Struct Scalar 

pub struct Scalar(pub(super) MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>);

Element in the NIST P-521 scalar field modulo n.

Trait impls

Much of the important functionality is provided by traits from the [ff] crate:

• [Field] represents elements of finite fields and provides:
  • [Field::random] generate a random field element
  • double, square, and invert operations
  • Bounds for Add, Sub, Mul, and Neg (and *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.

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

Tuple Fields

0: MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>

Implementations

impl Scalar

pub const ZERO: Self

Zero element.

pub const ONE: Self

Multiplicative identity.

pub fn from_bytes( repr: &MontyFieldBytes<ScalarParams, { ScalarParams::LIMBS }>, ) -> CtOption<Self>

Create a Scalar from a canonical byte representation using the field’s configured byte order.

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

Decode Scalar from a byte slice using the field’s configured byte order.

pub const fn from_hex_vartime(hex: &str) -> Self

Decode a Scalar from hex-encoded bytes using the field’s configured byte order.

This is primarily intended for defining constants using hex literals.

Panics

• When hex is malformed
• When input is the wrong length
• If input overflows the modulus

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) -> MontyFieldBytes<ScalarParams, { ScalarParams::LIMBS }>

Returns the big-endian encoding of this Scalar.

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 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.

impl Scalar

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

Decode Scalar from Uint 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 const fn to_canonical(self) -> Uint

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

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 multiply(&self, rhs: &Self) -> Self

Multiply elements.

pub const fn neg(&self) -> Self

Negate element.

pub const fn square(&self) -> Self

Compute modular square.

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

Compute Scalar inversion: 1 / self.

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

Compute Scalar inversion: 1 / self in variable-time.

pub const fn const_invert(&self) -> Self

Compute field inversion as a const fn. Panics if self is zero.

This is mainly intended for inverting constants at compile 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<{ <$params>::LIMBS }> for Scalar

const LIMBS: usize = ScalarParams::LIMBS

Number of limbs required to encode the Montgomery form

const PARAMS: FixedMontyParams<{ 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 ConstantTimeGreater for Scalar

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

Determine whether self > other.

impl ConstantTimeLess for Scalar

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

Determine whether self < other.

impl CtEq for Scalar

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

Determine if self is equal to other in constant-time.

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

Determine if self is NOT equal to other in constant-time.

impl CtGt for Scalar

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

Compute whether self > other in constant time.

impl CtLt for Scalar

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

Compute whether self < other in constant time.

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_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 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_random<R: TryRng + ?Sized>(rng: &mut R) -> Result<Self, R::Error>

Returns an element chosen uniformly at random using a user-provided fallible 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: Rng + ?Sized,

Returns an element chosen uniformly at random using a user-provided infallible 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 MontyFieldBytes<ScalarParams, { ScalarParams::LIMBS }>

fn from(fe: &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(fe: &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<MontyFieldElement<ScalarParams, { <$params>::LIMBS }>> for Scalar

fn from(fe: MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>) -> 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 MontyFieldBytes<ScalarParams, { ScalarParams::LIMBS }>

fn from(fe: Scalar) -> Self

Converts to this type from the input type.

impl From<Scalar> for MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>

fn from(fe: Scalar) -> MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>

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(fe: 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 Generate for Scalar

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

Generate random key using the provided TryCryptoRng.

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

where R: CryptoRng + ?Sized,

Generate random key using the provided CryptoRng.

fn try_generate() -> Result<Self, Error>

Randomly generate a value of this type using the system’s ambient cryptographically secure random number generator.

fn generate() -> Self

Randomly generate a value of this type using the system’s ambient cryptographically secure random number generator.

impl Invert for Scalar

type Output = CtOption<Scalar>

Output of the inversion.

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

Computes the inverse.

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

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 MulVartime<&<NistP521 as CurveArithmetic>::AffinePoint> for &Scalar

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

Multiply self by rhs in variable-time.

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

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

Multiply self by rhs in variable-time.

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

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

Multiply self by rhs in variable-time.

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

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

Multiply self by rhs in variable-time.

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

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

Multiply self by rhs in variable-time.

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

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

Multiply self by rhs in variable-time.

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

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

Multiply self by rhs in variable-time.

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

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

Multiply self by rhs in variable-time.

impl Neg for &Scalar

type Output = Scalar

The resulting type after applying the - operator.

fn neg(self) -> Scalar

Performs the unary - 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

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

const NUM_BITS: u32 = ::primefield::MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>::NUM_BITS

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

const CAPACITY: u32 = ::primefield::MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>::CAPACITY

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

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 = ::primefield::MontyFieldElement<ScalarParams, { ScalarParams::LIMBS }>::S

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, <ScalarParams as MontyFieldParams<{ <$params>::LIMBS }>>::ByteSize>

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

fn from_repr(bytes: Self::Repr) -> 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) -> Self::Repr

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 PrimeFieldExt for Scalar

const REPR_ENDIANNESS: ByteOrder

Endianness used when encoding ff::PrimeField::Repr.

fn to_be_repr(&self) -> Self::Repr

Encode self using a big endian representation.

fn to_le_repr(&self) -> Self::Repr

Encode self using a little endian representation.

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

fn reduce(w: &Uint) -> Self

Reduces self modulo Modulus.

impl Retrieve for Scalar

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

The original type.

fn retrieve(&self) -> Uint

Convert the number back from the optimized representation.

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: &Uint) -> Result<Self>

Performs the conversion.

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

impl FieldExt for Scalar