curve25519-dalek-source/src/scalar.rs

// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2017 Isis Lovecruft, Henry de Valence
// Portions Copyright 2017 Brian Smith
// See LICENSE for licensing information.
//
// Authors:
// - Isis Agora Lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>
// - Brian Smith <brian@briansmith.org>

//! Arithmetic on scalars (integers mod the group order).
//!
//! Both the Ristretto group and the Ed25519 basepoint have prime order
//! \\( \ell = 2\^{252} + 27742317777372353535851937790883648493 \\).
//!
//! This code is intended to be useful with both the Ristretto group
//! (where everything is done modulo \\( \ell \\)), and the X/Ed25519
//! setting, which mandates specific bit-twiddles that are not
//! well-defined modulo \\( \ell \\).
//!
//! To create a `Scalar` from a supposedly canonical encoding, use
//! `Scalar::from_canonical_bytes`.
//!
//! To create a `Scalar` by reducing a \\(256\\)-bit integer mod \\( \ell \\),
//! use `Scalar::from_bytes_mod_order`.
//!
//! To create a `Scalar` by reducing a \\(512\\)-bit integer mod \\( \ell \\),
//! use `Scalar::from_bytes_mod_order_wide`.
//!
//! To create a `Scalar` with a specific bit-pattern (e.g., for
//! compatibility with X25519 "clamping"), use `Scalar::from_bits`.
//!
//! All arithmetic on `Scalars` is done modulo \\( \ell \\).

use core::fmt::Debug;
use core::ops::Neg;
use core::ops::{Add, AddAssign};
use core::ops::{Sub, SubAssign};
use core::ops::{Mul, MulAssign};
use core::ops::{Index};
use core::cmp::{Eq, PartialEq};
use core::iter::{Product, Sum};
use core::borrow::Borrow;

use rand::{Rng, CryptoRng};

use digest::Digest;
use generic_array::typenum::U64;

use subtle::Choice;
use subtle::ConditionallyAssignable;
use subtle::ConstantTimeEq;

use backend;
use constants;

/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.
///
/// This is a type alias for one of the scalar types in the `backend`
/// module.
#[cfg(feature = "u64_backend")]
type UnpackedScalar = backend::u64::scalar::Scalar64;

/// An `UnpackedScalar` represents an element of the field GF(l), optimized for speed.
///
/// This is a type alias for one of the scalar types in the `backend`
/// module.
#[cfg(feature = "u32_backend")]
type UnpackedScalar = backend::u32::scalar::Scalar32;


/// The `Scalar` struct holds an integer \\(s < 2\^{255} \\) which
/// represents an element of \\(\mathbb Z / \ell\\).
#[derive(Copy, Clone)]
pub struct Scalar {
    /// `bytes` is a little-endian byte encoding of an integer representing a scalar modulo the group order.
    ///
    /// # Invariant
    ///
    /// The integer representing this scalar must be bounded above by \\(2\^{255}\\), or equivalently the high bit of `bytes[31]` must be zero.
    ///
    /// This ensures that there is room for a carry bit when computing a NAF representation.
    // XXX This is pub(crate) so we can write literal constants.  If const fns were stable, we could make the Scalar constructors const fns and use those instead.
    pub(crate) bytes: [u8; 32],
}

impl Scalar {
    /// Construct a `Scalar` by reducing a 256-bit little-endian integer
    /// modulo the group order \\( \ell \\).
    pub fn from_bytes_mod_order(bytes: [u8; 32]) -> Scalar {
        // Temporarily allow s_unreduced.bytes > 2^255 ...
        let s_unreduced = Scalar{bytes: bytes};

        // Then reduce mod the group order and return the reduced representative.
        let s = s_unreduced.reduce();
        debug_assert_eq!(0u8, s[31] >> 7);

        s
    }

    /// Construct a `Scalar` by reducing a 512-bit little-endian integer
    /// modulo the group order \\( \ell \\).
    pub fn from_bytes_mod_order_wide(input: &[u8; 64]) -> Scalar {
        UnpackedScalar::from_bytes_wide(input).pack()
    }

    /// Attempt to construct a `Scalar` from a canonical byte representation.
    ///
    /// # Return
    ///
    /// - `Some(s)`, where `s` is the `Scalar` corresponding to `bytes`,
    ///   if `bytes` is a canonical byte representation;
    /// - `None` if `bytes` is not a canonical byte representation.
    pub fn from_canonical_bytes(bytes: [u8; 32]) -> Option<Scalar> {
        // Check that the high bit is not set
        if (bytes[31] >> 7) != 0u8 { return None; }
        let candidate = Scalar::from_bits(bytes);

        if candidate.is_canonical() {
            Some(candidate)
        } else {
            None
        }
    }

    /// Construct a `Scalar` from the low 255 bits of a 256-bit integer.
    ///
    /// This function is intended for applications like X25519 which
    /// require specific bit-patterns when performing scalar
    /// multiplication.
    pub fn from_bits(bytes: [u8; 32]) -> Scalar {
        let mut s = Scalar{bytes: bytes};
        // Ensure that s < 2^255 by masking the high bit
        s.bytes[31] &= 0b0111_1111;

        s
    }
}

impl Debug for Scalar {
    fn fmt(&self, f: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
        write!(f, "Scalar{{\n\tbytes: {:?},\n}}", &self.bytes)
    }
}

impl Eq for Scalar {}
impl PartialEq for Scalar {
    fn eq(&self, other: &Self) -> bool {
        self.ct_eq(other).unwrap_u8() == 1u8
    }
}

impl ConstantTimeEq for Scalar {
    fn ct_eq(&self, other: &Self) -> Choice {
        self.bytes.ct_eq(&other.bytes)
    }
}

impl Index<usize> for Scalar {
    type Output = u8;

    /// Index the bytes of the representative for this `Scalar`.  Mutation is not permitted.
    fn index(&self, _index: usize) -> &u8 {
        &(self.bytes[_index])
    }
}

impl<'b> MulAssign<&'b Scalar> for Scalar {
    fn mul_assign(&mut self, _rhs: &'b Scalar) {
        *self = UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack();
    }
}

define_mul_assign_variants!(LHS = Scalar, RHS = Scalar);

impl<'a, 'b> Mul<&'b Scalar> for &'a Scalar {
    type Output = Scalar;
    fn mul(self, _rhs: &'b Scalar) -> Scalar {
        UnpackedScalar::mul(&self.unpack(), &_rhs.unpack()).pack()
    }
}

define_mul_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);

impl<'b> AddAssign<&'b Scalar> for Scalar {
    fn add_assign(&mut self, _rhs: &'b Scalar) {
        *self = UnpackedScalar::add(&self.unpack(), &_rhs.unpack()).pack();
    }
}

define_add_assign_variants!(LHS = Scalar, RHS = Scalar);

impl<'a, 'b> Add<&'b Scalar> for &'a Scalar {
    type Output = Scalar;
    fn add(self, _rhs: &'b Scalar) -> Scalar {
        UnpackedScalar::add(&self.unpack(), &_rhs.unpack()).pack()
    }
}

define_add_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);

impl<'b> SubAssign<&'b Scalar> for Scalar {
    fn sub_assign(&mut self, _rhs: &'b Scalar) {
        *self = UnpackedScalar::sub(&self.unpack(), &_rhs.unpack()).pack();
    }
}

define_sub_assign_variants!(LHS = Scalar, RHS = Scalar);

impl<'a, 'b> Sub<&'b Scalar> for &'a Scalar {
    type Output = Scalar;
    fn sub(self, _rhs: &'b Scalar) -> Scalar {
        UnpackedScalar::sub(&self.unpack(), &_rhs.unpack()).pack()
    }
}

define_sub_variants!(LHS = Scalar, RHS = Scalar, Output = Scalar);

impl<'a> Neg for &'a Scalar {
    type Output = Scalar;
    fn neg(self) -> Scalar {
        &Scalar::zero() - self
    }
}

impl<'a> Neg for Scalar {
    type Output = Scalar;
    fn neg(self) -> Scalar {
        -&self
    }
}

impl ConditionallyAssignable for Scalar {
    fn conditional_assign(&mut self, other: &Scalar, choice: Choice) {
        for i in 0..32 {
            self.bytes[i].conditional_assign(&other.bytes[i], choice);
        }
    }
}

#[cfg(feature = "serde")]
use serde::{self, Serialize, Deserialize, Serializer, Deserializer};
#[cfg(feature = "serde")]
use serde::de::Visitor;

#[cfg(feature = "serde")]
impl Serialize for Scalar {
    fn serialize<S>(&self, serializer: S) -> Result<S::Ok, S::Error>
        where S: Serializer
    {
        serializer.serialize_bytes(self.reduce().as_bytes())
    }
}

#[cfg(feature = "serde")]
impl<'de> Deserialize<'de> for Scalar {
    fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>
        where D: Deserializer<'de>
    {
        struct ScalarVisitor;

        impl<'de> Visitor<'de> for ScalarVisitor {
            type Value = Scalar;

            fn expecting(&self, formatter: &mut ::core::fmt::Formatter) -> ::core::fmt::Result {
                formatter.write_str("a canonically-encoded 32-byte scalar value")
            }

            fn visit_bytes<E>(self, v: &[u8]) -> Result<Scalar, E>
                where E: serde::de::Error
            {
                if v.len() == 32 {
                    let mut bytes = [0u8;32];
                    bytes.copy_from_slice(v);

                    static ERRMSG: &'static str = "encoding was not canonical";

                    Scalar::from_canonical_bytes(bytes)
                        .ok_or(
                            serde::de::Error::invalid_value(
                                serde::de::Unexpected::Bytes(v),
                                &ERRMSG,
                            )
                        )
                } else {
                    Err(serde::de::Error::invalid_length(v.len(), &self))
                }
            }
        }

        deserializer.deserialize_bytes(ScalarVisitor)
    }
}

impl<T> Product<T> for Scalar
where
    T: Borrow<Scalar>
{
    fn product<I>(iter: I) -> Self
    where
        I: Iterator<Item = T>
    {
        iter.fold(Scalar::one(), |acc, item| acc * item.borrow())
    }
}

impl<T> Sum<T> for Scalar
where
    T: Borrow<Scalar>
{
    fn sum<I>(iter: I) -> Self
    where
        I: Iterator<Item = T>
    {
        iter.fold(Scalar::zero(), |acc, item| acc + item.borrow())
    }
}

impl Scalar {
    /// Return a `Scalar` chosen uniformly at random using a user-provided RNG.
    ///
    /// # Inputs
    ///
    /// * `rng`: any RNG which implements the `rand::CryptoRng` interface.
    ///
    /// # Returns
    ///
    /// A random scalar within ℤ/lℤ.
    #[cfg(feature = "std")]
    pub fn random<T: Rng + CryptoRng>(rng: &mut T) -> Self {
        let mut scalar_bytes = [0u8; 64];
        rng.fill(&mut scalar_bytes);
        Scalar::from_bytes_mod_order_wide(&scalar_bytes)
    }

    /// Hash a slice of bytes into a scalar.
    ///
    /// Takes a type parameter `D`, which is any `Digest` producing 64
    /// bytes (512 bits) of output.
    ///
    /// Convenience wrapper around `from_hash`.
    ///
    /// # Example
    ///
    /// ```
    /// # extern crate curve25519_dalek;
    /// # use curve25519_dalek::scalar::Scalar;
    /// extern crate sha2;
    /// use sha2::Sha512;
    ///
    /// # // Need fn main() here in comment so the doctest compiles
    /// # // See https://doc.rust-lang.org/book/documentation.html#documentation-as-tests
    /// # fn main() {
    /// let msg = "To really appreciate architecture, you may even need to commit a murder";
    /// let s = Scalar::hash_from_bytes::<Sha512>(msg.as_bytes());
    /// # }
    /// ```
    ///
    pub fn hash_from_bytes<D>(input: &[u8]) -> Scalar
        where D: Digest<OutputSize = U64> + Default
    {
        let mut hash = D::default();
        hash.input(input);
        Scalar::from_hash(hash)
    }

    /// Construct a scalar from an existing `Digest` instance.
    ///
    /// Use this instead of `hash_from_bytes` if it is more convenient
    /// to stream data into the `Digest` than to pass a single byte
    /// slice.
    pub fn from_hash<D>(hash: D) -> Scalar
        where D: Digest<OutputSize = U64> + Default
    {
        // XXX this seems clumsy
        let mut output = [0u8; 64];
        output.copy_from_slice(hash.result().as_slice());
        Scalar::from_bytes_mod_order_wide(&output)
    }

    /// Convert this `Scalar` to its underlying sequence of bytes.
    pub fn to_bytes(&self) -> [u8; 32] {
        self.bytes
    }

    /// View this `Scalar` as a sequence of bytes.
    pub fn as_bytes(&self) -> &[u8; 32] {
        &self.bytes
    }

    /// Construct the scalar \\( 0 \\).
    pub fn zero() -> Self {
        Scalar { bytes: [0u8; 32]}
    }

    /// Construct the scalar \\( 1 \\).
    pub fn one() -> Self {
        Scalar {
            bytes: [
                1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
            ],
        }
    }

    /// Construct a scalar from the given `u64`.
    pub fn from_u64(x: u64) -> Scalar {
        let mut s_bytes = [0u8; 32];
        for i in 0..8 {
            s_bytes[i] = (x >> (i*8)) as u8;
        }
        Scalar{ bytes: s_bytes }
    }

    /// Compute the multiplicative inverse of this scalar.
    pub fn invert(&self) -> Scalar {
        self.unpack().invert().pack()
    }

    /// Given a slice of nonzero (possibly secret) `Scalar`s,
    /// compute their inverses in a batch.
    ///
    /// # Return
    ///
    /// Each element of `inputs` is replaced by its inverse.
    ///
    /// The product of all inverses is returned.
    ///
    /// # Warning
    ///
    /// All input `Scalars` **MUST** be nonzero.  If you cannot
    /// *prove* that this is the case, you **SHOULD NOT USE THIS
    /// FUNCTION**.
    ///
    /// # Example
    ///
    /// ```
    /// # extern crate curve25519_dalek;
    /// # use curve25519_dalek::scalar::Scalar;
    /// # fn main() {
    ///
    /// let mut scalars = [
    ///     Scalar::from_u64(3),
    ///     Scalar::from_u64(5),
    ///     Scalar::from_u64(7),
    ///     Scalar::from_u64(11),
    /// ];
    ///
    /// let allinv = Scalar::batch_invert(&mut scalars);
    ///
    /// assert_eq!(allinv, Scalar::from_u64(3*5*7*11).invert());
    /// assert_eq!(scalars[0], Scalar::from_u64(3).invert());
    /// assert_eq!(scalars[1], Scalar::from_u64(5).invert());
    /// assert_eq!(scalars[2], Scalar::from_u64(7).invert());
    /// assert_eq!(scalars[3], Scalar::from_u64(11).invert());
    /// # }
    /// ```
    #[cfg(any(feature = "alloc", feature = "std"))]
    pub fn batch_invert(inputs: &mut [Scalar]) -> Scalar {
        // This code is essentially identical to the FieldElement
        // implementation, and is documented there.  Unfortunately,
        // it's not easy to write it generically, since here we want
        // to use `UnpackedScalar`s internally, and `Scalar`s
        // externally, but there's no corresponding distinction for
        // field elements.

        use clear_on_drop::ClearOnDrop;
        use clear_on_drop::clear::ZeroSafe;
        // Mark UnpackedScalars as zeroable.
        unsafe impl ZeroSafe for UnpackedScalar {}

        let n = inputs.len();
        let one: UnpackedScalar = Scalar::one().unpack().to_montgomery();

        // Wrap the scratch storage in a ClearOnDrop to wipe it when
        // we pass out of scope.
        let scratch_vec = vec![one; n];
        let mut scratch = ClearOnDrop::new(scratch_vec);

        // Keep an accumulator of all of the previous products
        let mut acc = Scalar::one().unpack().to_montgomery();

        // Pass through the input vector, recording the previous
        // products in the scratch space
        for (input, scratch) in inputs.iter_mut().zip(scratch.iter_mut()) {
            *scratch = acc;

            // Avoid unnecessary Montgomery multiplication in second pass by
            // keeping inputs in Montgomery form
            let tmp = input.unpack().to_montgomery();
            *input = tmp.pack();
            acc = UnpackedScalar::montgomery_mul(&acc, &tmp);
        }

        // acc is nonzero iff all inputs are nonzero
        debug_assert!(acc.pack() != Scalar::zero());

        // Compute the inverse of all products
        acc = acc.montgomery_invert().from_montgomery();

        // We need to return the product of all inverses later
        let ret = acc.pack();

        // Pass through the vector backwards to compute the inverses
        // in place
        for (input, scratch) in inputs.iter_mut().rev().zip(scratch.into_iter().rev()) {
            let tmp = UnpackedScalar::montgomery_mul(&acc, &input.unpack());
            *input = UnpackedScalar::montgomery_mul(&acc, &scratch).pack();
            acc = tmp;
        }

        ret
    }

    /// Get the bits of the scalar.
    pub(crate) fn bits(&self) -> [i8; 256] {
        let mut bits = [0i8; 256];
        for i in 0..256 {
            // As i runs from 0..256, the bottom 3 bits index the bit,
            // while the upper bits index the byte.
            bits[i] = ((self.bytes[i>>3] >> (i&7)) & 1u8) as i8;
        }
        bits
    }

    /// Compute a width-\\(w\\) "Non-Adjacent Form" of this scalar.
    ///
    /// A width-\\(w\\) NAF of a positive integer \\(k\\) is an expression
    /// $$
    /// k = \sum_{i=0}\^m n\_i 2\^i,
    /// $$
    /// where each nonzero
    /// coefficient \\(n\_i\\) is odd and bounded by \\(|n\_i| < 2\^{w-1}\\),
    /// \\(n\_{m-1}\\) is nonzero, and at most one of any \\(w\\) consecutive
    /// coefficients is nonzero.  (Hankerson, Menezes, Vanstone; def 3.32).
    ///
    /// The length of the NAF is at most one more than the length of
    /// the binary representation of \\(k\\).  This is why the
    /// `Scalar` type maintains an invariant that the top bit is
    /// \\(0\\), so that the NAF of a scalar has at most 256 digits.
    ///
    /// Intuitively, this is like a binary expansion, except that we
    /// allow some coefficients to grow in magnitude up to
    /// \\(2\^{w-1}\\) so that the nonzero coefficients are as sparse
    /// as possible.
    ///
    /// When doing scalar multiplication, we can then use a lookup
    /// table of precomputed multiples of a point to add the nonzero
    /// terms \\( k_i P \\).  Using signed digits cuts the table size
    /// in half, and using odd digits cuts the table size in half
    /// again.
    ///
    /// To compute a \\(w\\)-NAF, we use a modification of Algorithm 3.35 of HMV:
    ///
    /// 1. \\( i \gets 0 \\)
    /// 2. While \\( k \ge 1 \\):
    ///     1. If \\(k\\) is odd, \\( n_i \gets k \operatorname{mods} 2^w \\), \\( k \gets k - n_i \\).
    ///     2. If \\(k\\) is even, \\( n_i \gets 0 \\).
    ///     3. \\( k \gets k / 2 \\), \\( i \gets i + 1 \\).
    /// 3. Return \\( n_0, n_1, ... , \\)
    ///
    /// Here \\( \bar x = x \operatorname{mods} 2^w \\) means the
    /// \\( \bar x \\) with \\( \bar x \equiv x \pmod{2^w} \\) and
    /// \\( -2^{w-1} \leq \bar x < 2^w \\).
    ///
    /// We implement this by scanning across the bits of \\(k\\) from
    /// least-significant bit to most-significant-bit.
    /// Write the bits of \\(k\\) as
    /// $$
    /// k = \sum\_{i=0}\^m k\_i 2^i,
    /// $$
    /// and split the sum as
    /// $$
    /// k = \sum\_{i=0}^{w-1} k\_i 2^i + 2^w \sum\_{i=0} k\_{i+w} 2^i
    /// $$
    /// where the first part is \\( k \mod 2^w \\).
    ///
    /// If \\( k \mod 2^w\\) is odd, and \\( k \mod 2^w < 2^{w-1} \\), then we emit
    /// \\( n_0 = k \mod 2^w \\).  Instead of computing
    /// \\( k - n_0 \\), we just advance \\(w\\) bits and reindex.
    ///
    /// If \\( k \mod 2^w\\) is odd, and \\( k \mod 2^w \ge 2^{w-1} \\), then
    /// \\( n_0 = k \operatorname{mods} 2^w = k \mod 2^w - 2^w \\).
    /// The quantity \\( k - n_0 \\) is
    /// $$
    /// \begin{aligned}
    /// k - n_0 &= \sum\_{i=0}^{w-1} k\_i 2^i + 2^w \sum\_{i=0} k\_{i+w} 2^i
    ///          - \sum\_{i=0}^{w-1} k\_i 2^i + 2^w \\\\
    /// &= 2^w + 2^w \sum\_{i=0} k\_{i+w} 2^i
    /// \end{aligned}
    /// $$
    /// so instead of computing the subtraction, we can set a carry
    /// bit, advance \\(w\\) bits, and reindex.
    ///
    /// If \\( k \mod 2^w\\) is even, we emit \\(0\\), advance 1 bit
    /// and reindex.  In fact, by setting all digits to \\(0\\)
    /// initially, we don't need to emit anything.
    pub(crate) fn non_adjacent_form(&self, w: usize) -> [i8; 256] {
        // required by the NAF definition
        debug_assert!( w >= 2 );
        // required so that the NAF digits fit in i8
        debug_assert!( w <= 8 );

        use byteorder::{ByteOrder, LittleEndian};

        let mut naf = [0i8; 256];

        let mut x_u64 = [0u64; 5];
        LittleEndian::read_u64_into(&self.bytes, &mut x_u64[0..4]);

        let width = 1 << w;
        let window_mask = width - 1;

        let mut pos = 0;
        let mut carry = 0;
        while pos < 256 {
            // Construct a buffer of bits of the scalar, starting at bit `pos`
            let u64_idx = pos / 64;
            let bit_idx = pos % 64;
            let bit_buf: u64;
            if bit_idx < 64 - w {
                // This window's bits are contained in a single u64
                bit_buf = x_u64[u64_idx] >> bit_idx;
            } else {
                // Combine the current u64's bits with the bits from the next u64
                bit_buf = (x_u64[u64_idx] >> bit_idx) | (x_u64[1+u64_idx] << (64 - bit_idx));
            }

            // Add the carry into the current window
            let window = carry + (bit_buf & window_mask);

            if window & 1 == 0 {
                // If the window value is even, preserve the carry and continue.
                // Why is the carry preserved?
                // If carry == 0 and window & 1 == 0, then the next carry should be 0
                // If carry == 1 and window & 1 == 0, then bit_buf & 1 == 1 so the next carry should be 1
                pos += 1;
                continue;
            }

            if window < width/2 {
                carry = 0;
                naf[pos] = window as i8;
            } else {
                carry = 1;
                naf[pos] = (window as i8) - (width as i8);
            }

            pos += w;
        }

        naf
    }

    /// Write this scalar in radix 16, with coefficients in \\([-8,8)\\),
    /// i.e., compute \\(a\_i\\) such that
    /// $$
    ///    a = a\_0 + a\_1 16\^1 + \cdots + a_{63} 16\^{63},
    /// $$
    /// with \\(-8 \leq a_i < 8\\) for \\(0 \leq i < 63\\) and \\(-8 \leq a_{63} \leq 8\\).
    pub(crate) fn to_radix_16(&self) -> [i8; 64] {
        debug_assert!(self[31] <= 127);
        let mut output = [0i8; 64];

        // Step 1: change radix.
        // Convert from radix 256 (bytes) to radix 16 (nibbles)
        #[inline(always)]
        fn bot_half(x: u8) -> u8 { (x >> 0) & 15 }
        #[inline(always)]
        fn top_half(x: u8) -> u8 { (x >> 4) & 15 }

        for i in 0..32 {
            output[2*i  ] = bot_half(self[i]) as i8;
            output[2*i+1] = top_half(self[i]) as i8;
        }
        // Precondition note: since self[31] <= 127, output[63] <= 7

        // Step 2: recenter coefficients from [0,16) to [-8,8)
        for i in 0..63 {
            let carry    = (output[i] + 8) >> 4;
            output[i  ] -= carry << 4;
            output[i+1] += carry;
        }
        // Precondition note: output[63] is not recentered.  It
        // increases by carry <= 1.  Thus output[63] <= 8.

        output
    }

    /// Unpack this `Scalar` to an `UnpackedScalar` for faster arithmetic.
    pub(crate) fn unpack(&self) -> UnpackedScalar {
        UnpackedScalar::from_bytes(&self.bytes)
    }

    /// Reduce this `Scalar` modulo \\(\ell\\).
    #[allow(non_snake_case)]
    pub fn reduce(&self) -> Scalar {
        let x = self.unpack();
        let xR = UnpackedScalar::mul_internal(&x, &constants::R);
        let x_mod_l = UnpackedScalar::montgomery_reduce(&xR);
        x_mod_l.pack()
    }

    /// Check whether this `Scalar` is the canonical representative mod \\(\ell\\).
    ///
    /// This is intended for uses like input validation, where variable-time code is acceptable.
    ///
    /// ```
    /// # extern crate curve25519_dalek;
    /// # extern crate subtle;
    /// # use curve25519_dalek::scalar::Scalar;
    /// # use subtle::ConditionallyAssignable;
    /// # fn main() {
    /// // 2^255 - 1, since `from_bits` clears the high bit
    /// let _2_255_minus_1 = Scalar::from_bits([0xff;32]);
    /// assert!(!_2_255_minus_1.is_canonical());
    ///
    /// let reduced = _2_255_minus_1.reduce();
    /// assert!(reduced.is_canonical());
    /// # }
    /// ```
    pub fn is_canonical(&self) -> bool {
        *self == self.reduce()
    }
}

impl UnpackedScalar {
    /// Pack the limbs of this `UnpackedScalar` into a `Scalar`.
    fn pack(&self) -> Scalar {
        Scalar{ bytes: self.to_bytes() }
    }

    /// Inverts an UnpackedScalar in Montgomery form.
    pub fn montgomery_invert(&self) -> UnpackedScalar {
        // Uses the addition chain from
        // https://briansmith.org/ecc-inversion-addition-chains-01#curve25519_scalar_inversion
        let    _1 = self;
        let   _10 = _1.montgomery_square();
        let  _100 = _10.montgomery_square();
        let   _11 = UnpackedScalar::montgomery_mul(&_10,     &_1);
        let  _101 = UnpackedScalar::montgomery_mul(&_10,    &_11);
        let  _111 = UnpackedScalar::montgomery_mul(&_10,   &_101);
        let _1001 = UnpackedScalar::montgomery_mul(&_10,   &_111);
        let _1011 = UnpackedScalar::montgomery_mul(&_10,  &_1001);
        let _1111 = UnpackedScalar::montgomery_mul(&_100, &_1011);

        // _10000
        let mut y = UnpackedScalar::montgomery_mul(&_1111, &_1);

        #[inline]
        fn square_multiply(y: &mut UnpackedScalar, squarings: usize, x: &UnpackedScalar) {
            for _ in 0..squarings {
                *y = y.montgomery_square();
            }
            *y = UnpackedScalar::montgomery_mul(y, x);
        }

        square_multiply(&mut y, 123 + 3, &_101);
        square_multiply(&mut y,   2 + 2, &_11);
        square_multiply(&mut y,   1 + 4, &_1111);
        square_multiply(&mut y,   1 + 4, &_1111);
        square_multiply(&mut y,       4, &_1001);
        square_multiply(&mut y,       2, &_11);
        square_multiply(&mut y,   1 + 4, &_1111);
        square_multiply(&mut y,   1 + 3, &_101);
        square_multiply(&mut y,   3 + 3, &_101);
        square_multiply(&mut y,       3, &_111);
        square_multiply(&mut y,   1 + 4, &_1111);
        square_multiply(&mut y,   2 + 3, &_111);
        square_multiply(&mut y,   2 + 2, &_11);
        square_multiply(&mut y,   1 + 4, &_1011);
        square_multiply(&mut y,   2 + 4, &_1011);
        square_multiply(&mut y,   6 + 4, &_1001);
        square_multiply(&mut y,   2 + 2, &_11);
        square_multiply(&mut y,   3 + 2, &_11);
        square_multiply(&mut y,   3 + 2, &_11);
        square_multiply(&mut y,   1 + 4, &_1001);
        square_multiply(&mut y,   1 + 3, &_111);
        square_multiply(&mut y,   2 + 4, &_1111);
        square_multiply(&mut y,   1 + 4, &_1011);
        square_multiply(&mut y,       3, &_101);
        square_multiply(&mut y,   2 + 4, &_1111);
        square_multiply(&mut y,       3, &_101);
        square_multiply(&mut y,   1 + 2, &_11);

        y
    }

    /// Inverts an UnpackedScalar not in Montgomery form.
    pub fn invert(&self) -> UnpackedScalar {
        self.to_montgomery().montgomery_invert().from_montgomery()
    }
}

#[cfg(test)]
mod test {
    use super::*;
    use constants;

    /// x = 2238329342913194256032495932344128051776374960164957527413114840482143558222
    pub static X: Scalar = Scalar{
        bytes: [
            0x4e, 0x5a, 0xb4, 0x34, 0x5d, 0x47, 0x08, 0x84,
            0x59, 0x13, 0xb4, 0x64, 0x1b, 0xc2, 0x7d, 0x52,
            0x52, 0xa5, 0x85, 0x10, 0x1b, 0xcc, 0x42, 0x44,
            0xd4, 0x49, 0xf4, 0xa8, 0x79, 0xd9, 0xf2, 0x04,
        ],
    };
    /// 1/x = 6859937278830797291664592131120606308688036382723378951768035303146619657244
    pub static XINV: Scalar = Scalar{
        bytes: [
            0x1c, 0xdc, 0x17, 0xfc, 0xe0, 0xe9, 0xa5, 0xbb,
            0xd9, 0x24, 0x7e, 0x56, 0xbb, 0x01, 0x63, 0x47,
            0xbb, 0xba, 0x31, 0xed, 0xd5, 0xa9, 0xbb, 0x96,
            0xd5, 0x0b, 0xcd, 0x7a, 0x3f, 0x96, 0x2a, 0x0f,
        ],
    };
    /// y = 2592331292931086675770238855846338635550719849568364935475441891787804997264
    pub static Y: Scalar = Scalar{
        bytes: [
            0x90, 0x76, 0x33, 0xfe, 0x1c, 0x4b, 0x66, 0xa4,
            0xa2, 0x8d, 0x2d, 0xd7, 0x67, 0x83, 0x86, 0xc3,
            0x53, 0xd0, 0xde, 0x54, 0x55, 0xd4, 0xfc, 0x9d,
            0xe8, 0xef, 0x7a, 0xc3, 0x1f, 0x35, 0xbb, 0x05,
        ],
    };

    /// x*y = 5690045403673944803228348699031245560686958845067437804563560795922180092780
    static X_TIMES_Y: Scalar = Scalar{
        bytes: [
            0x6c, 0x33, 0x74, 0xa1, 0x89, 0x4f, 0x62, 0x21,
            0x0a, 0xaa, 0x2f, 0xe1, 0x86, 0xa6, 0xf9, 0x2c,
            0xe0, 0xaa, 0x75, 0xc2, 0x77, 0x95, 0x81, 0xc2,
            0x95, 0xfc, 0x08, 0x17, 0x9a, 0x73, 0x94, 0x0c,
        ],
    };

    /// sage: l = 2^252 + 27742317777372353535851937790883648493
    /// sage: big = 2^256 - 1
    /// sage: repr((big % l).digits(256))
    static CANONICAL_2_256_MINUS_1: Scalar = Scalar{
        bytes: [
              28, 149, 152, 141, 116,  49, 236, 214,
             112, 207, 125, 115, 244,  91, 239, 198,
             254, 255, 255, 255, 255, 255, 255, 255,
             255, 255, 255, 255, 255, 255, 255,  15,
        ],
    };

    static A_SCALAR: Scalar = Scalar{
        bytes: [
            0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d,
            0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8, 0x26, 0x4d,
            0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1,
            0x58, 0x9e, 0x7b, 0x7f, 0x23, 0x76, 0xef, 0x09,
        ],
    };

    static A_NAF: [i8; 256] =
        [0,13,0,0,0,0,0,0,0,7,0,0,0,0,0,0,-9,0,0,0,0,-11,0,0,0,0,3,0,0,0,0,1,
         0,0,0,0,9,0,0,0,0,-5,0,0,0,0,0,0,3,0,0,0,0,11,0,0,0,0,11,0,0,0,0,0,
         -9,0,0,0,0,0,-3,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,9,0,
         0,0,0,-15,0,0,0,0,-7,0,0,0,0,-9,0,0,0,0,0,5,0,0,0,0,13,0,0,0,0,0,-3,0,
         0,0,0,-11,0,0,0,0,-7,0,0,0,0,-13,0,0,0,0,11,0,0,0,0,-9,0,0,0,0,0,1,0,0,
         0,0,0,-15,0,0,0,0,1,0,0,0,0,7,0,0,0,0,0,0,0,0,5,0,0,0,0,0,13,0,0,0,
         0,0,0,11,0,0,0,0,0,15,0,0,0,0,0,-9,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,7,
         0,0,0,0,0,-15,0,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0];

    #[test]
    fn fuzzer_testcase_reduction() {
        // LE bytes of 24519928653854221733733552434404946937899825954937634815
        let a_bytes = [255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 0, 0, 0, 0, 0, 0, 0, 0, 0];
        // LE bytes of 4975441334397345751130612518500927154628011511324180036903450236863266160640
        let b_bytes = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 255, 210, 210, 210, 255, 255, 255, 255, 10];
        // LE bytes of 6432735165214683820902750800207468552549813371247423777071615116673864412038
        let c_bytes = [134, 171, 119, 216, 180, 128, 178, 62, 171, 132, 32, 62, 34, 119, 104, 193, 47, 215, 181, 250, 14, 207, 172, 93, 75, 207, 211, 103, 144, 204, 56, 14];

        let a = Scalar::from_bytes_mod_order(a_bytes);
        let b = Scalar::from_bytes_mod_order(b_bytes);
        let c = Scalar::from_bytes_mod_order(c_bytes);

        let mut tmp = [0u8; 64];

        // also_a = (a mod l)
        tmp[0..32].copy_from_slice(&a_bytes[..]);
        let also_a = Scalar::from_bytes_mod_order_wide(&tmp);

        // also_b = (b mod l)
        tmp[0..32].copy_from_slice(&b_bytes[..]);
        let also_b = Scalar::from_bytes_mod_order_wide(&tmp);

        let expected_c = &a * &b;
        let also_expected_c = &also_a * &also_b;

        assert_eq!(c, expected_c);
        assert_eq!(c, also_expected_c);
    }

    #[test]
    fn non_adjacent_form() {
        let naf = A_SCALAR.non_adjacent_form(5);
        for i in 0..256 {
            assert_eq!(naf[i], A_NAF[i]);
        }
    }

    #[test]
    fn from_unsigned() {
        let val = 0xdeadbeefdeadbeef;
        let s = Scalar::from_u64(val);
        assert_eq!(s[7], 0xde);
        assert_eq!(s[6], 0xad);
        assert_eq!(s[5], 0xbe);
        assert_eq!(s[4], 0xef);
        assert_eq!(s[3], 0xde);
        assert_eq!(s[2], 0xad);
        assert_eq!(s[1], 0xbe);
        assert_eq!(s[0], 0xef);
    }

    #[test]
    fn scalar_mul_by_one() {
        let test_scalar = &X * &Scalar::one();
        for i in 0..32 {
            assert!(test_scalar[i] == X[i]);
        }
    }

    #[test]
    fn impl_add() {
        let two = Scalar::from_u64(2);
        let one = Scalar::one();
        let should_be_two = &one + &one;
        assert_eq!(should_be_two, two);
    }

    #[allow(non_snake_case)]
    #[test]
    fn impl_mul() {
        let should_be_X_times_Y = &X * &Y;
        assert_eq!(should_be_X_times_Y, X_TIMES_Y);
    }

    #[allow(non_snake_case)]
    #[test]
    fn impl_product() {
        // Test that product works for non-empty iterators
        let X_Y_vector = vec![X, Y];
        let should_be_X_times_Y: Scalar = X_Y_vector.iter().product();
        assert_eq!(should_be_X_times_Y, X_TIMES_Y);

        // Test that product works for the empty iterator
        let one = Scalar::one();
        let empty_vector = vec![];
        let should_be_one: Scalar = empty_vector.iter().product();
        assert_eq!(should_be_one, one);

        // Test that product works for iterators where Item = Scalar
        let xs = [Scalar::from_u64(2); 10];
        let ys = [Scalar::from_u64(3); 10];
        // now zs is an iterator with Item = Scalar
        let zs = xs.iter().zip(ys.iter()).map(|(x,y)| x * y);

        let x_prod: Scalar = xs.iter().product();
        let y_prod: Scalar = ys.iter().product();
        let z_prod: Scalar = zs.product();

        assert_eq!(x_prod, Scalar::from_u64(1024));
        assert_eq!(y_prod, Scalar::from_u64(59049));
        assert_eq!(z_prod, Scalar::from_u64(60466176));
        assert_eq!(x_prod * y_prod, z_prod);

    }

    #[test]
    fn impl_sum() {

        // Test that sum works for non-empty iterators
        let two = Scalar::from_u64(2);
        let one_vector = vec![Scalar::one(), Scalar::one()];
        let should_be_two: Scalar = one_vector.iter().sum();
        assert_eq!(should_be_two, two);

        // Test that sum works for the empty iterator
        let zero = Scalar::zero();
        let empty_vector = vec![];
        let should_be_zero: Scalar = empty_vector.iter().sum();
        assert_eq!(should_be_zero, zero);

        // Test that sum works for owned types
        let xs = [Scalar::from_u64(1); 10];
        let ys = [Scalar::from_u64(2); 10];
        // now zs is an iterator with Item = Scalar
        let zs = xs.iter().zip(ys.iter()).map(|(x,y)| x + y);

        let x_sum: Scalar = xs.iter().sum();
        let y_sum: Scalar = ys.iter().sum();
        let z_sum: Scalar = zs.sum();

        assert_eq!(x_sum, Scalar::from_u64(10));
        assert_eq!(y_sum, Scalar::from_u64(20));
        assert_eq!(z_sum, Scalar::from_u64(30));
        assert_eq!(x_sum + y_sum, z_sum);
    }

    #[test]
    fn square() {
        let expected = &X * &X;
        let actual = X.unpack().square().pack();
        for i in 0..32 {
            assert!(expected[i] == actual[i]);
        }
    }

    #[test]
    fn reduce() {
        let biggest = Scalar::from_bytes_mod_order([0xff; 32]);
        assert_eq!(biggest, CANONICAL_2_256_MINUS_1);
    }

    #[test]
    fn from_bytes_mod_order_wide() {
        let mut bignum = [0u8; 64];
        // set bignum = x + 2^256x
        for i in 0..32 {
            bignum[   i] = X[i];
            bignum[32+i] = X[i];
        }
        // 3958878930004874126169954872055634648693766179881526445624823978500314864344
        // = x + 2^256x (mod l)
        let reduced = Scalar{
            bytes: [
                216, 154, 179, 139, 210, 121,   2,  71,
                 69,  99, 158, 216,  23, 173,  63, 100,
                204,   0,  91,  50, 219, 153,  57, 249,
                 28,  82,  31, 197, 100, 165, 192,   8,
            ],
        };
        let test_red = Scalar::from_bytes_mod_order_wide(&bignum);
        for i in 0..32 {
            assert!(test_red[i] == reduced[i]);
        }
    }

    #[allow(non_snake_case)]
    #[test]
    fn invert() {
        let inv_X = X.invert();
        assert_eq!(inv_X, XINV);
        let should_be_one = &inv_X * &X;
        assert_eq!(should_be_one, Scalar::one());
    }

    // Negating a scalar twice should result in the original scalar.
    #[allow(non_snake_case)]
    #[test]
    fn neg_twice_is_identity() {
        let negative_X = -&X;
        let should_be_X = -&negative_X;

        assert_eq!(should_be_X, X);
    }

    #[test]
    fn to_bytes_from_bytes_roundtrips() {
        let unpacked = X.unpack();
        let bytes = unpacked.to_bytes();
        let should_be_unpacked = UnpackedScalar::from_bytes(&bytes);

        assert_eq!(should_be_unpacked.0, unpacked.0);
    }

    #[test]
    fn montgomery_reduce_matches_from_bytes_mod_order_wide() {
        let mut bignum = [0u8; 64];

        // set bignum = x + 2^256x
        for i in 0..32 {
            bignum[   i] = X[i];
            bignum[32+i] = X[i];
        }
        // x + 2^256x (mod l)
        //         = 3958878930004874126169954872055634648693766179881526445624823978500314864344
        let expected = Scalar{
            bytes: [
                216, 154, 179, 139, 210, 121,   2,  71,
                 69,  99, 158, 216,  23, 173,  63, 100,
                204,   0,  91,  50, 219, 153,  57, 249,
                 28,  82,  31, 197, 100, 165, 192,   8
            ],
        };
        let reduced = Scalar::from_bytes_mod_order_wide(&bignum);

        // The reduced scalar should match the expected
        assert_eq!(reduced.bytes, expected.bytes);

        //  (x + 2^256x) * R
        let interim = UnpackedScalar::mul_internal(&UnpackedScalar::from_bytes_wide(&bignum),
                                                   &constants::R);
        // ((x + 2^256x) * R) / R  (mod l)
        let montgomery_reduced = UnpackedScalar::montgomery_reduce(&interim);

        // The Montgomery reduced scalar should match the reduced one, as well as the expected
        assert_eq!(montgomery_reduced.0, reduced.unpack().0);
        assert_eq!(montgomery_reduced.0, expected.unpack().0)
    }

    #[test]
    fn canonical_decoding() {
        // canonical encoding of 1667457891
        let canonical_bytes = [99, 99, 99, 99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,];

        // encoding of
        //   7265385991361016183439748078976496179028704920197054998554201349516117938192
        // = 28380414028753969466561515933501938171588560817147392552250411230663687203 (mod l)
        // non_canonical because unreduced mod l
        let non_canonical_bytes_because_unreduced = [16; 32];

        // encoding with high bit set, to check that the parser isn't pre-masking the high bit
        let non_canonical_bytes_because_highbit = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 128];

        assert!( Scalar::from_canonical_bytes(canonical_bytes).is_some() );
        assert!( Scalar::from_canonical_bytes(non_canonical_bytes_because_unreduced).is_none() );
        assert!( Scalar::from_canonical_bytes(non_canonical_bytes_because_highbit).is_none() );
    }

    #[test]
    #[cfg(feature = "serde")]
    fn serde_cbor_scalar_roundtrip() {
        // XXX remove serde_cbor
        use serde_cbor;
        let output = serde_cbor::to_vec(&X).unwrap();
        let parsed: Scalar = serde_cbor::from_slice(&output).unwrap();
        assert_eq!(parsed, X);
    }

    #[cfg(debug_assertions)]
    #[test]
    #[should_panic]
    fn batch_invert_with_a_zero_input_panics() {
        let mut xs = vec![Scalar::one(); 16];
        xs[3] = Scalar::zero();
        // This should panic in debug mode.
        Scalar::batch_invert(&mut xs);
    }

    #[test]
    fn batch_invert_empty() {
        assert_eq!(Scalar::one(), Scalar::batch_invert(&mut []));
    }

    #[test]
    fn batch_invert_consistency() {
        let mut x = Scalar::from_u64(1);
        let mut v1: Vec<_> = (0..16).map(|_| {let tmp = x; x = x + x; tmp}).collect();
        let v2 = v1.clone();

        let expected: Scalar = v1.iter().product();
        let expected = expected.invert();
        let ret = Scalar::batch_invert(&mut v1);
        assert_eq!(ret, expected);

        for (a, b) in v1.iter().zip(v2.iter()) {
            assert_eq!(a * b, Scalar::one());
        }
    }
}