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: U576) -> CtOption<Self>

Decode Scalar from U576 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) -> U576

Available on 32-bit only.

Translate Scalar out of the Montgomery domain, returning a U576 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(&self, exp: &[u64]) -> Self

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

This operation is variable time with respect to the exponent.

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

pub const fn shr_vartime(&self, shift: usize) -> Scalar

Available on 32-bit only.

Right shifts the scalar.

Note: not constant-time with respect to the shift parameter.

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<[u64; 9]> for Scalar

fn as_ref(&self) -> &[u64; 9]

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

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 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 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 random(rng: impl RngCore) -> Self

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 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<&Scalar> for FieldBytes

fn from(scalar: &Scalar) -> Self

Converts to this type from the input type.

impl From<&Scalar> for ScalarPrimitive<NistP521>

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

Converts to this type from the input type.

impl From<&Scalar> for U576

fn from(scalar: &Scalar) -> U576

Converts to this type from the input type.

impl From<&ScalarPrimitive<NistP521>> for Scalar

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

Converts to this type from the input type.

impl From<&SecretKey<NistP521>> for Scalar

fn from(secret_key: &SecretKey) -> Scalar

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 ScalarPrimitive<NistP521>

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

Converts to this type from the input type.

impl From<Scalar> for U576

fn from(scalar: Scalar) -> U576

Converts to this type from the input type.

impl From<ScalarPrimitive<NistP521>> for Scalar

fn from(w: ScalarPrimitive<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 FromOkm for Scalar

Available on crate feature hash2curve only.

type Length = UInt<UInt<UInt<UInt<UInt<UInt<UInt<UTerm, B1>, B1>, B0>, B0>, B0>, B1>, B0>

The number of bytes needed to convert to a field element.

fn from_okm(data: &GenericArray<u8, Self::Length>) -> Self

Convert a byte sequence into a field element.

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>

Field element type

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

Invert a field element.

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

Invert a field element in variable time.

impl IsHigh for Scalar

fn is_high(&self) -> Choice

Is this scalar greater than or equal to n / 2?

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 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: &Scalar) -> 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: &Scalar) -> 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 = "01fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386409"

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 = GenericArray<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<Uint<crypto_bigint::::uint::U576::{constant#0}>> for Scalar

type Bytes = GenericArray<u8, <NistP521 as Curve>::FieldBytesSize>

Bytes used as input to Reduce::reduce_bytes.

fn reduce(w: U576) -> Self

Perform a modular reduction, returning a field element.

fn reduce_bytes(bytes: &FieldBytes) -> Self

Interpret the given bytes as an integer and perform a modular reduction.

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 Shr<usize> for &Scalar

type Output = Scalar

The resulting type after applying the >> operator.

fn shr(self, rhs: usize) -> Self::Output

Performs the >> operation.

impl Shr<usize> for Scalar

type Output = Scalar

The resulting type after applying the >> operator.

fn shr(self, rhs: usize) -> Self::Output

Performs the >> operation.

impl ShrAssign<usize> for Scalar

fn shr_assign(&mut self, rhs: usize)

Performs the >>= operation.

impl SignPrimitive<NistP521> for Scalar

Available on crate features ecdsa-core and ecdsa only.

fn try_sign_prehashed<K>( &self, k: K, z: &GenericArray<u8, <C as Curve>::FieldBytesSize>, ) -> Result<(Signature<C>, Option<RecoveryId>), Error>

where K: AsRef<Self> + Invert<Output = CtOption<Self>>,

Try to sign the prehashed message.

fn try_sign_prehashed_rfc6979<D>( &self, z: &GenericArray<u8, <C as Curve>::FieldBytesSize>, ad: &[u8], ) -> Result<(Signature<C>, Option<RecoveryId>), Error>

where Self: From<ScalarPrimitive<C>> + Invert<Output = CtOption<Self>>, D: Digest<OutputSize = <C as Curve>::FieldBytesSize> + BlockSizeUser + FixedOutput + FixedOutputReset,

Try to sign the given message digest deterministically using the method described in RFC6979 for computing ECDSA ephemeral scalar k.

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<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: U576) -> Result<Self>

Performs the conversion.

impl Copy for Scalar

impl DefaultIsZeroes for Scalar

impl Eq for Scalar