Struct Scalar 

pub struct Scalar(/* private fields */);

Available on crate feature arithmetic only.

Scalars are elements in the finite field modulo n.

Trait impls

Much of the important functionality of scalars is provided by traits from the ff crate, which is re-exported as p521::elliptic_curve::ff:

• Field - represents elements of finite fields and provides:
  • Field::random - generate a random scalar
  • double, square, and invert operations
  • Bounds for Add, Sub, Mul, and Neg (as well as *Assign equivalents)
  • Bounds for ConditionallySelectable from the subtle crate
• PrimeField - represents elements of prime fields and provides:
  • from_repr/to_repr for converting field elements from/to big integers.
  • multiplicative_generator and root_of_unity constants.
• PrimeFieldBits - operations over field elements represented as bits (requires bits feature)

Please see the documentation for the relevant traits for more information.

Implementations

impl Scalar

pub const ZERO: Self

Zero element.

pub const ONE: Self

Multiplicative identity.

pub fn from_bytes(repr: &FieldBytes) -> CtOption<Self>

Create a Scalar from a canonical big-endian representation.

pub fn from_slice(slice: &[u8]) -> Result<Self>

Decode Scalar from a big endian byte slice.

pub fn from_uint(uint: Uint) -> CtOption<Self>

Decode Scalar from Uint converting it into Montgomery form:

w * R^2 * R^-1 mod p = wR mod p

pub const fn from_u64(w: u64) -> Self

Convert a u64 into a Scalar.

pub fn to_bytes(self) -> FieldBytes

Returns the big-endian encoding of this Scalar.

pub const fn to_canonical(self) -> Uint

Available on 64-bit only.

Translate Scalar out of the Montgomery domain, returning a Uint in canonical form.

pub fn is_odd(&self) -> Choice

Determine if this Scalar is odd in the SEC1 sense: self mod 2 == 1.

Returns

If odd, return Choice(1). Otherwise, return Choice(0).

pub fn is_even(&self) -> Choice

Determine if this Scalar is even in the SEC1 sense: self mod 2 == 0.

Returns

If even, return Choice(1). Otherwise, return Choice(0).

pub fn is_zero(&self) -> Choice

Determine if this Scalar is zero.

Returns

If zero, return Choice(1). Otherwise, return Choice(0).

pub const fn add(&self, rhs: &Self) -> Self

Add elements.

pub const fn double(&self) -> Self

Double element (add it to itself).

pub const fn sub(&self, rhs: &Self) -> Self

Subtract elements.

pub const fn neg(&self) -> Self

Negate element.

pub const fn multiply(&self, rhs: &Self) -> Self

Multiply elements.

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

Compute Scalar inversion: 1 / self.

pub const fn square(&self) -> Self

Compute modular square.

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

Compute modular square root.

pub const fn pow_vartime<const RHS_LIMBS: usize>( &self, exp: &Uint<RHS_LIMBS>, ) -> Self

Returns self^exp, where exp is a little-endian integer exponent.

This operation is variable time with respect to the exponent exp.

If the exponent is fixed, this operation is constant time.

pub const fn sqn_vartime(&self, n: usize) -> Self

Returns self^(2^n) mod p.

This operation is variable time with respect to the exponent n.

If the exponent is fixed, this operation is constant time.

Trait Implementations

impl Add<&Scalar> for &Scalar

type Output = Scalar

The resulting type after applying the + operator.

fn add(self, rhs: &Scalar) -> Scalar

Performs the + operation.

impl Add<&Scalar> for Scalar

type Output = Scalar

The resulting type after applying the + operator.

fn add(self, rhs: &Scalar) -> Scalar

Performs the + operation.

impl Add for Scalar

type Output = Scalar

The resulting type after applying the + operator.

fn add(self, rhs: Scalar) -> Scalar

Performs the + operation.

impl AddAssign<&Scalar> for Scalar

fn add_assign(&mut self, other: &Scalar)

Performs the += operation.

impl AddAssign for Scalar

fn add_assign(&mut self, other: Scalar)

Performs the += operation.

impl AsRef<Scalar> for Scalar

fn as_ref(&self) -> &Scalar

Converts this type into a shared reference of the (usually inferred) input type.

impl Clone for Scalar

fn clone(&self) -> Scalar

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl ConditionallySelectable for Scalar

fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self

Select a or b according to choice.

fn conditional_assign(&mut self, other: &Self, choice: Choice)

Conditionally assign other to self, according to choice.

fn conditional_swap(a: &mut Self, b: &mut Self, choice: Choice)

Conditionally swap self and other if choice == 1; otherwise, reassign both unto themselves.

impl ConstMontyParams<{ Uint::LIMBS }> for Scalar

const LIMBS: usize = 9usize

Number of limbs required to encode the Montgomery form

const PARAMS: MontyParams<{ Uint::LIMBS }> = ScalarParams::PARAMS

Montgomery parameters constant.

impl ConstantTimeEq for Scalar

fn ct_eq(&self, other: &Self) -> Choice

Determine if two items are equal.

fn ct_ne(&self, other: &Self) -> Choice

Determine if two items are NOT equal.

impl CtEq for Scalar

fn ct_eq(&self, other: &Self) -> Choice

Equality

fn ct_ne(&self, other: &Rhs) -> Choice

Inequality

impl CtSelect for Scalar

fn ct_select(&self, other: &Self, choice: Choice) -> Self

Select between self and other based on choice, returning a copy of the value.

fn ct_assign(&mut self, other: &Self, choice: Choice)

Conditionally assign other to self if choice is Choice::TRUE.

fn ct_swap(&mut self, other: &mut Self, choice: Choice)

Conditionally swap self and other if choice is Choice::TRUE.

impl Debug for Scalar

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

Formats the value using the given formatter.

impl Default for Scalar

fn default() -> Self

Returns the “default value” for a type.

impl<'de> Deserialize<'de> for Scalar

Available on crate feature serde only.

fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>

where D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl Field for Scalar

const ZERO: Self = Self::ZERO

The zero element of the field, the additive identity.

const ONE: Self = Self::ONE

The one element of the field, the multiplicative identity.

fn try_from_rng<R: TryRngCore + ?Sized>(rng: &mut R) -> Result<Self, R::Error>

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

fn is_zero(&self) -> Choice

Returns true iff this element is zero.

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(&self) -> CtOption<Self>

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

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

Computes:

fn random<R>(rng: &mut R) -> Self

where R: RngCore + ?Sized,

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

fn is_zero_vartime(&self) -> bool

Returns true iff this element is zero.

fn cube(&self) -> Self

Cubes this element.

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

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

fn pow<S>(&self, exp: S) -> Self

where S: AsRef<[u64]>,

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

impl From<&NonZeroScalar<NistP521>> for Scalar

fn from(scalar: &NonZeroScalar<NistP521>) -> Self

Converts to this type from the input type.

impl From<&Scalar> for FieldBytes

fn from(scalar: &Scalar) -> Self

Converts to this type from the input type.

impl From<&Scalar> for ScalarValue<NistP521>

fn from(scalar: &Scalar) -> ScalarValue<NistP521>

Converts to this type from the input type.

impl From<&Scalar> for U576

fn from(scalar: &Scalar) -> Uint

Converts to this type from the input type.

impl From<&ScalarValue<NistP521>> for Scalar

fn from(w: &ScalarValue<NistP521>) -> Scalar

Converts to this type from the input type.

impl From<&SecretKey<NistP521>> for Scalar

fn from(secret_key: &SecretKey<NistP521>) -> Scalar

Converts to this type from the input type.

impl From<NonZeroScalar<NistP521>> for Scalar

fn from(scalar: NonZeroScalar<NistP521>) -> Self

Converts to this type from the input type.

impl From<Scalar> for FieldBytes

fn from(scalar: Scalar) -> Self

Converts to this type from the input type.

impl From<Scalar> for ScalarValue<NistP521>

fn from(scalar: Scalar) -> ScalarValue<NistP521>

Converts to this type from the input type.

impl From<Scalar> for U576

fn from(scalar: Scalar) -> Uint

Converts to this type from the input type.

impl From<ScalarValue<NistP521>> for Scalar

fn from(w: ScalarValue<NistP521>) -> Self

Converts to this type from the input type.

impl From<u128> for Scalar

fn from(n: u128) -> Scalar

Converts to this type from the input type.

impl From<u32> for Scalar

fn from(n: u32) -> Scalar

Converts to this type from the input type.

impl From<u64> for Scalar

fn from(n: u64) -> Scalar

Converts to this type from the input type.

impl FromUintUnchecked for Scalar

type Uint = Uint<crypto_bigint::::uint::U576::{constant#0}>

Unsigned integer type (i.e. Curve::Uint)

fn from_uint_unchecked(uint: Self::Uint) -> Self

Instantiate scalar from an unsigned integer without checking whether the value overflows the field modulus.

impl Invert for Scalar

type Output = CtOption<Scalar>

Output of the inversion.

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

Computes the inverse.

fn invert_vartime(&self) -> Self::Output

Computes the inverse in variable-time.

impl IsHigh for Scalar

fn is_high(&self) -> Choice

Is this scalar greater than n / 2?

impl Mul<&<NistP521 as CurveArithmetic>::AffinePoint> for &Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: &AffinePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul<&<NistP521 as CurveArithmetic>::AffinePoint> for Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: &AffinePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul<&<NistP521 as CurveArithmetic>::ProjectivePoint> for &Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: &ProjectivePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul<&<NistP521 as CurveArithmetic>::ProjectivePoint> for Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: &ProjectivePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul<&Scalar> for &Scalar

type Output = Scalar

The resulting type after applying the * operator.

fn mul(self, rhs: &Scalar) -> Scalar

Performs the * operation.

impl Mul<&Scalar> for Scalar

type Output = Scalar

The resulting type after applying the * operator.

fn mul(self, rhs: &Scalar) -> Scalar

Performs the * operation.

impl Mul<<NistP521 as CurveArithmetic>::AffinePoint> for &Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: AffinePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul<<NistP521 as CurveArithmetic>::AffinePoint> for Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: AffinePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul<<NistP521 as CurveArithmetic>::ProjectivePoint> for &Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: ProjectivePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul<<NistP521 as CurveArithmetic>::ProjectivePoint> for Scalar

type Output = <NistP521 as CurveArithmetic>::ProjectivePoint

The resulting type after applying the * operator.

fn mul(self, rhs: ProjectivePoint<NistP521>) -> ProjectivePoint<NistP521>

Performs the * operation.

impl Mul for Scalar

type Output = Scalar

The resulting type after applying the * operator.

fn mul(self, rhs: Scalar) -> Scalar

Performs the * operation.

impl MulAssign<&Scalar> for Scalar

fn mul_assign(&mut self, other: &Scalar)

Performs the *= operation.

impl MulAssign for Scalar

fn mul_assign(&mut self, other: Scalar)

Performs the *= operation.

impl Neg for Scalar

type Output = Scalar

The resulting type after applying the - operator.

fn neg(self) -> Scalar

Performs the unary - operation.

impl Ord for Scalar

fn cmp(&self, other: &Self) -> Ordering

This method returns an Ordering between self and other.

fn max(self, other: Self) -> Self

where Self: Sized,

Compares and returns the maximum of two values.

fn min(self, other: Self) -> Self

where Self: Sized,

Compares and returns the minimum of two values.

fn clamp(self, min: Self, max: Self) -> Self

where Self: Sized,

Restrict a value to a certain interval.

impl PartialEq for Scalar

fn eq(&self, rhs: &Self) -> 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 PartialOrd for Scalar

fn partial_cmp(&self, other: &Self) -> Option<Ordering>

This method returns an ordering between self and other values if one exists.

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

Tests less than (for self and other) and is used by the < operator.

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

Tests less than or equal to (for self and other) and is used by the <= operator.

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

Tests greater than (for self and other) and is used by the > operator.

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

Tests greater than or equal to (for self and other) and is used by the >= operator.

impl PrimeField for Scalar

const MODULUS: &'static str = ORDER_HEX

Modulus of the field written as a string for debugging purposes.

const CAPACITY: u32 = 520u32

How many bits of information can be reliably stored in the field element.

const NUM_BITS: u32 = 521u32

How many bits are needed to represent an element of this field.

const TWO_INV: Self

Inverse of $2$ in the field.

const MULTIPLICATIVE_GENERATOR: Self

A fixed multiplicative generator of modulus - 1 order. This element must also be a quadratic nonresidue.

const S: u32 = 3u32

An integer s satisfying the equation 2^s * t = modulus - 1 with t odd.

const ROOT_OF_UNITY: Self

The 2^s root of unity.

const ROOT_OF_UNITY_INV: Self

Inverse of Self::ROOT_OF_UNITY.

const DELTA: Self

Generator of the t-order multiplicative subgroup.

type Repr = Array<u8, <NistP521 as Curve>::FieldBytesSize>

The prime field can be converted back and forth into this binary representation.

fn from_repr(bytes: FieldBytes) -> CtOption<Self>

Attempts to convert a byte representation of a field element into an element of this prime field, failing if the input is not canonical (is not smaller than the field’s modulus).

fn to_repr(&self) -> FieldBytes

Converts an element of the prime field into the standard byte representation for this field.

fn is_odd(&self) -> Choice

Returns true iff this element is odd.

fn from_str_vartime(s: &str) -> Option<Self>

Interpret a string of numbers as a (congruent) prime field element. Does not accept unnecessary leading zeroes or a blank string.

fn from_u128(v: u128) -> Self

Obtains a field element congruent to the integer v.

fn from_repr_vartime(repr: Self::Repr) -> Option<Self>

Attempts to convert a byte representation of a field element into an element of this prime field, failing if the input is not canonical (is not smaller than the field’s modulus).

fn is_even(&self) -> Choice

Returns true iff this element is even.

impl<'a> Product<&'a Scalar> for Scalar

fn product<I: Iterator<Item = &'a Self>>(iter: I) -> Self

Takes an iterator and generates Self from the elements by multiplying the items.

impl Product for Scalar

fn product<I: Iterator<Item = Self>>(iter: I) -> Self

Takes an iterator and generates Self from the elements by multiplying the items.

impl Reduce<Array<u8, <NistP521 as Curve>::FieldBytesSize>> for Scalar

fn reduce(bytes: &FieldBytes) -> Self

Reduces self modulo Modulus.

impl Reduce<Array<u8, UInt<UInt<UInt<UInt<UInt<UInt<UInt<UTerm, B1>, B1>, B0>, B0>, B0>, B1>, B0>>> for Scalar

Available on crate feature hash2curve only.

fn reduce(value: &Array<u8, U98>) -> Self

Reduces self modulo Modulus.

impl Reduce<Uint<crypto_bigint::::uint::U576::{constant#0}>> for Scalar

fn reduce(w: &Uint) -> Self

Reduces self modulo Modulus.

impl ReduceNonZero<Array<u8, <NistP521 as Curve>::FieldBytesSize>> for Scalar

fn reduce_nonzero(bytes: &FieldBytes) -> Self

Perform a modular reduction, returning a field element.

impl ReduceNonZero<Uint<crypto_bigint::::uint::U576::{constant#0}>> for Scalar

fn reduce_nonzero(w: &Uint) -> Self

Perform a modular reduction, returning a field element.

impl Serialize for Scalar

Available on crate feature serde only.

fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>

where S: Serializer,

Serialize this value into the given Serde serializer.

impl Sub<&Scalar> for &Scalar

type Output = Scalar

The resulting type after applying the - operator.

fn sub(self, rhs: &Scalar) -> Scalar

Performs the - operation.

impl Sub<&Scalar> for Scalar

type Output = Scalar

The resulting type after applying the - operator.

fn sub(self, rhs: &Scalar) -> Scalar

Performs the - operation.

impl Sub for Scalar

type Output = Scalar

The resulting type after applying the - operator.

fn sub(self, rhs: Scalar) -> Scalar

Performs the - operation.

impl SubAssign<&Scalar> for Scalar

fn sub_assign(&mut self, other: &Scalar)

Performs the -= operation.

impl SubAssign for Scalar

fn sub_assign(&mut self, other: Scalar)

Performs the -= operation.

impl<'a> Sum<&'a Scalar> for Scalar

fn sum<I: Iterator<Item = &'a Scalar>>(iter: I) -> Self

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

impl Sum for Scalar

fn sum<I: Iterator<Item = Self>>(iter: I) -> Self

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

impl TryFrom<Scalar> for NonZeroScalar<NistP521>

The constant-time alternative is available at elliptic_curve::NonZeroScalar<NistP521>::new().

type Error = Error

The type returned in the event of a conversion error.

fn try_from(scalar: Scalar) -> Result<Self>

Performs the conversion.

impl TryFrom<Uint<crypto_bigint::::uint::U576::{constant#0}>> for Scalar

type Error = Error

The type returned in the event of a conversion error.

fn try_from(w: Uint) -> Result<Self>

Performs the conversion.

impl Copy for Scalar

impl DefaultIsZeroes for Scalar

impl Eq for Scalar