p384/arithmetic/

scalar.rs

//! secp384r1 scalar field elements.

#![allow(clippy::unusual_byte_groupings)]

#[cfg_attr(target_pointer_width = "32", path = "scalar/p384_scalar_32.rs")]
#[cfg_attr(target_pointer_width = "64", path = "scalar/p384_scalar_64.rs")]
#[allow(
    clippy::identity_op,
    clippy::too_many_arguments,
    clippy::unnecessary_cast
)]
mod scalar_impl;

use self::scalar_impl::*;
use crate::{FieldBytes, NistP384, SecretKey, ORDER_HEX, U384};
use core::{
    iter::{Product, Sum},
    ops::{AddAssign, MulAssign, Neg, Shr, ShrAssign, SubAssign},
};
use elliptic_curve::{
    bigint::{self, ArrayEncoding, Limb},
    ff::PrimeField,
    ops::{Invert, Reduce},
    scalar::{FromUintUnchecked, IsHigh},
    subtle::{Choice, ConditionallySelectable, ConstantTimeEq, ConstantTimeGreater, CtOption},
    Curve as _, Error, Result, ScalarPrimitive,
};

#[cfg(feature = "bits")]
use {crate::ScalarBits, elliptic_curve::group::ff::PrimeFieldBits};

#[cfg(feature = "serde")]
use serdect::serde::{de, ser, Deserialize, Serialize};

#[cfg(doc)]
use core::ops::{Add, Mul, Sub};

/// 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`](https://docs.rs/ff/) crate, which is re-exported as
/// `p384::elliptic_curve::ff`:
///
/// - [`Field`](https://docs.rs/ff/latest/ff/trait.Field.html) -
///   represents elements of finite fields and provides:
///   - [`Field::random`](https://docs.rs/ff/latest/ff/trait.Field.html#tymethod.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`](https://docs.rs/ff/latest/ff/trait.PrimeField.html) -
///   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`](https://docs.rs/ff/latest/ff/trait.PrimeFieldBits.html) -
///   operations over field elements represented as bits (requires `bits` feature)
///
/// Please see the documentation for the relevant traits for more information.
///
/// # `serde` support
///
/// When the `serde` feature of this crate is enabled, the `Serialize` and
/// `Deserialize` traits are impl'd for this type.
///
/// The serialization is a fixed-width big endian encoding. When used with
/// textual formats, the binary data is encoded as hexadecimal.
#[derive(Clone, Copy, Debug, PartialOrd, Ord)]
pub struct Scalar(U384);

primeorder::impl_mont_field_element!(
    NistP384,
    Scalar,
    FieldBytes,
    U384,
    NistP384::ORDER,
    fiat_p384_scalar_montgomery_domain_field_element,
    fiat_p384_scalar_from_montgomery,
    fiat_p384_scalar_to_montgomery,
    fiat_p384_scalar_add,
    fiat_p384_scalar_sub,
    fiat_p384_scalar_mul,
    fiat_p384_scalar_opp,
    fiat_p384_scalar_square
);

impl Scalar {
    /// Compute [`Scalar`] inversion: `1 / self`.
    pub fn invert(&self) -> CtOption<Self> {
        CtOption::new(self.invert_unchecked(), !self.is_zero())
    }

    /// Returns the multiplicative inverse of self.
    ///
    /// Does not check that self is non-zero.
    const fn invert_unchecked(&self) -> Self {
        let words = impl_field_invert!(
            self.to_canonical().as_words(),
            Self::ONE.0.to_words(),
            Limb::BITS,
            bigint::nlimbs!(U384::BITS),
            fiat_p384_scalar_mul,
            fiat_p384_scalar_opp,
            fiat_p384_scalar_divstep_precomp,
            fiat_p384_scalar_divstep,
            fiat_p384_scalar_msat,
            fiat_p384_scalar_selectznz,
        );

        Self(U384::from_words(words))
    }

    /// Compute modular square root.
    pub fn sqrt(&self) -> CtOption<Self> {
        // p mod 4 = 3 -> compute sqrt(x) using x^((p+1)/4) =
        // x^9850501549098619803069760025035903451269934817616361666986726319906914849778315892349739077038073728388608413485661
        let t1 = *self;
        let t10 = t1.square();
        let t11 = *self * t10;
        let t101 = t10 * t11;
        let t111 = t10 * t101;
        let t1001 = t10 * t111;
        let t1011 = t10 * t1001;
        let t1101 = t10 * t1011;
        let t1111 = t10 * t1101;
        let t11110 = t1111.square();
        let t11111 = t1 * t11110;
        let t1111100 = t11111.sqn(2);
        let t11111000 = t1111100.square();
        let i14 = t11111000.square();
        let i20 = i14.sqn(5) * i14;
        let i31 = i20.sqn(10) * i20;
        let i58 = (i31.sqn(4) * t11111000).sqn(21) * i31;
        let i110 = (i58.sqn(3) * t1111100).sqn(47) * i58;
        let x194 = i110.sqn(95) * i110 * t1111;
        let i225 = ((x194.sqn(6) * t111).sqn(3) * t11).sqn(7);
        let i235 = ((t1101 * i225).sqn(6) * t1101).square() * t1;
        let i258 = ((i235.sqn(11) * t11111).sqn(2) * t1).sqn(8);
        let i269 = ((t1101 * i258).sqn(2) * t11).sqn(6) * t1011;
        let i286 = ((i269.sqn(4) * t111).sqn(6) * t11111).sqn(5);
        let i308 = ((t1011 * i286).sqn(10) * t1101).sqn(9) * t1101;
        let i323 = ((i308.sqn(4) * t1011).sqn(6) * t1001).sqn(3);
        let i340 = ((t1 * i323).sqn(7) * t1011).sqn(7) * t101;
        let i357 = ((i340.sqn(5) * t111).sqn(5) * t1111).sqn(5);
        let i369 = ((t1011 * i357).sqn(4) * t1011).sqn(5) * t111;
        let i387 = ((i369.sqn(3) * t11).sqn(7) * t11).sqn(6);
        let i397 = ((t1011 * i387).sqn(4) * t101).sqn(3) * t11;
        let i413 = ((i397.sqn(4) * t11).sqn(4) * t11).sqn(6);
        let i427 = ((t101 * i413).sqn(5) * t101).sqn(6) * t1011;
        let x = i427.sqn(3) * t101;
        CtOption::new(x, x.square().ct_eq(&t1))
    }

    fn sqn(&self, n: usize) -> Self {
        let mut x = *self;
        for _ in 0..n {
            x = x.square();
        }
        x
    }

    /// Right shifts the scalar.
    ///
    /// Note: not constant-time with respect to the `shift` parameter.
    pub const fn shr_vartime(&self, shift: usize) -> Scalar {
        Self(self.0.shr_vartime(shift))
    }
}

impl AsRef<Scalar> for Scalar {
    fn as_ref(&self) -> &Scalar {
        self
    }
}

impl FromUintUnchecked for Scalar {
    type Uint = U384;

    fn from_uint_unchecked(uint: Self::Uint) -> Self {
        Self::from_uint_unchecked(uint)
    }
}

impl Invert for Scalar {
    type Output = CtOption<Self>;

    fn invert(&self) -> CtOption<Self> {
        self.invert()
    }
}

impl IsHigh for Scalar {
    fn is_high(&self) -> Choice {
        const MODULUS_SHR1: U384 = NistP384::ORDER.shr_vartime(1);
        self.to_canonical().ct_gt(&MODULUS_SHR1)
    }
}

impl Shr<usize> for Scalar {
    type Output = Self;

    fn shr(self, rhs: usize) -> Self::Output {
        self.shr_vartime(rhs)
    }
}

impl Shr<usize> for &Scalar {
    type Output = Scalar;

    fn shr(self, rhs: usize) -> Self::Output {
        self.shr_vartime(rhs)
    }
}

impl ShrAssign<usize> for Scalar {
    fn shr_assign(&mut self, rhs: usize) {
        *self = *self >> rhs;
    }
}

impl PrimeField for Scalar {
    type Repr = FieldBytes;

    const MODULUS: &'static str = ORDER_HEX;
    const CAPACITY: u32 = 383;
    const NUM_BITS: u32 = 384;
    const TWO_INV: Self = Self::from_u64(2).invert_unchecked();
    const MULTIPLICATIVE_GENERATOR: Self = Self::from_u64(2);
    const S: u32 = 1;
    const ROOT_OF_UNITY: Self = Self::from_hex("ffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf581a0db248b0a77aecec196accc52972");
    const ROOT_OF_UNITY_INV: Self = Self::ROOT_OF_UNITY.invert_unchecked();
    const DELTA: Self = Self::from_u64(4);

    #[inline]
    fn from_repr(bytes: FieldBytes) -> CtOption<Self> {
        Self::from_bytes(&bytes)
    }

    #[inline]
    fn to_repr(&self) -> FieldBytes {
        self.to_bytes()
    }

    #[inline]
    fn is_odd(&self) -> Choice {
        self.is_odd()
    }
}

#[cfg(feature = "bits")]
impl PrimeFieldBits for Scalar {
    type ReprBits = fiat_p384_scalar_montgomery_domain_field_element;

    fn to_le_bits(&self) -> ScalarBits {
        self.to_canonical().to_words().into()
    }

    fn char_le_bits() -> ScalarBits {
        NistP384::ORDER.to_words().into()
    }
}

impl Reduce<U384> for Scalar {
    type Bytes = FieldBytes;

    fn reduce(w: U384) -> Self {
        let (r, underflow) = w.sbb(&NistP384::ORDER, Limb::ZERO);
        let underflow = Choice::from((underflow.0 >> (Limb::BITS - 1)) as u8);
        Self::from_uint_unchecked(U384::conditional_select(&w, &r, !underflow))
    }

    #[inline]
    fn reduce_bytes(bytes: &FieldBytes) -> Self {
        Self::reduce(U384::from_be_byte_array(*bytes))
    }
}

impl From<ScalarPrimitive<NistP384>> for Scalar {
    fn from(w: ScalarPrimitive<NistP384>) -> Self {
        Scalar::from(&w)
    }
}

impl From<&ScalarPrimitive<NistP384>> for Scalar {
    fn from(w: &ScalarPrimitive<NistP384>) -> Scalar {
        Scalar::from_uint_unchecked(*w.as_uint())
    }
}

impl From<Scalar> for ScalarPrimitive<NistP384> {
    fn from(scalar: Scalar) -> ScalarPrimitive<NistP384> {
        ScalarPrimitive::from(&scalar)
    }
}

impl From<&Scalar> for ScalarPrimitive<NistP384> {
    fn from(scalar: &Scalar) -> ScalarPrimitive<NistP384> {
        ScalarPrimitive::new(scalar.into()).unwrap()
    }
}

impl From<Scalar> for FieldBytes {
    fn from(scalar: Scalar) -> Self {
        scalar.to_repr()
    }
}

impl From<&Scalar> for FieldBytes {
    fn from(scalar: &Scalar) -> Self {
        scalar.to_repr()
    }
}

impl From<Scalar> for U384 {
    fn from(scalar: Scalar) -> U384 {
        U384::from(&scalar)
    }
}

impl From<&Scalar> for U384 {
    fn from(scalar: &Scalar) -> U384 {
        scalar.to_canonical()
    }
}

impl From<&SecretKey> for Scalar {
    fn from(secret_key: &SecretKey) -> Scalar {
        *secret_key.to_nonzero_scalar()
    }
}

impl TryFrom<U384> for Scalar {
    type Error = Error;

    fn try_from(w: U384) -> Result<Self> {
        Option::from(Self::from_uint(w)).ok_or(Error)
    }
}

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

#[cfg(feature = "serde")]
impl<'de> Deserialize<'de> for Scalar {
    fn deserialize<D>(deserializer: D) -> core::result::Result<Self, D::Error>
    where
        D: de::Deserializer<'de>,
    {
        Ok(ScalarPrimitive::deserialize(deserializer)?.into())
    }
}

#[cfg(test)]
mod tests {
    use super::Scalar;
    use crate::FieldBytes;
    use elliptic_curve::ff::PrimeField;
    use primeorder::impl_primefield_tests;

    /// t = (modulus - 1) >> S
    const T: [u64; 6] = [
        0x76760cb5666294b9,
        0xac0d06d9245853bd,
        0xe3b1a6c0fa1b96ef,
        0xffffffffffffffff,
        0xffffffffffffffff,
        0x7fffffffffffffff,
    ];

    impl_primefield_tests!(Scalar, T);

    #[test]
    fn from_to_bytes_roundtrip() {
        let k: u64 = 42;
        let mut bytes = FieldBytes::default();
        bytes[40..].copy_from_slice(k.to_be_bytes().as_ref());

        let scalar = Scalar::from_repr(bytes).unwrap();
        assert_eq!(bytes, scalar.to_bytes());
    }

    /// Basic tests that multiplication works.
    #[test]
    fn multiply() {
        let one = Scalar::ONE;
        let two = one + one;
        let three = two + one;
        let six = three + three;
        assert_eq!(six, two * three);

        let minus_two = -two;
        let minus_three = -three;
        assert_eq!(two, -minus_two);

        assert_eq!(minus_three * minus_two, minus_two * minus_three);
        assert_eq!(six, minus_two * minus_three);
    }

    /// Basic tests that scalar inversion works.
    #[test]
    fn invert() {
        let one = Scalar::ONE;
        let three = one + one + one;
        let inv_three = three.invert().unwrap();
        assert_eq!(three * inv_three, one);

        let minus_three = -three;
        let inv_minus_three = minus_three.invert().unwrap();
        assert_eq!(inv_minus_three, -inv_three);
        assert_eq!(three * inv_minus_three, -one);
    }

    /// Basic tests that sqrt works.
    #[test]
    fn sqrt() {
        for &n in &[1u64, 4, 9, 16, 25, 36, 49, 64] {
            let scalar = Scalar::from(n);
            let sqrt = scalar.sqrt().unwrap();
            assert_eq!(sqrt.square(), scalar);
        }
    }
}