Struct Scalar 

pub struct Scalar([u64; 9]);

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.

Tuple Fields

0: [u64; 9]

Implementations

impl Scalar

pub const ZERO: Self

Zero element.

pub const ONE: Self

Multiplicative identity.

const BYTES: usize = 66usize

Number of bytes in the serialized representation.

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(crate) const fn from_hex(hex: &str) -> Self

Parse a Scalar from big endian hex-encoded bytes.

Does not perform a check that the field element does not overflow the order.

This method is primarily intended for defining internal constants.

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

Convert a u64 into a Scalar.

pub(crate) const fn from_uint_unchecked(w: U576) -> Self

Decode Scalar from U576 converting it into Montgomery form.

Does not perform a check that the field element does not overflow the order.

Used incorrectly this can lead to invalid results!

pub fn to_bytes(self) -> FieldBytes

Returns the big-endian encoding of this Scalar.

pub const fn to_canonical(self) -> U576

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.

const fn invert_unchecked(&self) -> Self

Compute Scalar inversion: 1 / self.

Does not check that self is non-zero.

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

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 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 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 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

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