self::MulBase], scalars: &[u32]) -> Self

Performs multi-scalar multiplication when the scalars are known to be u32-sized. The default implementation is faster than Self::msm_bigint.

fn msm_u64(bases: &[Self::MulBase], scalars: &[u64]) -> Self

Performs multi-scalar multiplication when the scalars are known to be u64-sized. The default implementation is faster than Self::msm_bigint.

fn msm_chunks<I, J>(bases_stream: &J, scalars_stream: &I) -> Self

where I: Iterable + ?Sized, <I as Iterable>::Item: Borrow<Self::ScalarField>, J: Iterable, <J as Iterable>::Item: Borrow<Self::MulBase>,

Streaming multi-scalar multiplication algorithm with hard-coded chunk size.

impl<P> Zero for Projective<P>

where P: DOCurveConfig,

fn zero() -> Projective<P>

Returns one of the representants for the identity, namely the point-at-infinity (1,1,0,0).

The other representant N=(-1,1,0,0) of the identity could also be returned, but the implementation of formulas only requires one representant.

fn is_zero(&self) -> bool

Returns true if self is equal to the additive identity.

fn set_zero(&mut self)

Sets self to the additive identity element of Self, 0.

impl<P> Zeroize for Projective<P>

where P: DOCurveConfig,

fn zeroize(&mut self)

Zero out this object from memory using Rust intrinsics which ensure the zeroization operation is not “optimized away” by the compiler.

impl<P> Copy for Projective<P>

where P: DOCurveConfig, <P as CurveConfig>::BaseField: Copy,

impl<P> Eq for Projective<P>

where P: DOCurveConfig,