//! Traits provided by this crate

mod sealed;

pub use num_traits::{
    ConstZero, WrappingAdd, WrappingMul, WrappingNeg, WrappingShl, WrappingShr, WrappingSub,
};

pub(crate) use sealed::PrecomputeInverterWithAdjuster;

use crate::{Limb, NonZero, Odd, Reciprocal};
use core::fmt::{self, Debug};
use core::ops::{
    Add, AddAssign, BitAnd, BitAndAssign, BitOr, BitOrAssign, BitXor, BitXorAssign, Div, DivAssign,
    Mul, MulAssign, Neg, Not, Rem, Shl, ShlAssign, Shr, ShrAssign, Sub, SubAssign,
};
use subtle::{
    Choice, ConditionallySelectable, ConstantTimeEq, ConstantTimeGreater, ConstantTimeLess,
    CtOption,
};

#[cfg(feature = "rand_core")]
use rand_core::CryptoRngCore;

/// Integers whose representation takes a bounded amount of space.
pub trait Bounded {
    /// Size of this integer in bits.
    const BITS: u32;

    /// Size of this integer in bytes.
    const BYTES: usize;
}

/// Trait for types which are conditionally selectable in constant time, similar to (and blanket impl'd for) `subtle`'s
/// [`ConditionallySelectable`] trait, but without the `Copy` bound which allows it to be impl'd for heap allocated
/// types such as `BoxedUint`.
///
/// It also provides generic implementations of conditional assignment and conditional swaps.
pub trait ConstantTimeSelect: Clone {
    /// Select `a` or `b` according to `choice`.
    ///
    /// # Returns
    /// - `a` if `choice == Choice(0)`;
    /// - `b` if `choice == Choice(1)`.
    fn ct_select(a: &Self, b: &Self, choice: Choice) -> Self;

    /// Conditionally assign `other` to `self`, according to `choice`.
    #[inline]
    fn ct_assign(&mut self, other: &Self, choice: Choice) {
        *self = Self::ct_select(self, other, choice);
    }

    /// Conditionally swap `self` and `other` if `choice == 1`; otherwise, reassign both unto themselves.
    #[inline]
    fn ct_swap(a: &mut Self, b: &mut Self, choice: Choice) {
        let t: Self = a.clone();
        a.ct_assign(b, choice);
        b.ct_assign(&t, choice);
    }
}

impl<T: ConditionallySelectable> ConstantTimeSelect for T {
    #[inline(always)]
    fn ct_select(a: &Self, b: &Self, choice: Choice) -> Self {
        T::conditional_select(a, b, choice)
    }

    #[inline(always)]
    fn ct_assign(&mut self, other: &Self, choice: Choice) {
        self.conditional_assign(other, choice)
    }

    #[inline(always)]
    fn ct_swap(a: &mut Self, b: &mut Self, choice: Choice) {
        T::conditional_swap(a, b, choice)
    }
}

/// Integer trait: represents common functionality of integer types provided by this crate.
pub trait Integer:
    'static
    + Add<Output = Self>
    + for<'a> Add<&'a Self, Output = Self>
    + AddMod<Output = Self>
    + AsRef<[Limb]>
    + BitAnd<Output = Self>
    + for<'a> BitAnd<&'a Self, Output = Self>
    + BitAndAssign
    + for<'a> BitAndAssign<&'a Self>
    + BitOr<Output = Self>
    + for<'a> BitOr<&'a Self, Output = Self>
    + BitOrAssign
    + for<'a> BitOrAssign<&'a Self>
    + BitXor<Output = Self>
    + for<'a> BitXor<&'a Self, Output = Self>
    + BitXorAssign
    + for<'a> BitXorAssign<&'a Self>
    + BitOps
    + CheckedAdd
    + CheckedSub
    + CheckedMul
    + CheckedDiv
    + Clone
    + ConstantTimeEq
    + ConstantTimeGreater
    + ConstantTimeLess
    + ConstantTimeSelect
    + Debug
    + Default
    + Div<NonZero<Self>, Output = Self>
    + for<'a> Div<&'a NonZero<Self>, Output = Self>
    + DivAssign<NonZero<Self>>
    + for<'a> DivAssign<&'a NonZero<Self>>
    + DivRemLimb
    + Eq
    + From<u8>
    + From<u16>
    + From<u32>
    + From<u64>
    + From<Limb>
    + Mul<Output = Self>
    + for<'a> Mul<&'a Self, Output = Self>
    + MulMod<Output = Self>
    + NegMod<Output = Self>
    + Not<Output = Self>
    + Ord
    + Rem<NonZero<Self>, Output = Self>
    + for<'a> Rem<&'a NonZero<Self>, Output = Self>
    + RemLimb
    + Send
    + Sized
    + Shl<u32, Output = Self>
    + ShlAssign<u32>
    + ShlVartime
    + Shr<u32, Output = Self>
    + ShrAssign<u32>
    + ShrVartime
    + Sub<Output = Self>
    + for<'a> Sub<&'a Self, Output = Self>
    + SubMod<Output = Self>
    + Sync
    + SquareRoot
    + WrappingAdd
    + WrappingSub
    + WrappingMul
    + WrappingNeg
    + WrappingShl
    + WrappingShr
    + Zero
{
    /// The corresponding Montgomery representation,
    /// optimized for the performance of modular operations at the price of a conversion overhead.
    type Monty: Monty<Integer = Self>;

    /// The value `1`.
    fn one() -> Self;

    /// The value `1` with the same precision as `other`.
    fn one_like(other: &Self) -> Self {
        Self::from_limb_like(Limb::ONE, other)
    }

    /// Returns an integer with the first limb set to `limb`, and the same precision as `other`.
    fn from_limb_like(limb: Limb, other: &Self) -> Self;

    /// Number of limbs in this integer.
    fn nlimbs(&self) -> usize;

    /// Is this integer value an odd number?
    ///
    /// # Returns
    ///
    /// If odd, returns `Choice(1)`. Otherwise, returns `Choice(0)`.
    fn is_odd(&self) -> Choice {
        self.as_ref()
            .first()
            .map(|limb| limb.is_odd())
            .unwrap_or_else(|| Choice::from(0))
    }

    /// Is this integer value an even number?
    ///
    /// # Returns
    ///
    /// If even, returns `Choice(1)`. Otherwise, returns `Choice(0)`.
    fn is_even(&self) -> Choice {
        !self.is_odd()
    }
}

/// Fixed-width integers.
pub trait FixedInteger: Bounded + ConditionallySelectable + Constants + Copy + Integer {
    /// The number of limbs used on this platform.
    const LIMBS: usize;
}

/// Compute the greatest common divisor of two integers.
pub trait Gcd<Rhs = Self>: Sized {
    /// Output type.
    type Output;

    /// Compute the greatest common divisor of `self` and `rhs`.
    ///
    /// Returns none unless `self` is odd (`rhs` may be even or odd)`.
    fn gcd(&self, rhs: &Rhs) -> Self::Output;
}

/// Trait impl'd by precomputed modular inverters obtained via the [`PrecomputeInverter`] trait.
pub trait Inverter {
    /// Output of an inversion.
    type Output;

    /// Compute a modular inversion, returning `None` if the result is undefined (i.e. if `value` is zero or isn't
    /// prime relative to the modulus).
    fn invert(&self, value: &Self::Output) -> CtOption<Self::Output>;
}

/// Obtain a precomputed inverter for efficiently computing modular inversions for a given modulus.
pub trait PrecomputeInverter {
    /// Inverter type for integers of this size.
    type Inverter: Inverter<Output = Self::Output> + Sized;

    /// Output produced by the inverter.
    type Output;

    /// Obtain a precomputed inverter for `&self` as the modulus, using `Self::one()` as an adjusting parameter.
    ///
    /// Returns `None` if `self` is even.
    fn precompute_inverter(&self) -> Self::Inverter;
}

/// Zero values.
pub trait Zero: ConstantTimeEq + Sized {
    /// The value `0`.
    fn zero() -> Self;

    /// Determine if this value is equal to zero.
    ///
    /// # Returns
    ///
    /// If zero, returns `Choice(1)`. Otherwise, returns `Choice(0)`.
    #[inline]
    fn is_zero(&self) -> Choice {
        self.ct_eq(&Self::zero())
    }

    /// Set `self` to its additive identity, i.e. `Self::zero`.
    #[inline]
    fn set_zero(&mut self) {
        *self = Zero::zero();
    }

    /// Return the value `0` with the same precision as `other`.
    fn zero_like(other: &Self) -> Self
    where
        Self: Clone,
    {
        let mut ret = other.clone();
        ret.set_zero();
        ret
    }
}

impl<T: ConstZero + ConstantTimeEq> Zero for T {
    #[inline(always)]
    fn zero() -> T {
        Self::ZERO
    }
}

/// Trait for associating constant values with a type.
pub trait Constants: ConstZero {
    /// The value `1`.
    const ONE: Self;

    /// Maximum value this integer can express.
    const MAX: Self;
}

/// Random number generation support.
#[cfg(feature = "rand_core")]
pub trait Random: Sized {
    /// Generate a cryptographically secure random value.
    fn random(rng: &mut impl CryptoRngCore) -> Self;
}

/// Possible errors of the methods in [`RandomBits`] trait.
#[cfg(feature = "rand_core")]
#[derive(Debug)]
pub enum RandomBitsError {
    /// An error of the internal RNG library.
    RandCore(rand_core::Error),
    /// The requested `bits_precision` does not match the size of the integer
    /// corresponding to the type (in the cases where this is set in compile time).
    BitsPrecisionMismatch {
        /// The requested precision.
        bits_precision: u32,
        /// The compile-time size of the integer.
        integer_bits: u32,
    },
    /// The requested `bit_length` is larger than `bits_precision`.
    BitLengthTooLarge {
        /// The requested bit length of the random number.
        bit_length: u32,
        /// The requested precision.
        bits_precision: u32,
    },
}

#[cfg(feature = "rand_core")]
impl fmt::Display for RandomBitsError {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        match self {
            Self::RandCore(err) => write!(f, "{}", err),
            Self::BitsPrecisionMismatch {
                bits_precision,
                integer_bits,
            } => write!(
                f,
                concat![
                    "The requested `bits_precision` ({}) does not match ",
                    "the size of the integer corresponding to the type ({})"
                ],
                bits_precision, integer_bits
            ),
            Self::BitLengthTooLarge {
                bit_length,
                bits_precision,
            } => write!(
                f,
                "The requested `bit_length` ({}) is larger than `bits_precision` ({}).",
                bit_length, bits_precision
            ),
        }
    }
}

#[cfg(feature = "std")]
impl std::error::Error for RandomBitsError {}

/// Random bits generation support.
#[cfg(feature = "rand_core")]
pub trait RandomBits: Sized {
    /// Generate a cryptographically secure random value in range `[0, 2^bit_length)`.
    ///
    /// A wrapper for [`RandomBits::try_random_bits`] that panics on error.
    fn random_bits(rng: &mut impl CryptoRngCore, bit_length: u32) -> Self {
        Self::try_random_bits(rng, bit_length).expect("try_random_bits() failed")
    }

    /// Generate a cryptographically secure random value in range `[0, 2^bit_length)`.
    ///
    /// This method is variable time wrt `bit_length`.
    fn try_random_bits(
        rng: &mut impl CryptoRngCore,
        bit_length: u32,
    ) -> Result<Self, RandomBitsError>;

    /// Generate a cryptographically secure random value in range `[0, 2^bit_length)`,
    /// returning an integer with the closest available size to `bits_precision`
    /// (if the implementing type supports runtime sizing).
    ///
    /// A wrapper for [`RandomBits::try_random_bits_with_precision`] that panics on error.
    fn random_bits_with_precision(
        rng: &mut impl CryptoRngCore,
        bit_length: u32,
        bits_precision: u32,
    ) -> Self {
        Self::try_random_bits_with_precision(rng, bit_length, bits_precision)
            .expect("try_random_bits_with_precision() failed")
    }

    /// Generate a cryptographically secure random value in range `[0, 2^bit_length)`,
    /// returning an integer with the closest available size to `bits_precision`
    /// (if the implementing type supports runtime sizing).
    ///
    /// This method is variable time wrt `bit_length`.
    fn try_random_bits_with_precision(
        rng: &mut impl CryptoRngCore,
        bit_length: u32,
        bits_precision: u32,
    ) -> Result<Self, RandomBitsError>;
}

/// Modular random number generation support.
#[cfg(feature = "rand_core")]
pub trait RandomMod: Sized + Zero {
    /// Generate a cryptographically secure random number which is less than
    /// a given `modulus`.
    ///
    /// This function uses rejection sampling, a method which produces an
    /// unbiased distribution of in-range values provided the underlying
    /// CSRNG is unbiased, but runs in variable-time.
    ///
    /// The variable-time nature of the algorithm should not pose a security
    /// issue so long as the underlying random number generator is truly a
    /// CSRNG, where previous outputs are unrelated to subsequent
    /// outputs and do not reveal information about the RNG's internal state.
    fn random_mod(rng: &mut impl CryptoRngCore, modulus: &NonZero<Self>) -> Self;
}

/// Compute `self + rhs mod p`.
pub trait AddMod<Rhs = Self> {
    /// Output type.
    type Output;

    /// Compute `self + rhs mod p`.
    ///
    /// Assumes `self` and `rhs` are `< p`.
    fn add_mod(&self, rhs: &Rhs, p: &Self) -> Self::Output;
}

/// Compute `self - rhs mod p`.
pub trait SubMod<Rhs = Self> {
    /// Output type.
    type Output;

    /// Compute `self - rhs mod p`.
    ///
    /// Assumes `self` and `rhs` are `< p`.
    fn sub_mod(&self, rhs: &Rhs, p: &Self) -> Self::Output;
}

/// Compute `-self mod p`.
pub trait NegMod {
    /// Output type.
    type Output;

    /// Compute `-self mod p`.
    #[must_use]
    fn neg_mod(&self, p: &Self) -> Self::Output;
}

/// Compute `self * rhs mod p`.
pub trait MulMod<Rhs = Self> {
    /// Output type.
    type Output;

    /// Compute `self * rhs mod p`.
    fn mul_mod(&self, rhs: &Rhs, p: &Self) -> Self::Output;
}

/// Compute `1 / self mod p`.
pub trait InvMod: Sized {
    /// Compute `1 / self mod p`.
    fn inv_mod(&self, p: &Self) -> CtOption<Self>;
}

/// Checked addition.
pub trait CheckedAdd<Rhs = Self>: Sized {
    /// Perform checked addition, returning a [`CtOption`] which `is_some` only if the operation
    /// did not overflow.
    fn checked_add(&self, rhs: &Rhs) -> CtOption<Self>;
}

/// Checked division.
pub trait CheckedDiv<Rhs = Self>: Sized {
    /// Perform checked division, returning a [`CtOption`] which `is_some` only if the divisor is
    /// non-zero.
    fn checked_div(&self, rhs: &Rhs) -> CtOption<Self>;
}

/// Checked multiplication.
pub trait CheckedMul<Rhs = Self>: Sized {
    /// Perform checked multiplication, returning a [`CtOption`] which `is_some`
    /// only if the operation did not overflow.
    fn checked_mul(&self, rhs: &Rhs) -> CtOption<Self>;
}

/// Checked subtraction.
pub trait CheckedSub<Rhs = Self>: Sized {
    /// Perform checked subtraction, returning a [`CtOption`] which `is_some`
    /// only if the operation did not underflow.
    fn checked_sub(&self, rhs: &Rhs) -> CtOption<Self>;
}

/// Concatenate two numbers into a "wide" double-width value, using the `hi` value as the most
/// significant portion of the resulting value.
pub trait Concat: ConcatMixed<Self, MixedOutput = Self::Output> {
    /// Concatenated output: twice the width of `Self`.
    type Output: Integer;

    /// Concatenate the two halves, with `self` as least significant and `hi` as the most significant.
    fn concat(&self, hi: &Self) -> Self::Output {
        self.concat_mixed(hi)
    }
}

/// Concatenate two numbers into a "wide" combined-width value, using the `hi` value as the most
/// significant value.
pub trait ConcatMixed<Hi: ?Sized = Self> {
    /// Concatenated output: combination of `Self` and `Hi`.
    type MixedOutput: Integer;

    /// Concatenate the two values, with `self` as least significant and `hi` as the most
    /// significant.
    fn concat_mixed(&self, hi: &Hi) -> Self::MixedOutput;
}

/// Split a number in half, returning the least significant half followed by the most significant.
pub trait Split: SplitMixed<Self::Output, Self::Output> {
    /// Split output: low/high components of the value.
    type Output;

    /// Split this number in half, returning its low and high components respectively.
    fn split(&self) -> (Self::Output, Self::Output) {
        self.split_mixed()
    }
}

/// Split a number into parts, returning the least significant part followed by the most
/// significant.
pub trait SplitMixed<Lo, Hi> {
    /// Split this number into parts, returning its low and high components respectively.
    fn split_mixed(&self) -> (Lo, Hi);
}

/// Encoding support.
pub trait Encoding: Sized {
    /// Byte array representation.
    type Repr: AsRef<[u8]>
        + AsMut<[u8]>
        + Copy
        + Clone
        + Sized
        + for<'a> TryFrom<&'a [u8], Error = core::array::TryFromSliceError>;

    /// Decode from big endian bytes.
    fn from_be_bytes(bytes: Self::Repr) -> Self;

    /// Decode from little endian bytes.
    fn from_le_bytes(bytes: Self::Repr) -> Self;

    /// Encode to big endian bytes.
    fn to_be_bytes(&self) -> Self::Repr;

    /// Encode to little endian bytes.
    fn to_le_bytes(&self) -> Self::Repr;
}

/// Possible errors in variable-time integer decoding methods.
#[derive(Clone, Copy, Debug, Eq, PartialEq)]
pub enum DecodeError {
    /// The input value was empty.
    Empty,

    /// The input was not consistent with the format restrictions.
    InvalidDigit,

    /// Input size is too small to fit in the given precision.
    InputSize,

    /// The deserialized number is larger than the given precision.
    Precision,
}

impl fmt::Display for DecodeError {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        match self {
            Self::Empty => write!(f, "empty value provided"),
            Self::InvalidDigit => {
                write!(f, "invalid digit character")
            }
            Self::InputSize => write!(f, "input size is too small to fit in the given precision"),
            Self::Precision => write!(
                f,
                "the deserialized number is larger than the given precision"
            ),
        }
    }
}

#[cfg(feature = "std")]
impl std::error::Error for DecodeError {}

/// Support for optimized squaring
pub trait Square {
    /// Computes the same as `self * self`, but may be more efficient.
    fn square(&self) -> Self;
}

/// Support for optimized squaring in-place
pub trait SquareAssign {
    /// Computes the same as `self * self`, but may be more efficient.
    /// Writes the result in `self`.
    fn square_assign(&mut self);
}

/// Support for calucaling square roots.
pub trait SquareRoot {
    /// Computes `floor(sqrt(self))`.
    fn sqrt(&self) -> Self;

    /// Computes `floor(sqrt(self))`, variable time in `self`.
    fn sqrt_vartime(&self) -> Self;
}

/// Support for optimized division by a single limb.
pub trait DivRemLimb: Sized {
    /// Computes `self / rhs` using a pre-made reciprocal,
    /// returns the quotient (q) and remainder (r).
    fn div_rem_limb(&self, rhs: NonZero<Limb>) -> (Self, Limb) {
        self.div_rem_limb_with_reciprocal(&Reciprocal::new(rhs))
    }

    /// Computes `self / rhs`, returns the quotient (q) and remainder (r).
    fn div_rem_limb_with_reciprocal(&self, reciprocal: &Reciprocal) -> (Self, Limb);
}

/// Support for optimized division by a single limb.
pub trait RemLimb: Sized {
    /// Computes `self % rhs` using a pre-made reciprocal.
    fn rem_limb(&self, rhs: NonZero<Limb>) -> Limb {
        self.rem_limb_with_reciprocal(&Reciprocal::new(rhs))
    }

    /// Computes `self % rhs`.
    fn rem_limb_with_reciprocal(&self, reciprocal: &Reciprocal) -> Limb;
}

/// Bit counting and bit operations.
pub trait BitOps {
    /// Precision of this integer in bits.
    fn bits_precision(&self) -> u32;

    /// `floor(log2(self.bits_precision()))`.
    fn log2_bits(&self) -> u32 {
        u32::BITS - self.bits_precision().leading_zeros() - 1
    }

    /// Precision of this integer in bytes.
    fn bytes_precision(&self) -> usize;

    /// Calculate the number of bits needed to represent this number.
    fn bit(&self, index: u32) -> Choice;

    /// Sets the bit at `index` to 0 or 1 depending on the value of `bit_value`.
    fn set_bit(&mut self, index: u32, bit_value: Choice);

    /// Calculate the number of bits required to represent a given number.
    fn bits(&self) -> u32 {
        self.bits_precision() - self.leading_zeros()
    }

    /// Calculate the number of trailing zeros in the binary representation of this number.
    fn trailing_zeros(&self) -> u32;

    /// Calculate the number of trailing ones in the binary representation of this number.
    fn trailing_ones(&self) -> u32;

    /// Calculate the number of leading zeros in the binary representation of this number.
    fn leading_zeros(&self) -> u32;

    /// Returns `true` if the bit at position `index` is set, `false` otherwise.
    ///
    /// # Remarks
    /// This operation is variable time with respect to `index` only.
    fn bit_vartime(&self, index: u32) -> bool;

    /// Calculate the number of bits required to represent a given number in variable-time with
    /// respect to `self`.
    fn bits_vartime(&self) -> u32 {
        self.bits_precision() - self.leading_zeros_vartime()
    }

    /// Sets the bit at `index` to 0 or 1 depending on the value of `bit_value`,
    /// variable time in `self`.
    fn set_bit_vartime(&mut self, index: u32, bit_value: bool);

    /// Calculate the number of leading zeros in the binary representation of this number.
    fn leading_zeros_vartime(&self) -> u32 {
        self.bits_precision() - self.bits_vartime()
    }

    /// Calculate the number of trailing zeros in the binary representation of this number in
    /// variable-time with respect to `self`.
    fn trailing_zeros_vartime(&self) -> u32;

    /// Calculate the number of trailing ones in the binary representation of this number,
    /// variable time in `self`.
    fn trailing_ones_vartime(&self) -> u32;
}

/// Constant-time exponentiation.
pub trait Pow<Exponent> {
    /// Raises to the `exponent` power.
    fn pow(&self, exponent: &Exponent) -> Self;
}

impl<T: PowBoundedExp<Exponent>, Exponent: Bounded> Pow<Exponent> for T {
    fn pow(&self, exponent: &Exponent) -> Self {
        self.pow_bounded_exp(exponent, Exponent::BITS)
    }
}

/// Constant-time exponentiation with exponent of a bounded bit size.
pub trait PowBoundedExp<Exponent> {
    /// Raises to the `exponent` power,
    /// with `exponent_bits` representing the number of (least significant) bits
    /// to take into account for the exponent.
    ///
    /// NOTE: `exponent_bits` may be leaked in the time pattern.
    fn pow_bounded_exp(&self, exponent: &Exponent, exponent_bits: u32) -> Self;
}

/// Performs modular multi-exponentiation using Montgomery's ladder.
///
/// See: Straus, E. G. Problems and solutions: Addition chains of vectors. American Mathematical Monthly 71 (1964), 806–808.
pub trait MultiExponentiate<Exponent, BasesAndExponents>: Pow<Exponent> + Sized
where
    BasesAndExponents: AsRef<[(Self, Exponent)]> + ?Sized,
{
    /// Calculates `x1 ^ k1 * ... * xn ^ kn`.
    fn multi_exponentiate(bases_and_exponents: &BasesAndExponents) -> Self;
}

impl<T, Exponent, BasesAndExponents> MultiExponentiate<Exponent, BasesAndExponents> for T
where
    T: MultiExponentiateBoundedExp<Exponent, BasesAndExponents>,
    Exponent: Bounded,
    BasesAndExponents: AsRef<[(Self, Exponent)]> + ?Sized,
{
    fn multi_exponentiate(bases_and_exponents: &BasesAndExponents) -> Self {
        Self::multi_exponentiate_bounded_exp(bases_and_exponents, Exponent::BITS)
    }
}

/// Performs modular multi-exponentiation using Montgomery's ladder.
/// `exponent_bits` represents the number of bits to take into account for the exponent.
///
/// See: Straus, E. G. Problems and solutions: Addition chains of vectors. American Mathematical Monthly 71 (1964), 806–808.
///
/// NOTE: this value is leaked in the time pattern.
pub trait MultiExponentiateBoundedExp<Exponent, BasesAndExponents>: Pow<Exponent> + Sized
where
    BasesAndExponents: AsRef<[(Self, Exponent)]> + ?Sized,
{
    /// Calculates `x1 ^ k1 * ... * xn ^ kn`.
    fn multi_exponentiate_bounded_exp(
        bases_and_exponents: &BasesAndExponents,
        exponent_bits: u32,
    ) -> Self;
}

/// Constant-time inversion.
pub trait Invert: Sized {
    /// Output of the inversion.
    type Output;

    /// Computes the inverse.
    fn invert(&self) -> Self::Output;
}

/// Widening multiply: returns a value with a number of limbs equal to the sum of the inputs.
pub trait WideningMul<Rhs = Self>: Sized {
    /// Output of the widening multiplication.
    type Output: Integer;

    /// Perform widening multiplication.
    fn widening_mul(&self, rhs: Rhs) -> Self::Output;
}

/// Left shifts, variable time in `shift`.
pub trait ShlVartime: Sized {
    /// Computes `self << shift`.
    ///
    /// Returns `None` if `shift >= self.bits_precision()`.
    fn overflowing_shl_vartime(&self, shift: u32) -> CtOption<Self>;

    /// Computes `self << shift` in a panic-free manner, masking off bits of `shift`
    /// which would cause the shift to exceed the type's width.
    fn wrapping_shl_vartime(&self, shift: u32) -> Self;
}

/// Right shifts, variable time in `shift`.
pub trait ShrVartime: Sized {
    /// Computes `self >> shift`.
    ///
    /// Returns `None` if `shift >= self.bits_precision()`.
    fn overflowing_shr_vartime(&self, shift: u32) -> CtOption<Self>;

    /// Computes `self >> shift` in a panic-free manner, masking off bits of `shift`
    /// which would cause the shift to exceed the type's width.
    fn wrapping_shr_vartime(&self, shift: u32) -> Self;
}

/// A representation of an integer optimized for the performance of modular operations.
pub trait Monty:
    'static
    + Clone
    + Debug
    + Eq
    + Sized
    + Send
    + Sync
    + Add<Output = Self>
    + for<'a> Add<&'a Self, Output = Self>
    + AddAssign
    + for<'a> AddAssign<&'a Self>
    + Sub<Output = Self>
    + for<'a> Sub<&'a Self, Output = Self>
    + SubAssign
    + for<'a> SubAssign<&'a Self>
    + Mul<Output = Self>
    + for<'a> Mul<&'a Self, Output = Self>
    + MulAssign
    + for<'a> MulAssign<&'a Self>
    + Neg<Output = Self>
    + PowBoundedExp<Self::Integer>
    + Square
    + SquareAssign
{
    /// The original integer type.
    type Integer: Integer<Monty = Self>;

    /// The precomputed data needed for this representation.
    type Params: 'static + Clone + Debug + Eq + Sized + Send + Sync;

    /// Create the precomputed data for Montgomery representation of integers modulo `modulus`,
    /// variable time in `modulus`.
    fn new_params_vartime(modulus: Odd<Self::Integer>) -> Self::Params;

    /// Convert the value into the representation using precomputed data.
    fn new(value: Self::Integer, params: Self::Params) -> Self;

    /// Returns zero in this representation.
    fn zero(params: Self::Params) -> Self;

    /// Returns one in this representation.
    fn one(params: Self::Params) -> Self;

    /// Returns the parameter struct used to initialize this object.
    fn params(&self) -> &Self::Params;

    /// Access the value in Montgomery form.
    fn as_montgomery(&self) -> &Self::Integer;

    /// Performs division by 2, that is returns `x` such that `x + x = self`.
    fn div_by_2(&self) -> Self;
}