betrusted-curve25519-dalek-source/curve25519-dalek/src/montgomery.rs

// -*- mode: rust; -*-
//
// This file is part of curve25519-dalek.
// Copyright (c) 2016-2021 isis lovecruft
// Copyright (c) 2016-2019 Henry de Valence
// See LICENSE for licensing information.
//
// Authors:
// - isis agora lovecruft <isis@patternsinthevoid.net>
// - Henry de Valence <hdevalence@hdevalence.ca>

//! Scalar multiplication on the Montgomery form of Curve25519.
//!
//! To avoid notational confusion with the Edwards code, we use
//! variables \\( u, v \\) for the Montgomery curve, so that “Montgomery
//! \\(u\\)” here corresponds to “Montgomery \\(x\\)” elsewhere.
//!
//! Montgomery arithmetic works not on the curve itself, but on the
//! \\(u\\)-line, which discards sign information and unifies the curve
//! and its quadratic twist.  See [_Montgomery curves and their
//! arithmetic_][costello-smith] by Costello and Smith for more details.
//!
//! The `MontgomeryPoint` struct contains the affine \\(u\\)-coordinate
//! \\(u\_0(P)\\) of a point \\(P\\) on either the curve or the twist.
//! Here the map \\(u\_0 : \mathcal M \rightarrow \mathbb F\_p \\) is
//! defined by \\(u\_0((u,v)) = u\\); \\(u\_0(\mathcal O) = 0\\).  See
//! section 5.4 of Costello-Smith for more details.
//!
//! # Scalar Multiplication
//!
//! Scalar multiplication on `MontgomeryPoint`s is provided by the `*`
//! operator, which implements the Montgomery ladder.
//!
//! # Edwards Conversion
//!
//! The \\(2\\)-to-\\(1\\) map from the Edwards model to the Montgomery
//! \\(u\\)-line is provided by `EdwardsPoint::to_montgomery()`.
//!
//! To lift a `MontgomeryPoint` to an `EdwardsPoint`, use
//! `MontgomeryPoint::to_edwards()`, which takes a sign parameter.
//! This function rejects `MontgomeryPoints` which correspond to points
//! on the twist.
//!
//! [costello-smith]: https://eprint.iacr.org/2017/212.pdf

// We allow non snake_case names because coordinates in projective space are
// traditionally denoted by the capitalisation of their respective
// counterparts in affine space.  Yeah, you heard me, rustc, I'm gonna have my
// affine and projective cakes and eat both of them too.
#![allow(non_snake_case)]

use core::{
    hash::{Hash, Hasher},
    ops::{Mul, MulAssign},
};

use crate::constants::{APLUS2_OVER_FOUR, MONTGOMERY_A, MONTGOMERY_A_NEG};

use crate::edwards::{CompressedEdwardsY, EdwardsPoint};
use crate::field::FieldElement;
use crate::scalar::{clamp_integer, Scalar};

use crate::traits::Identity;

use subtle::Choice;
use subtle::ConstantTimeEq;
use subtle::{ConditionallyNegatable, ConditionallySelectable};

#[cfg(feature = "zeroize")]
use zeroize::Zeroize;

/// Holds the \\(u\\)-coordinate of a point on the Montgomery form of
/// Curve25519 or its twist.
#[derive(Copy, Clone, Debug, Default)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
pub struct MontgomeryPoint(pub [u8; 32]);

/// Equality of `MontgomeryPoint`s is defined mod p.
impl ConstantTimeEq for MontgomeryPoint {
    fn ct_eq(&self, other: &MontgomeryPoint) -> Choice {
        let self_fe = FieldElement::from_bytes(&self.0);
        let other_fe = FieldElement::from_bytes(&other.0);

        self_fe.ct_eq(&other_fe)
    }
}

impl PartialEq for MontgomeryPoint {
    fn eq(&self, other: &MontgomeryPoint) -> bool {
        self.ct_eq(other).into()
    }
}

impl Eq for MontgomeryPoint {}

// Equal MontgomeryPoints must hash to the same value. So we have to get them into a canonical
// encoding first
impl Hash for MontgomeryPoint {
    fn hash<H: Hasher>(&self, state: &mut H) {
        // Do a round trip through a `FieldElement`. `as_bytes` is guaranteed to give a canonical
        // 32-byte encoding
        let canonical_bytes = FieldElement::from_bytes(&self.0).as_bytes();
        canonical_bytes.hash(state);
    }
}

impl Identity for MontgomeryPoint {
    /// Return the group identity element, which has order 4.
    fn identity() -> MontgomeryPoint {
        MontgomeryPoint([0u8; 32])
    }
}

#[cfg(feature = "zeroize")]
impl Zeroize for MontgomeryPoint {
    fn zeroize(&mut self) {
        self.0.zeroize();
    }
}

impl MontgomeryPoint {
    /// Fixed-base scalar multiplication (i.e. multiplication by the base point).
    pub fn mul_base(scalar: &Scalar) -> Self {
        EdwardsPoint::mul_base(scalar).to_montgomery()
    }

    /// Multiply this point by `clamp_integer(bytes)`. For a description of clamping, see
    /// [`clamp_integer`].
    pub fn mul_clamped(self, bytes: [u8; 32]) -> Self {
        // We have to construct a Scalar that is not reduced mod l, which breaks scalar invariant
        // #2. But #2 is not necessary for correctness of variable-base multiplication. All that
        // needs to hold is invariant #1, i.e., the scalar is less than 2^255. This is guaranteed
        // by clamping.
        // Further, we don't do any reduction or arithmetic with this clamped value, so there's no
        // issues arising from the fact that the curve point is not necessarily in the prime-order
        // subgroup.
        let s = Scalar {
            bytes: clamp_integer(bytes),
        };
        s * self
    }

    /// Multiply the basepoint by `clamp_integer(bytes)`. For a description of clamping, see
    /// [`clamp_integer`].
    pub fn mul_base_clamped(bytes: [u8; 32]) -> Self {
        // See reasoning in Self::mul_clamped why it is OK to make an unreduced Scalar here. We
        // note that fixed-base multiplication is also defined for all values of `bytes` less than
        // 2^255.
        let s = Scalar {
            bytes: clamp_integer(bytes),
        };
        Self::mul_base(&s)
    }

    //TODO: understand _clamped multiplication_ and ensure we are doing it correctly and
    //compatibily as this has changed since our fork.
    /// Given `self` \\( = u\_0(P) \\), and a big-endian bit representation of an integer
    /// \\(n\\), return \\( u\_0(\[n\]P) \\). This is constant time in the length of `bits`.
    ///
    /// **NOTE:** You probably do not want to use this function. Almost every protocol built on
    /// Curve25519 uses _clamped multiplication_, explained
    /// [here](https://neilmadden.blog/2020/05/28/whats-the-curve25519-clamping-all-about/).
    /// When in doubt, use [`Self::mul_clamped`].
    pub fn mul_bits_be(&self, bits: impl Iterator<Item = bool>) -> MontgomeryPoint {
        // Algorithm 8 of Costello-Smith 2017
        let affine_u = FieldElement::from_bytes(&self.0);
        let mut x0 = ProjectivePoint::identity();
        let mut x1 = ProjectivePoint {
            U: affine_u,
            W: FieldElement::ONE,
        };

        // Go through the bits from most to least significant, using a sliding window of 2
        let mut prev_bit = false;
        for cur_bit in bits {
            let choice: u8 = (prev_bit ^ cur_bit) as u8;

            debug_assert!(choice == 0 || choice == 1);

            ProjectivePoint::conditional_swap(&mut x0, &mut x1, choice.into());
            differential_add_and_double(&mut x0, &mut x1, &affine_u);

            prev_bit = cur_bit;
        }
        // The final value of prev_bit above is scalar.bits()[0], i.e., the LSB of scalar
        ProjectivePoint::conditional_swap(&mut x0, &mut x1, Choice::from(prev_bit as u8));
        // Don't leave the bit in the stack
        #[cfg(feature = "zeroize")]
        prev_bit.zeroize();

        x0.as_affine()
    }

    /// View this `MontgomeryPoint` as an array of bytes.
    pub const fn as_bytes(&self) -> &[u8; 32] {
        &self.0
    }

    /// Convert this `MontgomeryPoint` to an array of bytes.
    pub const fn to_bytes(&self) -> [u8; 32] {
        self.0
    }

    /// Attempt to convert to an `EdwardsPoint`, using the supplied
    /// choice of sign for the `EdwardsPoint`.
    ///
    /// # Inputs
    ///
    /// * `sign`: a `u8` donating the desired sign of the resulting
    ///   `EdwardsPoint`.  `0` denotes positive and `1` negative.
    ///
    /// # Return
    ///
    /// * `Some(EdwardsPoint)` if `self` is the \\(u\\)-coordinate of a
    /// point on (the Montgomery form of) Curve25519;
    ///
    /// * `None` if `self` is the \\(u\\)-coordinate of a point on the
    /// twist of (the Montgomery form of) Curve25519;
    ///
    pub fn to_edwards(&self, sign: u8) -> Option<EdwardsPoint> {
        // To decompress the Montgomery u coordinate to an
        // `EdwardsPoint`, we apply the birational map to obtain the
        // Edwards y coordinate, then do Edwards decompression.
        //
        // The birational map is y = (u-1)/(u+1).
        //
        // The exceptional points are the zeros of the denominator,
        // i.e., u = -1.
        //
        // But when u = -1, v^2 = u*(u^2+486662*u+1) = 486660.
        //
        // Since this is nonsquare mod p, u = -1 corresponds to a point
        // on the twist, not the curve, so we can reject it early.

        let u = FieldElement::from_bytes(&self.0);

        if u == FieldElement::MINUS_ONE {
            return None;
        }

        let one = FieldElement::ONE;

        let y = &(&u - &one) * &(&u + &one).invert();

        let mut y_bytes = y.as_bytes();
        y_bytes[31] ^= sign << 7;

        CompressedEdwardsY(y_bytes).decompress()
    }
}

/// Perform the Elligator2 mapping to a Montgomery point.
///
/// See <https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-10#section-6.7.1>
//
// TODO Determine how much of the hash-to-group API should be exposed after the CFRG
//      draft gets into a more polished/accepted state.
#[allow(unused)]
pub(crate) fn elligator_encode(r_0: &FieldElement) -> MontgomeryPoint {
    let one = FieldElement::ONE;
    let d_1 = &one + &r_0.square2(); /* 2r^2 */

    let d = &MONTGOMERY_A_NEG * &(d_1.invert()); /* A/(1+2r^2) */

    let d_sq = &d.square();
    let au = &MONTGOMERY_A * &d;

    let inner = &(d_sq + &au) + &one;
    let eps = &d * &inner; /* eps = d^3 + Ad^2 + d */

    let (eps_is_sq, _eps) = FieldElement::sqrt_ratio_i(&eps, &one);

    let zero = FieldElement::ZERO;
    let Atemp = FieldElement::conditional_select(&MONTGOMERY_A, &zero, eps_is_sq); /* 0, or A if nonsquare*/
    let mut u = &d + &Atemp; /* d, or d+A if nonsquare */
    u.conditional_negate(!eps_is_sq); /* d, or -d-A if nonsquare */

    MontgomeryPoint(u.as_bytes())
}

/// A `ProjectivePoint` holds a point on the projective line
/// \\( \mathbb P(\mathbb F\_p) \\), which we identify with the Kummer
/// line of the Montgomery curve.
#[derive(Copy, Clone, Debug)]
pub(crate) struct ProjectivePoint {
    pub U: FieldElement,
    pub W: FieldElement,
}

impl Identity for ProjectivePoint {
    fn identity() -> ProjectivePoint {
        ProjectivePoint {
            U: FieldElement::ONE,
            W: FieldElement::ZERO,
        }
    }
}

impl Default for ProjectivePoint {
    fn default() -> ProjectivePoint {
        ProjectivePoint::identity()
    }
}

impl ConditionallySelectable for ProjectivePoint {
    fn conditional_select(
        a: &ProjectivePoint,
        b: &ProjectivePoint,
        choice: Choice,
    ) -> ProjectivePoint {
        ProjectivePoint {
            U: FieldElement::conditional_select(&a.U, &b.U, choice),
            W: FieldElement::conditional_select(&a.W, &b.W, choice),
        }
    }
}

impl ProjectivePoint {
    /// Dehomogenize this point to affine coordinates.
    ///
    /// # Return
    ///
    /// * \\( u = U / W \\) if \\( W \neq 0 \\);
    /// * \\( 0 \\) if \\( W \eq 0 \\);
    #[cfg(not(curve25519_dalek_backend = "u32e_backend"))]
    pub fn as_affine(&self) -> MontgomeryPoint {
        //TODO: consider making this a seperate feature. Something like "panic_on_sw_eval" which would
        //be ameniable to upstreaming
        #[cfg(all(not(test), curve25519_dalek_backend = "u32e_backend"))] // due to issue https://github.com/rust-lang/rust/issues/59168, you will have to manually comment this out when running a test on the full system and not just this crate.
        log::warn!("sw as_affine being used - check for build config errors!");
        let u = &self.U * &self.W.invert();
        MontgomeryPoint(u.as_bytes())
    }
    #[allow(dead_code)]
    #[cfg(curve25519_dalek_backend = "u32e_backend")]
    pub fn as_affine(&self) -> MontgomeryPoint {
        let mcode = assemble_engine25519!(
            start:
                // W.invert() in %21
                // U in %29
                // W in %30
                // result in %31
                // loop counter in %28

                // from FieldElement.invert()
                    // let (t19, t3) = self.pow22501();   // t19: 249..0 ; t3: 3,1,0
                    // let t0  = self.square();           // 1         e_0 = 2^1
                    mul %0, %30, %30  // self is W, e.g. %30
                    // let t1  = t0.square().square();    // 3         e_1 = 2^3
                    mul %1, %0, %0
                    mul %1, %1, %1
                    // let t2  = self * &t1;              // 3,0       e_2 = 2^3 + 2^0
                    mul %2, %30, %1
                    // let t3  = &t0 * &t2;               // 3,1,0
                    mul %3, %0, %2
                    // let t4  = t3.square();             // 4,2,1
                    mul %4, %3, %3
                    // let t5  = &t2 * &t4;               // 4,3,2,1,0
                    mul %5, %2, %4

                    // let t6  = t5.pow2k(5);             // 9,8,7,6,5
                    psa %28, #5       // coincidentally, constant #5 is the number 5
                    mul %6, %5, %5
                pow2k_5:
                    sub %28, %28, #1  // %28 = %28 - 1
                    brz pow2k_5_exit, %28
                    mul %6, %6, %6
                    brz pow2k_5, #0
                pow2k_5_exit:
                    // let t7  = &t6 * &t5;               // 9,8,7,6,5,4,3,2,1,0
                    mul %7, %6, %5

                    // let t8  = t7.pow2k(10);            // 19..10
                    psa %28, #6        // constant #6 is the number 10
                    mul %8, %7, %7
                pow2k_10:
                    sub %28, %28, #1
                    brz pow2k_10_exit, %28
                    mul %8, %8, %8
                    brz pow2k_10, #0
                pow2k_10_exit:
                    // let t9  = &t8 * &t7;               // 19..0
                    mul %9, %8, %7

                    // let t10 = t9.pow2k(20);            // 39..20
                    psa %28, #7         // constant #7 is the number 20
                    mul %10, %9, %9
                pow2k_20:
                    sub %28, %28, #1
                    brz pow2k_20_exit, %28
                    mul %10, %10, %10
                    brz pow2k_20, #0
                pow2k_20_exit:
                    // let t11 = &t10 * &t9;              // 39..0
                    mul %11, %10, %9

                    // let t12 = t11.pow2k(10);           // 49..10
                    psa %28, #6         // constant #6 is the number 10
                    mul %12, %11, %11
                pow2k_10b:
                    sub %28, %28, #1
                    brz pow2k_10b_exit, %28
                    mul %12, %12, %12
                    brz pow2k_10b, #0
                pow2k_10b_exit:
                    // let t13 = &t12 * &t7;              // 49..0
                    mul %13, %12, %7

                    // let t14 = t13.pow2k(50);           // 99..50
                    psa %28, #8         // constant #8 is the number 50
                    mul %14, %13, %13
                pow2k_50a:
                    sub %28, %28, #1
                    brz pow2k_50a_exit, %28
                    mul %14, %14, %14
                    brz pow2k_50a, #0
                pow2k_50a_exit:
                    // let t15 = &t14 * &t13;             // 99..0
                    mul %15, %14, %13

                    // let t16 = t15.pow2k(100);          // 199..100
                    psa %28, #9         // constant #9 is the number 100
                    mul %16, %15, %15
                pow2k_100:
                    sub %28, %28, #1
                    brz pow2k_100_exit, %28
                    mul %16, %16, %16
                    brz pow2k_100, #0
                pow2k_100_exit:
                    // let t17 = &t16 * &t15;             // 199..0
                    mul %17, %16, %15

                    // let t18 = t17.pow2k(50);           // 249..50
                    psa %28, #8         // constant #8 is the number 50
                    mul %18, %17, %17
                pow2k_50b:
                    sub %28, %28, #1
                    brz pow2k_50b_exit, %28
                    mul %18, %18, %18
                    brz pow2k_50b, #0
                pow2k_50b_exit:
                    // let t19 = &t18 * &t13;             // 249..0
                    mul %19, %18, %13
                    //(t19, t3) // just a return value, values are already there, do nothing

                    //let t20 = t19.pow2k(5);            // 254..5
                    psa %28, #5
                    mul %20, %19, %19
                pow2k_5_last:
                    sub %28, %28, #1
                    brz pow2k_5_last_exit, %28
                    mul %20, %20, %20
                    brz pow2k_5_last, #0
                pow2k_5_last_exit:

                    //let t21 = &t20 * &t3;              // 254..5,3,1,0
                    mul %21, %20, %3

                // u = &self.U * &self.W.invert()
                mul %31, %29, %21
                fin  // finish execution
        );

        use crate::backend::serial::u32e::*;
        match ensure_engine() {
            Ok(_) => {
                // safety: these were called after ensure_engine()
                let mut ucode_hw = unsafe { get_ucode() };
                let rf_hw = unsafe { get_rf() };

                copy_to_rf(self.U.as_bytes(), 29, rf_hw, 0);
                copy_to_rf(self.W.as_bytes(), 30, rf_hw, 0);

                let r = MontgomeryPoint(run_job(&mut ucode_hw, &rf_hw, &mcode, 0));
                #[cfg(feature="auto-release")]
                free_engine();
                r
            }
            _ => {
                #[cfg(feature="warn-fallback")]
                log::warn!("Hardware acceleration unavailable, falling back to software");
                let u = &self.U * &self.W.invert();
                MontgomeryPoint(u.as_bytes())
            }
        }
    }
}

/// Perform the double-and-add step of the Montgomery ladder.
///
/// Given projective points
/// \\( (U\_P : W\_P) = u(P) \\),
/// \\( (U\_Q : W\_Q) = u(Q) \\),
/// and the affine difference
/// \\(      u\_{P-Q} = u(P-Q) \\), set
/// $$
///     (U\_P : W\_P) \gets u(\[2\]P)
/// $$
/// and
/// $$
///     (U\_Q : W\_Q) \gets u(P + Q).
/// $$
#[cfg(not(curve25519_dalek_backend = "u32e_backend"))]
#[rustfmt::skip] // keep alignment of explanatory comments
pub(crate) fn differential_add_and_double(
    P: &mut ProjectivePoint,
    Q: &mut ProjectivePoint,
    affine_PmQ: &FieldElement,
) {
    let t0 = &P.U + &P.W;
    let t1 = &P.U - &P.W;
    let t2 = &Q.U + &Q.W;
    let t3 = &Q.U - &Q.W;

    let t4 = t0.square();   // (U_P + W_P)^2 = U_P^2 + 2 U_P W_P + W_P^2
    let t5 = t1.square();   // (U_P - W_P)^2 = U_P^2 - 2 U_P W_P + W_P^2

    let t6 = &t4 - &t5;     // 4 U_P W_P

    let t7 = &t0 * &t3;     // (U_P + W_P) (U_Q - W_Q) = U_P U_Q + W_P U_Q - U_P W_Q - W_P W_Q
    let t8 = &t1 * &t2;     // (U_P - W_P) (U_Q + W_Q) = U_P U_Q - W_P U_Q + U_P W_Q - W_P W_Q

    let t9  = &t7 + &t8;    // 2 (U_P U_Q - W_P W_Q)
    let t10 = &t7 - &t8;    // 2 (W_P U_Q - U_P W_Q)

    let t11 =  t9.square(); // 4 (U_P U_Q - W_P W_Q)^2
    let t12 = t10.square(); // 4 (W_P U_Q - U_P W_Q)^2

    let t13 = &APLUS2_OVER_FOUR * &t6; // (A + 2) U_P U_Q

    let t14 = &t4 * &t5;    // ((U_P + W_P)(U_P - W_P))^2 = (U_P^2 - W_P^2)^2
    let t15 = &t13 + &t5;   // (U_P - W_P)^2 + (A + 2) U_P W_P

    let t16 = &t6 * &t15;   // 4 (U_P W_P) ((U_P - W_P)^2 + (A + 2) U_P W_P)

    let t17 = affine_PmQ * &t12; // U_D * 4 (W_P U_Q - U_P W_Q)^2
    let t18 = t11;               // W_D * 4 (U_P U_Q - W_P W_Q)^2

    P.U = t14;  // U_{P'} = (U_P + W_P)^2 (U_P - W_P)^2
    P.W = t16;  // W_{P'} = (4 U_P W_P) ((U_P - W_P)^2 + ((A + 2)/4) 4 U_P W_P)
    Q.U = t18;  // U_{Q'} = W_D * 4 (U_P U_Q - W_P W_Q)^2
    Q.W = t17;  // W_{Q'} = U_D * 4 (W_P U_Q - U_P W_Q)^2
}

#[allow(dead_code)] // absorbed into mul, but might be useful later on as a subroutine for something else
#[cfg(curve25519_dalek_backend = "u32e_backend")]
pub(crate) fn differential_add_and_double(
    P: &mut ProjectivePoint,
    Q: &mut ProjectivePoint,
    affine_PmQ: &FieldElement,
) {
    let mcode = assemble_engine25519!(
        start:
            // P.U in %20
            // P.W in %21
            // Q.U in %22
            // Q.W in %23
            // affine_PmQ in %24
            // %30 is the TRD scratch register
            // %29 is the subtraction temporary value register

            // let t0 = &P.U + &P.W;
            add %0, %20, %21
            trd %30, %0
            sub %0, %0, %30
            // let t1 = &P.U - &P.W;
            sub %21, #3, %21    // negate &P.W using #FIELDPRIME (#3)
            add %1, %20, %21
            trd %30, %1
            sub %1, %1, %30
            // let t2 = &Q.U + &Q.W;
            add %2, %22, %23
            trd %30, %2
            sub %2, %2, %30
            // let t3 = &Q.U - &Q.W;
            sub %23, #3, %23
            add %3, %22, %23
            trd %30, %3
            sub %3, %3, %30
            // let t4 = t0.square();   // (U_P + W_P)^2 = U_P^2 + 2 U_P W_P + W_P^2
            mul %4, %0, %0
            // let t5 = t1.square();   // (U_P - W_P)^2 = U_P^2 - 2 U_P W_P + W_P^2
            mul %5, %1, %1
            // let t6 = &t4 - &t5;     // 4 U_P W_P
            sub %29, #3, %5
            add %6, %4, %29
            trd %30, %6
            sub %6, %6, %30
            // let t7 = &t0 * &t3;     // (U_P + W_P) (U_Q - W_Q) = U_P U_Q + W_P U_Q - U_P W_Q - W_P W_Q
            mul %7, %0, %3
            // let t8 = &t1 * &t2;     // (U_P - W_P) (U_Q + W_Q) = U_P U_Q - W_P U_Q + U_P W_Q - W_P W_Q
            mul %8, %1, %2
            // let t9  = &t7 + &t8;    // 2 (U_P U_Q - W_P W_Q)
            add %9, %7, %8
            trd %30, %9
            sub %9, %9, %30
            // let t10 = &t7 - &t8;    // 2 (W_P U_Q - U_P W_Q)
            sub %29, #3, %8
            add %10, %7, %29
            trd %30, %10
            sub %10, %10, %30
            // let t11 =  t9.square(); // 4 (U_P U_Q - W_P W_Q)^2
            mul %11, %9, %9
            // let t12 = t10.square(); // 4 (W_P U_Q - U_P W_Q)^2
            mul %12, %10, %10
            // let t13 = &APLUS2_OVER_FOUR * &t6; // (A + 2) U_P U_Q
            mul %13, #4, %6   // #4 is A+2/4
            // let t14 = &t4 * &t5;    // ((U_P + W_P)(U_P - W_P))^2 = (U_P^2 - W_P^2)^2
            mul %14, %4, %5
            // let t15 = &t13 + &t5;   // (U_P - W_P)^2 + (A + 2) U_P W_P
            add %15, %13, %5
            trd %30, %15
            sub %15, %15, %30
            // let t16 = &t6 * &t15;   // 4 (U_P W_P) ((U_P - W_P)^2 + (A + 2) U_P W_P)
            mul %16, %6, %15
            // let t17 = affine_PmQ * &t12; // U_D * 4 (W_P U_Q - U_P W_Q)^2
            mul %17, %24, %12    // affine_PmQ loaded into %24

            ///// these can be eliminated down the road, but included for 1:1 algorithm correspodence to reference in early testing
            // let t18 = t11;               // W_D * 4 (U_P U_Q - W_P W_Q)^2
            psa %18, %11
            // P.U = t14;  // U_{P'} = (U_P + W_P)^2 (U_P - W_P)^2
            psa %20, %14
            // P.W = t16;  // W_{P'} = (4 U_P W_P) ((U_P - W_P)^2 + ((A + 2)/4) 4 U_P W_P)
            psa %21, %16
            // Q.U = t18;  // U_{Q'} = W_D * 4 (U_P U_Q - W_P W_Q)^2
            psa %22, %18
            // Q.W = t17;  // W_{Q'} = U_D * 4 (W_P U_Q - U_P W_Q)^2
            psa %23, %17

            fin  // finish execution
    );
    use crate::backend::serial::u32e::*;
    match ensure_engine() {
        Ok(_) => {
            // safety: these were called after ensure_engine()
            let mut ucode_hw = unsafe { get_ucode() };
            let rf_hw = unsafe { get_rf() };

            // P.U in %20
            // P.W in %21
            // Q.U in %22
            // Q.W in %23
            // affine_PmQ in %24
            copy_to_rf(P.U.as_bytes(), 20, rf_hw, 0);
            copy_to_rf(P.W.as_bytes(), 21, rf_hw, 0);
            copy_to_rf(Q.U.as_bytes(), 22, rf_hw, 0);
            copy_to_rf(Q.W.as_bytes(), 23, rf_hw, 0);
            copy_to_rf(affine_PmQ.as_bytes(), 24, rf_hw, 0);

            // start the run
            run_job(&mut ucode_hw, &rf_hw, &mcode, 0);

            P.U = FieldElement::from_bytes(&copy_from_rf(20, &rf_hw, 0));
            P.W = FieldElement::from_bytes(&copy_from_rf(21, &rf_hw, 0));
            Q.U = FieldElement::from_bytes(&copy_from_rf(22, &rf_hw, 0));
            Q.W = FieldElement::from_bytes(&copy_from_rf(23, &rf_hw, 0));
            #[cfg(feature="auto-release")]
            free_engine();
        }
        _ => {
            #[cfg(feature="warn-fallback")]
            log::warn!("Hardware acceleration unavailable, falling back to software");
            let t0 = &P.U + &P.W;
            let t1 = &P.U - &P.W;
            let t2 = &Q.U + &Q.W;
            let t3 = &Q.U - &Q.W;

            let t4 = t0.square();   // (U_P + W_P)^2 = U_P^2 + 2 U_P W_P + W_P^2
            let t5 = t1.square();   // (U_P - W_P)^2 = U_P^2 - 2 U_P W_P + W_P^2

            let t6 = &t4 - &t5;     // 4 U_P W_P

            let t7 = &t0 * &t3;     // (U_P + W_P) (U_Q - W_Q) = U_P U_Q + W_P U_Q - U_P W_Q - W_P W_Q
            let t8 = &t1 * &t2;     // (U_P - W_P) (U_Q + W_Q) = U_P U_Q - W_P U_Q + U_P W_Q - W_P W_Q

            let t9  = &t7 + &t8;    // 2 (U_P U_Q - W_P W_Q)
            let t10 = &t7 - &t8;    // 2 (W_P U_Q - U_P W_Q)

            let t11 =  t9.square(); // 4 (U_P U_Q - W_P W_Q)^2
            let t12 = t10.square(); // 4 (W_P U_Q - U_P W_Q)^2

            let t13 = &APLUS2_OVER_FOUR * &t6; // (A + 2) U_P U_Q

            let t14 = &t4 * &t5;    // ((U_P + W_P)(U_P - W_P))^2 = (U_P^2 - W_P^2)^2
            let t15 = &t13 + &t5;   // (U_P - W_P)^2 + (A + 2) U_P W_P

            let t16 = &t6 * &t15;   // 4 (U_P W_P) ((U_P - W_P)^2 + (A + 2) U_P W_P)

            let t17 = affine_PmQ * &t12; // U_D * 4 (W_P U_Q - U_P W_Q)^2
            let t18 = t11;               // W_D * 4 (U_P U_Q - W_P W_Q)^2

            P.U = t14;  // U_{P'} = (U_P + W_P)^2 (U_P - W_P)^2
            P.W = t16;  // W_{P'} = (4 U_P W_P) ((U_P - W_P)^2 + ((A + 2)/4) 4 U_P W_P)
            Q.U = t18;  // U_{Q'} = W_D * 4 (U_P U_Q - W_P W_Q)^2
            Q.W = t17;  // W_{Q'} = U_D * 4 (W_P U_Q - U_P W_Q)^2
        }
    }
}

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

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

/// Multiply this `MontgomeryPoint` by a `Scalar`.
impl Mul<&Scalar> for &MontgomeryPoint {
    type Output = MontgomeryPoint;

    /// Given `self` \\( = u\_0(P) \\), and a `Scalar` \\(n\\), return \\( u\_0([n]P) \\).
    #[cfg(curve25519_dalek_backend = "u32e_backend")]
    fn mul(self, scalar: &Scalar) -> MontgomeryPoint {
        use crate::backend::serial::u32e::*;

        log::debug!("hw mont");
        // Algorithm 8 of Costello-Smith 2017
        let affine_u = FieldElement::from_bytes(&self.0);
        let x0 = ProjectivePoint::identity();
        let x1 = ProjectivePoint {
            U: affine_u,
            W: FieldElement::ONE,
        };

        // for now, prefer to use the fully-accelerated version where this code is local to the server
        // instead of transmitting it every call with the data...gives about a 2x performance speedup
        let mcode = assemble_engine25519!(
            start:
                // P.U in %20
                // P.W in %21
                // Q.U in %22
                // Q.W in %23
                // affine_PmQ in %24
                // %30 is the TRD scratch register and cswap dummy
                // %29 is the subtraction temporary value register and k_t
                // x0.U in %25
                // x0.W in %26
                // x1.U in %27
                // x1.W in %28
                // %19 is the loop counter, starts with 254 (if 0, loop runs exactly once)
                // %31 is the scalar
                // %18 is the swap variable
                psa %18, #0

                // for i in (0..255).rev()
            mainloop:
                // let choice: u8 = (bits[i + 1] ^ bits[i]) as u8;
                // ProjectivePoint::conditional_swap(&mut x0, &mut x1, choice.into());
                xbt %29, %31        // orignally[k_t = (k>>t) & 1] now[k_t = k[254]]
                shl %31, %31        // k = k<<1
                xor %18, %18, %29   // swap ^= k_t

                // cswap x0.U (%25), x1.U (%27)
                xor %30, %25, %27
                msk %30, %18, %30
                xor %25, %30, %25
                xor %27, %30, %27
                // cswap x0.W (%26), x1.W (%28)
                xor %30, %26, %28
                msk %30, %18, %30
                xor %26, %30, %26
                xor %28, %30, %28

                psa %18, %29  // swap = k_t

                    // differential_add_and_double(&mut x0, &mut x1, &affine_u);
                    psa %20, %25
                    psa %21, %26
                    psa %22, %27
                    psa %23, %28
                    // affine_u is already in %24

                    // let t0 = &P.U + &P.W;
                    add %0, %20, %21
                    trd %30, %0
                    sub %0, %0, %30
                    // let t1 = &P.U - &P.W;
                    sub %21, #3, %21    // negate &P.W using #FIELDPRIME (#3)
                    add %1, %20, %21
                    trd %30, %1
                    sub %1, %1, %30
                    // let t2 = &Q.U + &Q.W;
                    add %2, %22, %23
                    trd %30, %2
                    sub %2, %2, %30
                    // let t3 = &Q.U - &Q.W;
                    sub %23, #3, %23
                    add %3, %22, %23
                    trd %30, %3
                    sub %3, %3, %30
                    // let t4 = t0.square();   // (U_P + W_P)^2 = U_P^2 + 2 U_P W_P + W_P^2
                    mul %4, %0, %0
                    // let t5 = t1.square();   // (U_P - W_P)^2 = U_P^2 - 2 U_P W_P + W_P^2
                    mul %5, %1, %1
                    // let t6 = &t4 - &t5;     // 4 U_P W_P
                    sub %29, #3, %5
                    add %6, %4, %29
                    trd %30, %6
                    sub %6, %6, %30
                    // let t7 = &t0 * &t3;     // (U_P + W_P) (U_Q - W_Q) = U_P U_Q + W_P U_Q - U_P W_Q - W_P W_Q
                    mul %7, %0, %3
                    // let t8 = &t1 * &t2;     // (U_P - W_P) (U_Q + W_Q) = U_P U_Q - W_P U_Q + U_P W_Q - W_P W_Q
                    mul %8, %1, %2
                    // let t9  = &t7 + &t8;    // 2 (U_P U_Q - W_P W_Q)
                    add %9, %7, %8
                    trd %30, %9
                    sub %9, %9, %30
                    // let t10 = &t7 - &t8;    // 2 (W_P U_Q - U_P W_Q)
                    sub %29, #3, %8
                    add %10, %7, %29
                    trd %30, %10
                    sub %10, %10, %30
                    // let t11 =  t9.square(); // 4 (U_P U_Q - W_P W_Q)^2
                    mul %11, %9, %9
                    // let t12 = t10.square(); // 4 (W_P U_Q - U_P W_Q)^2
                    mul %12, %10, %10
                    // let t13 = &APLUS2_OVER_FOUR * &t6; // (A + 2) U_P U_Q
                    mul %13, #4, %6   // #4 is A+2/4
                    // let t14 = &t4 * &t5;    // ((U_P + W_P)(U_P - W_P))^2 = (U_P^2 - W_P^2)^2
                    mul %14, %4, %5
                    // let t15 = &t13 + &t5;   // (U_P - W_P)^2 + (A + 2) U_P W_P
                    add %15, %13, %5
                    trd %30, %15
                    sub %15, %15, %30
                    // let t16 = &t6 * &t15;   // 4 (U_P W_P) ((U_P - W_P)^2 + (A + 2) U_P W_P)
                    mul %16, %6, %15
                    // let t17 = affine_PmQ * &t12; // U_D * 4 (W_P U_Q - U_P W_Q)^2
                    mul %17, %24, %12    // affine_PmQ loaded into %24

                    ///// these can be eliminated down the road, but included for 1:1 algorithm correspodence to reference in early testing
                    // P.U = t14;  // U_{P'} = (U_P + W_P)^2 (U_P - W_P)^2
                    psa %20, %14
                    // P.W = t16;  // W_{P'} = (4 U_P W_P) ((U_P - W_P)^2 + ((A + 2)/4) 4 U_P W_P)
                    psa %21, %16
                    // let t18 = t11;               // W_D * 4 (U_P U_Q - W_P W_Q)^2
                    // Q.U = t18;  // U_{Q'} = W_D * 4 (U_P U_Q - W_P W_Q)^2
                    psa %22, %11   // collapsed two to save a register
                    // Q.W = t17;  // W_{Q'} = U_D * 4 (W_P U_Q - U_P W_Q)^2
                    psa %23, %17

                    ///// 'return' arguments for next iteration, can be optimized out later
                    psa %25, %20
                    psa %26, %21
                    psa %27, %22
                    psa %28, %23

                brz end, %19     // if loop counter is 0, quit
                sub %19, %19, #1 // subtract one from the loop counter and run again
                brz mainloop, #0    // go back to the top
            end:
                // ProjectivePoint::conditional_swap(&mut x0, &mut x1, Choice::from(bits[0] as u8));
                // cswap x0.U (%25), x1.U (%27)
                xor %30, %25, %27
                msk %30, %18, %30
                xor %25, %30, %25
                xor %27, %30, %27
                // cswap x0.W (%26), x1.W (%28)
                xor %30, %26, %28
                msk %30, %18, %30
                xor %26, %30, %26
                xor %28, %30, %28

                // AFFINE SPLICE -- pass arguments to the affine block
                psa %29, %25
                psa %30, %26
                // W.invert() in %21
                // U in %29
                // W in %30
                // result in %31
                // loop counter in %28

                // from FieldElement.invert()
                    // let (t19, t3) = self.pow22501();   // t19: 249..0 ; t3: 3,1,0
                    // let t0  = self.square();           // 1         e_0 = 2^1
                    mul %0, %30, %30  // self is W, e.g. %30
                    // let t1  = t0.square().square();    // 3         e_1 = 2^3
                    mul %1, %0, %0
                    mul %1, %1, %1
                    // let t2  = self * &t1;              // 3,0       e_2 = 2^3 + 2^0
                    mul %2, %30, %1
                    // let t3  = &t0 * &t2;               // 3,1,0
                    mul %3, %0, %2
                    // let t4  = t3.square();             // 4,2,1
                    mul %4, %3, %3
                    // let t5  = &t2 * &t4;               // 4,3,2,1,0
                    mul %5, %2, %4

                    // let t6  = t5.pow2k(5);             // 9,8,7,6,5
                    psa %28, #5       // coincidentally, constant #5 is the number 5
                    mul %6, %5, %5
                pow2k_5:
                    sub %28, %28, #1  // %28 = %28 - 1
                    brz pow2k_5_exit, %28
                    mul %6, %6, %6
                    brz pow2k_5, #0
                pow2k_5_exit:
                    // let t7  = &t6 * &t5;               // 9,8,7,6,5,4,3,2,1,0
                    mul %7, %6, %5

                    // let t8  = t7.pow2k(10);            // 19..10
                    psa %28, #6        // constant #6 is the number 10
                    mul %8, %7, %7
                pow2k_10:
                    sub %28, %28, #1
                    brz pow2k_10_exit, %28
                    mul %8, %8, %8
                    brz pow2k_10, #0
                pow2k_10_exit:
                    // let t9  = &t8 * &t7;               // 19..0
                    mul %9, %8, %7

                    // let t10 = t9.pow2k(20);            // 39..20
                    psa %28, #7         // constant #7 is the number 20
                    mul %10, %9, %9
                pow2k_20:
                    sub %28, %28, #1
                    brz pow2k_20_exit, %28
                    mul %10, %10, %10
                    brz pow2k_20, #0
                pow2k_20_exit:
                    // let t11 = &t10 * &t9;              // 39..0
                    mul %11, %10, %9

                    // let t12 = t11.pow2k(10);           // 49..10
                    psa %28, #6         // constant #6 is the number 10
                    mul %12, %11, %11
                pow2k_10b:
                    sub %28, %28, #1
                    brz pow2k_10b_exit, %28
                    mul %12, %12, %12
                    brz pow2k_10b, #0
                pow2k_10b_exit:
                    // let t13 = &t12 * &t7;              // 49..0
                    mul %13, %12, %7

                    // let t14 = t13.pow2k(50);           // 99..50
                    psa %28, #8         // constant #8 is the number 50
                    mul %14, %13, %13
                pow2k_50a:
                    sub %28, %28, #1
                    brz pow2k_50a_exit, %28
                    mul %14, %14, %14
                    brz pow2k_50a, #0
                pow2k_50a_exit:
                    // let t15 = &t14 * &t13;             // 99..0
                    mul %15, %14, %13

                    // let t16 = t15.pow2k(100);          // 199..100
                    psa %28, #9         // constant #9 is the number 100
                    mul %16, %15, %15
                pow2k_100:
                    sub %28, %28, #1
                    brz pow2k_100_exit, %28
                    mul %16, %16, %16
                    brz pow2k_100, #0
                pow2k_100_exit:
                    // let t17 = &t16 * &t15;             // 199..0
                    mul %17, %16, %15

                    // let t18 = t17.pow2k(50);           // 249..50
                    psa %28, #8         // constant #8 is the number 50
                    mul %18, %17, %17
                pow2k_50b:
                    sub %28, %28, #1
                    brz pow2k_50b_exit, %28
                    mul %18, %18, %18
                    brz pow2k_50b, #0
                pow2k_50b_exit:
                    // let t19 = &t18 * &t13;             // 249..0
                    mul %19, %18, %13
                    //(t19, t3) // just a return value, values are already there, do nothing

                    //let t20 = t19.pow2k(5);            // 254..5
                    psa %28, #5
                    mul %20, %19, %19
                pow2k_5_last:
                    sub %28, %28, #1
                    brz pow2k_5_last_exit, %28
                    mul %20, %20, %20
                    brz pow2k_5_last, #0
                pow2k_5_last_exit:

                    //let t21 = &t20 * &t3;              // 254..5,3,1,0
                    mul %21, %20, %3

                // u = &self.U * &self.W.invert()
                mul %31, %29, %21
                fin  // finish execution
        );

        let window = 0;
        match ensure_engine() {
            Ok(_) => {
                // safety: these were called after ensure_engine()
                let mut ucode_hw = unsafe { get_ucode() };
                let mut rf_hw = unsafe { get_rf() };

                copy_to_rf(x0.U.as_bytes(), 25, &mut rf_hw, window);
                copy_to_rf(x0.W.as_bytes(), 26, &mut rf_hw, window);
                copy_to_rf(x1.U.as_bytes(), 27, &mut rf_hw, window);
                copy_to_rf(x1.W.as_bytes(), 28, &mut rf_hw, window);
                copy_to_rf(affine_u.as_bytes(), 24, &mut rf_hw, window);
                copy_to_rf(scalar.bytes, 31, &mut rf_hw, window);
                copy_to_rf(
                    [
                        254, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
                        0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
                        0x00, 0x00, 0x00, 0x00,
                    ],
                    19,
                    &mut rf_hw,
                    window,
                ); // 254 as loop counter

                let r = MontgomeryPoint(run_job(&mut ucode_hw, &rf_hw, &mcode, window));
                #[cfg(feature="auto-release")]
                free_engine();
                r
            }
            _ => {
                #[cfg(feature="warn-fallback")]
                log::warn!("Hardware acceleration unavailable, falling back to software");
                // We multiply by the integer representation of the given Scalar. By scalar invariant #1,
                // the MSB is 0, so we can skip it.
                self.mul_bits_be(scalar.bits_le().rev().skip(1))
            }
        }
    }

    /// Given `self` \\( = u\_0(P) \\), and a `Scalar` \\(n\\), return \\( u\_0(\[n\]P) \\)
    #[cfg(not(curve25519_dalek_backend = "u32e_backend"))]
    fn mul(self, scalar: &Scalar) -> MontgomeryPoint {
        #[cfg(all(not(test), curve25519_dalek_backend = "u32e_backend"))] // due to issue https://github.com/rust-lang/rust/issues/59168, you will have to manually comment this out when running a test on the full system and not just this crate.
        log::warn!("sw montgomery multiply being used - check for build config errors!");
        // We multiply by the integer representation of the given Scalar. By scalar invariant #1,
        // the MSB is 0, so we can skip it.
        self.mul_bits_be(scalar.bits_le().rev().skip(1))
    }
}

impl MulAssign<&Scalar> for MontgomeryPoint {
    fn mul_assign(&mut self, scalar: &Scalar) {
        *self = (self as &MontgomeryPoint) * scalar;
    }
}

impl Mul<&MontgomeryPoint> for &Scalar {
    type Output = MontgomeryPoint;

    fn mul(self, point: &MontgomeryPoint) -> MontgomeryPoint {
        point * self
    }
}

// ------------------------------------------------------------------------
// Tests
// ------------------------------------------------------------------------

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

    #[cfg(feature = "alloc")]
    use alloc::vec::Vec;

    use rand_core::{CryptoRng, RngCore};

    #[test]
    fn identity_in_different_coordinates() {
        let id_projective = ProjectivePoint::identity();
        let id_montgomery = id_projective.as_affine();

        assert!(id_montgomery == MontgomeryPoint::identity());
    }

    #[test]
    fn identity_in_different_models() {
        assert!(EdwardsPoint::identity().to_montgomery() == MontgomeryPoint::identity());
    }

    #[test]
    #[cfg(feature = "serde")]
    fn serde_bincode_basepoint_roundtrip() {
        use bincode;

        let encoded = bincode::serialize(&constants::X25519_BASEPOINT).unwrap();
        let decoded: MontgomeryPoint = bincode::deserialize(&encoded).unwrap();

        assert_eq!(encoded.len(), 32);
        assert_eq!(decoded, constants::X25519_BASEPOINT);

        let raw_bytes = constants::X25519_BASEPOINT.as_bytes();
        let bp: MontgomeryPoint = bincode::deserialize(raw_bytes).unwrap();
        assert_eq!(bp, constants::X25519_BASEPOINT);
    }

    /// Test Montgomery -> Edwards on the X/Ed25519 basepoint
    #[test]
    fn basepoint_montgomery_to_edwards() {
        // sign bit = 0 => basepoint
        assert_eq!(
            constants::ED25519_BASEPOINT_POINT,
            constants::X25519_BASEPOINT.to_edwards(0).unwrap()
        );
        // sign bit = 1 => minus basepoint
        assert_eq!(
            -constants::ED25519_BASEPOINT_POINT,
            constants::X25519_BASEPOINT.to_edwards(1).unwrap()
        );
    }

    /// Test Edwards -> Montgomery on the X/Ed25519 basepoint
    #[test]
    fn basepoint_edwards_to_montgomery() {
        assert_eq!(
            constants::ED25519_BASEPOINT_POINT.to_montgomery(),
            constants::X25519_BASEPOINT
        );
    }

    /// Check that Montgomery -> Edwards fails for points on the twist.
    #[test]
    fn montgomery_to_edwards_rejects_twist() {
        let one = FieldElement::ONE;

        // u = 2 corresponds to a point on the twist.
        let two = MontgomeryPoint((&one + &one).as_bytes());

        assert!(two.to_edwards(0).is_none());

        // u = -1 corresponds to a point on the twist, but should be
        // checked explicitly because it's an exceptional point for the
        // birational map.  For instance, libsignal will accept it.
        let minus_one = MontgomeryPoint((-&one).as_bytes());

        assert!(minus_one.to_edwards(0).is_none());
    }

    #[test]
    fn eq_defined_mod_p() {
        let mut u18_bytes = [0u8; 32];
        u18_bytes[0] = 18;
        let u18 = MontgomeryPoint(u18_bytes);
        let u18_unred = MontgomeryPoint([255; 32]);

        assert_eq!(u18, u18_unred);
    }

    /// Returns a random point on the prime-order subgroup
    fn rand_prime_order_point(mut rng: impl RngCore + CryptoRng) -> EdwardsPoint {
        let s: Scalar = Scalar::random(&mut rng);
        EdwardsPoint::mul_base(&s)
    }

    /// Given a bytestring that's little-endian at the byte level, return an iterator over all the
    /// bits, in little-endian order.
    fn bytestring_bits_le(x: &[u8]) -> impl DoubleEndedIterator<Item = bool> + Clone + '_ {
        let bitlen = x.len() * 8;
        (0..bitlen).map(|i| {
            // As i runs from 0..256, the bottom 3 bits index the bit, while the upper bits index
            // the byte. Since self.bytes is little-endian at the byte level, this iterator is
            // little-endian on the bit level
            ((x[i >> 3] >> (i & 7)) & 1u8) == 1
        })
    }

    #[test]
    fn montgomery_ladder_matches_edwards_scalarmult() {
        let mut csprng = rand_core::OsRng;

        for _ in 0..100 {
            let p_edwards = rand_prime_order_point(&mut csprng);
            let p_montgomery: MontgomeryPoint = p_edwards.to_montgomery();

            let s: Scalar = Scalar::random(&mut csprng);
            let expected = s * p_edwards;
            let result = s * p_montgomery;

            assert_eq!(result, expected.to_montgomery())
        }
    }

    // Tests that, on the prime-order subgroup, MontgomeryPoint::mul_bits_be is the same as
    // multiplying by the Scalar representation of the same bits
    #[test]
    fn montgomery_mul_bits_be() {
        let mut csprng = rand_core::OsRng;

        for _ in 0..100 {
            // Make a random prime-order point P
            let p_edwards = rand_prime_order_point(&mut csprng);
            let p_montgomery: MontgomeryPoint = p_edwards.to_montgomery();

            // Make a random integer b
            let mut bigint = [0u8; 64];
            csprng.fill_bytes(&mut bigint[..]);
            let bigint_bits_be = bytestring_bits_le(&bigint).rev();

            // Check that bP is the same whether calculated as scalar-times-edwards or
            // integer-times-montgomery.
            let expected = Scalar::from_bytes_mod_order_wide(&bigint) * p_edwards;
            let result = p_montgomery.mul_bits_be(bigint_bits_be);
            assert_eq!(result, expected.to_montgomery())
        }
    }

    // Tests that MontgomeryPoint::mul_bits_be is consistent on any point, even ones that might be
    // on the curve's twist. Specifically, this tests that b₁(b₂P) == b₂(b₁P) for random
    // integers b₁, b₂ and random (curve or twist) point P.
    #[test]
    fn montgomery_mul_bits_be_twist() {
        let mut csprng = rand_core::OsRng;

        for _ in 0..100 {
            // Make a random point P on the curve or its twist
            let p_montgomery = {
                let mut buf = [0u8; 32];
                csprng.fill_bytes(&mut buf);
                MontgomeryPoint(buf)
            };

            // Compute two big integers b₁ and b₂
            let mut bigint1 = [0u8; 64];
            let mut bigint2 = [0u8; 64];
            csprng.fill_bytes(&mut bigint1[..]);
            csprng.fill_bytes(&mut bigint2[..]);

            // Compute b₁P and b₂P
            let bigint1_bits_be = bytestring_bits_le(&bigint1).rev();
            let bigint2_bits_be = bytestring_bits_le(&bigint2).rev();
            let prod1 = p_montgomery.mul_bits_be(bigint1_bits_be.clone());
            let prod2 = p_montgomery.mul_bits_be(bigint2_bits_be.clone());

            // Check that b₁(b₂P) == b₂(b₁P)
            assert_eq!(
                prod1.mul_bits_be(bigint2_bits_be),
                prod2.mul_bits_be(bigint1_bits_be)
            );
        }
    }

    /// Check that mul_base_clamped and mul_clamped agree
    #[test]
    fn mul_base_clamped() {
        let mut csprng = rand_core::OsRng;

        // Test agreement on a large integer. Even after clamping, this is not reduced mod l.
        let a_bytes = [0xff; 32];
        assert_eq!(
            MontgomeryPoint::mul_base_clamped(a_bytes),
            constants::X25519_BASEPOINT.mul_clamped(a_bytes)
        );

        // Test agreement on random integers
        for _ in 0..100 {
            // This will be reduced mod l with probability l / 2^256 ≈ 6.25%
            let mut a_bytes = [0u8; 32];
            csprng.fill_bytes(&mut a_bytes);

            assert_eq!(
                MontgomeryPoint::mul_base_clamped(a_bytes),
                constants::X25519_BASEPOINT.mul_clamped(a_bytes)
            );
        }
    }

    #[cfg(feature = "alloc")]
    const ELLIGATOR_CORRECT_OUTPUT: [u8; 32] = [
        0x5f, 0x35, 0x20, 0x00, 0x1c, 0x6c, 0x99, 0x36, 0xa3, 0x12, 0x06, 0xaf, 0xe7, 0xc7, 0xac,
        0x22, 0x4e, 0x88, 0x61, 0x61, 0x9b, 0xf9, 0x88, 0x72, 0x44, 0x49, 0x15, 0x89, 0x9d, 0x95,
        0xf4, 0x6e,
    ];

    #[test]
    #[cfg(feature = "alloc")]
    fn montgomery_elligator_correct() {
        let bytes: Vec<u8> = (0u8..32u8).collect();
        let bits_in: [u8; 32] = (&bytes[..]).try_into().expect("Range invariant broken");

        let fe = FieldElement::from_bytes(&bits_in);
        let eg = elligator_encode(&fe);
        assert_eq!(eg.to_bytes(), ELLIGATOR_CORRECT_OUTPUT);
    }

    #[test]
    fn montgomery_elligator_zero_zero() {
        let zero = [0u8; 32];
        let fe = FieldElement::from_bytes(&zero);
        let eg = elligator_encode(&fe);
        assert_eq!(eg.to_bytes(), zero);
    }
}