crrl 0.9.0

//! Edwards25519 curve implementation.
//!
//! This module implements generic group operations on the twisted
//! Edwards curve of equation `-x^2 + y^2 = 1 + d*x^2*y^2`, over the
//! finite field GF(2^255 - 19), for the constant `d` = -121665/121666.
//! This curve is described in [RFC 7748]. The signature algorithm
//! Ed25519, which operates on that curve, is described in [RFC 8032].
//!
//! The curve has order `8*L` for a given prime integer `L` (which is
//! slightly greater than 2^252). A conventional base point is defined,
//! that generates the subgroup of order `L`. Points used in public keys
//! and signatures are normally part of that subgroup, though this is not
//! verified (the function `Point::is_in_subgroup()` can be used to
//! validate that a given point is in that subgroup).
//!
//! A curve point is represented by the `Point` structure. `Point`
//! instances can be used in additions and subtractions with the usual
//! `+` and `-` operators; all combinations of raw values and references
//! are accepted, as well as compound assignment operators `+=` and `-=`.
//! Specialized functions are available, in particular for point doubling
//! (`Point::double()`) and for sequences of successive doublings
//! (`Point::xdouble()`), the latter using some extra optimizations.
//! Multiplication by an integer (`u64` type) or a scalar (`Scalar`
//! structure) is also accepted, using the `*` and `*=` operators.
//! Scalars are integers modulo `L`. The `Scalar` structure represents
//! such a value; it implements all usual arithmetic operators (`+`, `-`,
//! `*` and `/`, as well as `+=`, `-=`, `*=` and `/=`).
//!
//! Scalars can be encoded over 32 bytes (using unsigned little-endian
//! convention) and decoded back. Encoding is always canonical, and
//! decoding always verifies that the value is indeed in the canonical
//! range.
//!
//! Points can be encoded over 32 bytes, and decoded back. As with
//! scalars, encoding is always canonical, and verified upon decoding.
//!
//! The `PrivateKey` structure represents a private key for the Ed25519
//! signature algorithm. It is instantiated from a 32-byte seed; the seed
//! MUST have been generated with a cryptographically secure generator
//! (this library does not include provisions for this generation step).
//! Following [RFC 8032], the seed is derived into a secret scalar, and
//! an extra private value used for deterministic signature generation.
//! The private key allows signature generation with the Ed25519,
//! Ed25519ctx and Ed25519ph variants (in the third case, the pre-hashed
//! message is provided by the caller). The `PublicKey` structure
//! represents a public key, i.e. a curve point (and its 32-byte encoding
//! as an additional field). Signature verification functions are
//! available on `PublicKey`, again for Ed25519, Ed25519ctx and
//! Ed25519ph.
//!
//! # Ed25519 Edge Cases
//!
//! It is known that there is a great amount of variation about how
//! existing Ed25519 implementations handle signatures and public keys
//! which are slightly out of the strict standard formats, e.g. when
//! non-canonical encodings are used. See [Taming the many
//! EdDSAs][taming] for some analysis. In the notations of that paper,
//! this implementation is a strict RFC 8032 / FIPS 186-5 variant; it
//! accepts test vectors 0 to 5, and reject vectors 6 to 11 (see table 5
//! in the paper). In other words:
//!
//!   - Canonical encoding of both points and scalars is enforced.
//!   Non-canonical encodings are rejected.
//!
//!   - The cofactored verification equation is used (i.e. including the
//!   multiplication by 8).
//!
//!   - Points outside of the subgroup of order `L`, including low-order
//!   points and the identity point, are accepted both for the `R`
//!   component of the signatures, and for public keys.
//!
//!   - The `S` component of a signature is accepted as long as it is
//!   canonically encoded (i.e. in the 0 to `L-1` range). Zero is
//!   accepted. The full 32 bytes are used: the three top bits of the
//!   last byte, though always of value 0, are checked.
//!
//! # Truncated Signatures
//!
//! The `PublicKey::verify_trunc_*()` functions support _truncated
//! signatures_: a 64-byte signature is provided, but the last few bits
//! are considered to have been reused for encoding other data, and thus
//! are ignored. The verification function then tries to recompute the
//! complete, original, untruncated signature. This process is safe since
//! neither truncation nor reconstruction involve usage of the private
//! key, and the original signature is obtained as an outcome of the
//! process. Up to 32 bits (i.e. four whole bytes) can be rebuilt by this
//! implementation, which corresponds to shrinking the minimum signature
//! encoding size from 64 down to 60 bytes.
//!
//! Signature reconstruction cost increases with the number of ignored
//! bits; when 32 bits are ignored, the verification cost is about 25 to
//! 35 times the cost of verifying an untruncated signature.
//!
//! [RFC 7748]: https://datatracker.ietf.org/doc/html/rfc7748
//! [RFC 8032]: https://datatracker.ietf.org/doc/html/rfc8032
//! [taming]: https://eprint.iacr.org/2020/1244

// Projective/fractional coordinates traditionally use uppercase letters,
// using lowercase only for affine coordinates.
#![allow(non_snake_case)]

use core::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};
use core::convert::TryFrom;
use super::field::{GF25519, ModInt256};
use sha2::{Sha512, Digest};
use super::{CryptoRng, RngCore};

/// A point on the twisted Edwards curve edwards25519.
#[derive(Clone, Copy, Debug)]
pub struct Point {
    pub(crate) X: GF25519,
    pub(crate) Y: GF25519,
    pub(crate) Z: GF25519,
    pub(crate) T: GF25519,
}

/// Integers modulo L = 2^252 + 27742317777372353535851937790883648493.
///
/// L is the prime order of the subgroup of interest in edwards25519.
/// The complete curve contains 8*L points.
pub type Scalar = ModInt256<0x5812631A5CF5D3ED, 0x14DEF9DEA2F79CD6,
                            0x0000000000000000, 0x1000000000000000>;

impl Scalar {
    /// Encodes a scalar element into bytes (little-endian).
    pub fn encode(self) -> [u8; 32] {
        self.encode32()
    }
}

impl Point {

    /// The group neutral (identity point) in the curve.
    ///
    /// Affine coordinates of the neutral are (0,1).
    pub const NEUTRAL: Self = Self {
        X: GF25519::ZERO,
        Y: GF25519::ONE,
        Z: GF25519::ONE,
        T: GF25519::ZERO,
    };

    /// The point of order 2 on the curve.
    ///
    /// Affine coordinate of this point are (0,-1).
    const ORDER2: Self = Self {
        X: GF25519::ZERO,
        Y: GF25519::w64be(
            0x7FFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF,
            0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFEC),
        Z: GF25519::ONE,
        T: GF25519::ZERO,
    };

    /// The conventional base point in the curve.
    ///
    /// This point generates the subgroup of prime order L (integers
    /// modulo L are represented by the `Scalar` type).
    pub const BASE: Self = Self {
        X: GF25519::w64be(
            0x216936D3CD6E53FE, 0xC0A4E231FDD6DC5C,
            0x692CC7609525A7B2, 0xC9562D608F25D51A),
        Y: GF25519::w64be(
            0x6666666666666666, 0x6666666666666666,
            0x6666666666666666, 0x6666666666666658),
        Z: GF25519::ONE,
        T: GF25519::w64be(
            0x67875F0FD78B7665, 0x66EA4E8E64ABE37D,
            0x20F09F80775152F5, 0x6DDE8AB3A5B7DDA3),
    };

    /// Curve equation parameter d = -121665 / 121666.
    pub(crate) const D: GF25519 = GF25519::w64be(
        0x52036CEE2B6FFE73,
        0x8CC740797779E898,
        0x00700A4D4141D8AB,
        0x75EB4DCA135978A3,
    );

    /// Double of the curve equation parameter: 2*d
    const D2: GF25519 = GF25519::w64be(
        0x2406D9DC56DFFCE7,
        0x198E80F2EEF3D130,
        0x00E0149A8283B156,
        0xEBD69B9426B2F159,
    );

    /// 2^((p-1)/4), which is a square root of -1 in GF(2^255-19)
    pub(crate) const SQRT_M1: GF25519 = GF25519::w64be(
        0x2B8324804FC1DF0B,
        0x2B4D00993DFBD7A7,
        0x2F431806AD2FE478,
        0xC4EE1B274A0EA0B0,
    );

    /// Tries to decode a point from bytes.
    ///
    /// If the source slice has not length exactly 32 bytes, then
    /// decoding fails. If the source bytes are not a valid, canonical
    /// encoding of a curve point, then decoding fails. On success,
    /// 0xFFFFFFFF is returned; on failure, 0x00000000 is returned. On
    /// failure, this point is set to the neutral.
    ///
    /// If the source length is exactly 32 bytes, then the decoding
    /// outcome (success or failure) should remain hidden from
    /// timing-based side channels.
    pub fn set_decode(&mut self, buf: &[u8]) -> u32 {
        // We follow all steps from RFC 8032, section 5.1.3.

        if buf.len() != 32 {
            *self = Self::NEUTRAL;
            return 0;
        }

        // Extract and clear the sign-of-x bit.
        let mut bb = [0u8; 32];
        bb[..].copy_from_slice(buf);
        let sign_x = bb[31] >> 7;
        bb[31] &= 0x7F;

        // Decode y. This may fail if the source value is not in the
        // proper 0..p-1 range.
        let (mut y, mut r) = GF25519::decode32(&bb[..]);

        // Recompute a candidate x.

        // u = y^2 - 1
        // v = d*y^2 + 1
        let y2 = y.square();
        let u = y2 - GF25519::ONE;
        let v = Self::D * y2 + GF25519::ONE;

        // t = u*v^7
        let v2 = v.square();
        let v3 = v2 * v;
        let t = u * v3 * v2.square();

        // x = u*v^3*t^((p-5)/8)
        // (p-5)/8 = 2^252 - 3
        let t2 = t * t.square();
        let t5 = (t2.xsquare(2) * t2).square() * t;
        let t10 = t5.xsquare(5) * t5;
        let t25 = (t10.xsquare(10) * t10).xsquare(5) * t5;
        let t50 = t25.xsquare(25) * t25;
        let t125 = (t50.xsquare(50) * t50).xsquare(25) * t25;
        let t250 = t125.xsquare(125) * t125;
        let mut x = t250.xsquare(2) * t * u * v3;

        // If v*x^2 == u, then x is correct.
        // If v*x^2 == -u, then we must replace x with x*2^((p-1)/4).
        // If neither holds, then there is no solution.
        // Note that:
        //   - if u == 0, then x == 0 and it does not matter whether we
        //     multiply x by 2^((p-1)/4) or not;
        //   - if u != 0, then v*x^2 cannot be equal to u and -u at the
        //     same time.
        let w = x.square() * v;
        let r1 = w.equals(u);
        let r2 = w.equals(-u);
        r &= r1 | r2;
        x.set_cond(&(x * Self::SQRT_M1), r2);

        // If the sign bit of x does not match the specified bit, then
        // negate x. This may induce a failure if x == 0 and the requested
        // sign bit is 1.
        let nx = (((x.encode()[0] & 0x01) ^ sign_x) as u32).wrapping_neg();
        r &= !(x.iszero() & nx);
        x.set_cond(&-x, nx);

        // If the process failed, then set (x,y) to (0,1).
        x.set_cond(&GF25519::ZERO, !r);
        y.set_cond(&GF25519::ONE, !r);

        // Produce extended coordinates.
        self.X = x;
        self.Y = y;
        self.Z = GF25519::ONE;
        self.T = x * y;
        r
    }

    /// Tries to decode a point from some bytes.
    ///
    /// Decoding succeeds only if the source slice has length exactly 32
    /// bytes, and contains the canonical encoding of a valid curve
    /// point. Since this method returns an `Option<Point>`, it
    /// inherently leaks (through timing-based side channels) whether
    /// decoding succeeded or not; to avoid that, consider using
    /// `set_decode()`. The decoded point itself, however, does not leak.
    pub fn decode(buf: &[u8]) -> Option<Point> {
        let mut P = Point::NEUTRAL;
        if P.set_decode(buf) != 0 {
            Some(P)
        } else {
            None
        }
    }

    /// Encodes this point into exactly 32 bytes.
    ///
    /// Encoding is always canonical.
    pub fn encode(self) -> [u8; 32] {
        let iZ = GF25519::ONE / self.Z;
        let (x, y) = (self.X * iZ, self.Y * iZ);
        let mut r = y.encode();
        r[31] |= x.encode()[0] << 7;
        r
    }

    /// Creates a point by converting a point in Duif coordinates.
    fn from_duif(P: &PointDuif) -> Self {
        let X = (P.ypx - P.ymx).half();
        let Y = (P.ypx + P.ymx).half();
        let Z = GF25519::ONE;
        let T = X * Y;
        Self { X, Y, Z, T }
    }

    /// Adds another point (`rhs`) to this point.
    fn set_add(&mut self, rhs: &Self) {
        let (X1, Y1, Z1, T1) = (&self.X, &self.Y, &self.Z, &self.T);
        let (X2, Y2, Z2, T2) = (&rhs.X, &rhs.Y, &rhs.Z, &rhs.T);

        // Formulas from RFC 8032, section 5.1.4.
        let A = (Y1.sub_noreduce(X1)) * (Y2.sub_noreduce(X2));
        let B = (Y1.add_noreduce(X1)) * (Y2.add_noreduce(X2));
        let C = T1 * Self::D2 * T2;
        let D = Z1.mul2_noreduce() * Z2;
        let E = B.sub_noreduce(&A);
        let F = D.sub_noreduce(&C);
        let G = D.add_noreduce(&C);
        let H = B.add_noreduce(&A);
        self.X = E * F;
        self.Y = G * H;
        self.T = E * H;
        self.Z = F * G;

        // Note: since d is a fraction of small values (d = -121665/121666),
        // the multiplication by constant 2*d (Self::D2) could be replaced
        // with a mul_small(), provided that A, B and D are scaled up with
        // other mul_small() calls:
        //    A = (Y1 - X1) * (Y2 - X2) * 121666
        //    B = (Y1 + X1) * (Y2 + X2) * 121666
        //    C = T1 * T2 * (2 * 121665)
        //    D = Z1 * Z2 * (2 * 121666)
        //    E = B - A
        //    F = D + C   <- note the change of sign of C
        //    G = D - C   <- also here
        //    H = B + A
        // In practice, the performance difference with the implemented
        // method is negligible.
    }

    /// Specialized point addition routine when the other operand is in
    /// affine Duif coordinates (used in the pregenerated tables for
    /// multiples of the base point).
    fn set_add_duif(&mut self, rhs: &PointDuif) {
        let (X1, Y1, Z1, T1) = (&self.X, &self.Y, &self.Z, &self.T);
        let (ypx, ymx, t2d) = (&rhs.ypx, &rhs.ymx, &rhs.t2d);

        // Formulas from RFC 8032, section 5.1.4.
        let A = (Y1.sub_noreduce(X1)) * ymx;
        let B = (Y1.add_noreduce(X1)) * ypx;
        let C = T1 * t2d;
        let E = B.sub_noreduce(&A);
        let (G, F) = Z1.mul2add_mul2sub_noreduce(&C);
        let H = B.add_noreduce(&A);
        self.X = E * F;
        self.Y = G * H;
        self.T = E * H;
        self.Z = F * G;
    }

    /// Specialized point subtraction routine when the other operand is in
    /// affine Duif coordinates (used in the pregenerated tables for
    /// multiples of the base point).
    fn set_sub_duif(&mut self, rhs: &PointDuif) {
        let (X1, Y1, Z1, T1) = (&self.X, &self.Y, &self.Z, &self.T);
        let (ypx, ymx, t2d) = (&rhs.ypx, &rhs.ymx, &rhs.t2d);

        // This is the same code as set_add_duif() except that we
        // merge the negation of the second operand:
        //   ypx and ymx are swapped
        //   t2d is negated
        let A = (Y1.sub_noreduce(X1)) * ypx;
        let B = (Y1.add_noreduce(X1)) * ymx;
        let C = T1 * t2d;
        let E = B.sub_noreduce(&A);
        let (F, G) = Z1.mul2add_mul2sub_noreduce(&C);
        let H = B.add_noreduce(&A);
        self.X = E * F;
        self.Y = G * H;
        self.T = E * H;
        self.Z = F * G;
    }

    /// Doubles this point (in place).
    pub fn set_double(&mut self) {
        let (X, Y, Z) = (&self.X, &self.Y, &self.Z);

        // Formulas from RFC 8032, section 5.1.4 (special doubling case).
        let A = X.square();
        let B = Y.square();
        let (H, E) = A.add_addsub_noreduce(&B, &(X.add_noreduce(Y)).square());
        let (G, F) = A.sub_subadd2_noreduce(&B, &Z.square());
        self.X = E * F;
        self.Y = G * H;
        self.T = E * H;
        self.Z = F * G;
    }

    /// Doubles this point.
    #[inline(always)]
    pub fn double(self) -> Self {
        let mut r = self;
        r.set_double();
        r
    }

    /// Doubles this point n times (in place).
    pub fn set_xdouble(&mut self, n: u32) {
        if n == 0 {
            return;
        }

        // We use the formulas from RFC 8032, section 5.1.4. In the
        // doubling case, we notice that the formulas don't need the T
        // coordinate on input, though they can produce it on output.
        // We thus omit the computation of T for all intermediate doublings,
        // making it only for the last doubling. Total cost is 4*n
        // squarings and 3*n+1 multiplications.
        let (X, Y, Z) = (&mut self.X, &mut self.Y, &mut self.Z);

        // Formulas from RFC 8032, section 5.1.4 (special doubling case).
        for i in 0..n {
            let A = X.square();
            let B = Y.square();
            let (H, E) = A.add_addsub_noreduce(
                &B, &(X.add_noreduce(Y)).square());
            let (G, F) = A.sub_subadd2_noreduce(&B, &Z.square());
            *Y = G * H;
            *X = E * F;
            *Z = F * G;
            if i == n - 1 {
                self.T = E * H;
            }
        }
    }

    /// Doubles this point n times.
    #[inline(always)]
    pub fn xdouble(self, n: u32) -> Self {
        let mut r = self;
        r.set_xdouble(n);
        r
    }

    /// Negates this point (in place).
    #[inline(always)]
    pub fn set_neg(&mut self) {
        self.X.set_neg();
        self.T.set_neg();
    }

    /// Subtract another point (`rhs`) from this point.
    fn set_sub(&mut self, rhs: &Self) {
        self.set_add(&-rhs);
    }

    /// Multiplies this point by a small integer.
    ///
    /// This operation is constant-time with regard to the source point,
    /// but NOT with regard to the multiplier; the multiplier `n` MUST
    /// NOT be secret.
    pub fn set_mul_small(&mut self, n: u64) {
        if n == 0 {
            *self = Self::NEUTRAL;
            return;
        }
        if n == 1 {
            return;
        }

        let nlen = 64 - n.leading_zeros();
        let T = *self;
        let mut ndbl = 0u32;
        for i in (0..(nlen - 1)).rev() {
            ndbl += 1;
            if ((n >> i) & 1) == 0 {
                continue;
            }
            self.set_xdouble(ndbl);
            ndbl = 0;
            self.set_add(&T);
        }
        self.set_xdouble(ndbl);
    }

    /// Compares two points for equality.
    ///
    /// Returned value is 0xFFFFFFFF if the two points are equal,
    /// 0x00000000 otherwise.
    #[inline]
    pub fn equals(self, rhs: Self) -> u32 {
        (self.X * rhs.Z).equals(rhs.X * self.Z)
        & (self.Y * rhs.Z).equals(rhs.Y * self.Z)
    }

    /// Tests whether this point is the neutral (identity point on the
    /// curve).
    ///
    /// Returned value is 0xFFFFFFFF for the neutral, 0x00000000
    /// otherwise.
    #[inline(always)]
    pub fn isneutral(self) -> u32 {
        // The neutral is the only point with y == 1.
        self.Y.equals(self.Z)
    }

    /// Tests whether this point is a low-order point, i.e. a point of
    /// order 1, 2, 4 or 8.
    ///
    /// Returned value is 0xFFFFFFFF for a low order point, 0x00000000
    /// otherwise. The curve neutral point (group identity) is a
    /// low-order point.
    pub fn has_low_order(self) -> u32 {
        // Only the neutral (0, 1) has order 1.
        // Only the point (0, -1) has order 2.
        // Only points (i, 0) and (-i, 0) have order 4 (with i^2 = -1).
        // E[4], i.e. the subgroup of points of order 1, 2 or 4, is
        // exactly the set of curve points that have either x = 0 or y = 0.
        //
        // E[8] contains E[4], and all points whose double is in E[4].
        // We can thus look at the doubling formulas to see how x = 0 or
        // y = 0 can be obtained. From RFC 8032:
        //     A = X^2
        //     B = Y^2
        //     C = 2*Z^2
        //     H = A + B          = X^2 + Y^2
        //     E = H - (X + Y)^2  = -2*X*Y
        //     G = A - B
        //     F = C + G
        //     X3 = E*F
        //     Y3 = G*H
        //     T3 = E*H
        //     Z3 = F*G
        // Z3 is always non-zero, so we are only interested in cases where
        // either E or H is zero. E = 0 if and only if X = 0 or Y = 0.
        // H = 0 if and only if X^2 = -Y^2, i.e. X = i*Y or X = -i*Y. We can
        // thus compute i*Y, and compare X with 0, Y with 0, X with i*Y and
        // -X with i*Y.
        let X = self.X;
        let Y = self.Y;
        let iY = Y * Self::SQRT_M1;
        X.iszero() | Y.iszero() | iY.equals(X) | iY.equals(-X)
    }

    /// Tests whether this point is in the proper subgroup of order `L`.
    ///
    /// Returned value is 0xFFFFFFFF for a point in the prime-order
    /// subgroup, 0x00000000 otherwise. Note that the curve neutral point
    /// (group identity) is also part of that subgroup.
    ///
    /// This function is relatively expensive (about 40% of the cost of
    /// multiplying a random point with a scalar).
    pub fn is_in_subgroup(self) -> u32 {
        // Reference: https://eprint.iacr.org/2022/1164
        //
        // We first suppose that this point is not one of the low-order
        // points, i.e. this point can be written as P+Q with Q being a
        // low-order point (8*Q = 0), and P being in the proper subgroup
        // and distinct from the neutral. We want to establish whether
        // Q = 0; this is the case if and only if this point can be halved
        // three times successively.
        //
        // We map to a Weierstraß curve.
        // Namely, if the affine coordinate are (x,y) in the twisted
        // Edwards curve a*x^2 + y^2 = 1 + d*x^2*y^2, then we compute:
        //   u = (a - d)*(1 + y)/(1 - y)
        //   w = 2/x
        //   v = u*w
        // and we get the equation:
        //   v^2 = u*(u^2 + A*u + B)
        // with A = 2*(a + d) and B = (a - d)^2. We do not actually need
        // the v coordinate; we will use (u,w). The equation is then:
        //   u*w^2 = u^2 + A*u + B
        //
        // To compute point halvings, we consider the following curves.
        // We write "Curve(A,B)" to designate the curve of equation
        // u*w^2 = u^2 + A*u + B. For some given A and B, we also
        // define:
        //   Ap = -2*A
        //   Bp = A^2 - 4*B
        //   As = -2*Ap = 4*A
        //   Bs = Ap^2 - 4*Bp = 16*B
        // We then define these functions:
        //   psi1: Curve(A,B)   --> Curve(Ap,Bp)
        //         (u, w)       |-> (w^2, -(u - B/u)/w)
        //   psi2: Curve(Ap,Bp) --> Curve(As,Bs)
        //         (u, w)       |-> (w^2, -(u - Bp/u)/w)
        //   iso:  Curve(As,Bs) --> Curve(A,B)
        //         (u, w)       |-> (u/4, w/2)
        // iso() is an isomorphism (i.e. Curve(As,Bs) is isomorphic to
        // Curve(A,B)). psi1() and psi2() are isogenies; they are obtained
        // by applying Vélu's formulas over the subgroup {inf,(0,0)} (i.e.
        // the subgroup generated by the 2-torsion point (0,0)). Thus, for
        // any point P in Curve(A,B), we have:
        //   2*P = iso(psi2(psi1(P)))
        // To halve a point, we thus only need to invert iso, psi2 and psi1,
        // in that order.
        //
        // The inverse of iso() is trivial.
        //
        // To compute (up, wp) such that psi2(up, wp) = (us, ws):
        //  - Compute wp = sqrt(us) (if up is not a square, there is no
        //    solution, the point is not a double).
        //  - Compute up = (us - Ap - wp*ws)/2
        // Note that if there is a solution, then there are two solutions,
        // (up, wp) and (Bp/up, -wp). With Curve25519, Bp is NOT a square;
        // thus exactly one of up and Bp/up is a square.
        //
        // To compute (u, w) such that psi1(u, w) = (up, wp):
        //  - Compute w = sqrt(up). As noted above, exactly one of the
        //    two solutions for up is a square, so we need to choose and
        //    use that one.
        //  - Compute u = (up - A - w*wp)/2
        //
        // A source point is in the proper prime-order subgroup if and only
        // if it can be halved three times. We thus apply the process above
        // three times. Since we only need to know whether the point P is in
        // the subgroup, but not actually compute P/8, we can replace the
        // third halving with only a test that the halving is possible, i.e.
        // whether its u coordinate is a square (only the inversion of psi2()
        // may fail; if u is a square then us will be a square and psi2()
        // inversion will succeed).
        //
        // To avoid divisions, we in fact work in isomorphic curves, as the
        // need be. We maintain a non-zero value e; the map
        // (u, w) |-> (u*e^2, w*e) is an isomorphism from Curve(A,B)
        // into Curve(A*e^2, B*e^4).
        //
        // The low-order points can yield some exceptional cases for this
        // process, so we handle them separately, as a corrective final
        // step; we can efficiently identify such points (see has_low_order()).

        // a - d
        const A_MINUS_D: GF25519 = GF25519::w64be(
            0x2DFC9311D490018C, 0x7338BF8688861767,
            0xFF8FF5B2BEBE2754, 0x8A14B235ECA68749);

        // A0 = 2*(a + d)
        const A0: GF25519 = GF25519::w64be(
            0x2406D9DC56DFFCE7, 0x198E80F2EEF3D130,
            0x00E0149A8283B156, 0xEBD69B9426B2F157);

        /* unused
        // B0 = (a - d)^2
        const B0: GF25519 = GF25519::w64be(
            0x21766733A42C1C0F, 0x7FF9D5153082F14B,
            0xD45E7361B5257C47, 0x095A91D292532FE5);
        */

        // Ap0 = -2*A
        const AP0: GF25519 = GF25519::w64be(
            0x37F24C4752400631, 0xCCE2FE1A22185D9F,
            0xFE3FD6CAFAF89D52, 0x2852C8D7B29A1D3F);

        /* unused
        // BP0 = A^2 - 4*B
        const BP0: GF25519 = GF25519::w64be(
            0x5FC9311D490018C7, 0x338BF8688861767F,
            0xF8FF5B2BEBE27548, 0xA14B235ECA6874FF);
        */

        // sqrt(2*(A0^2 - 4*B0))
        const SQRT_2BP0: GF25519 = GF25519::w64be(
            0x184E375B980A76DF, 0xECCFA7E460864B8D,
            0xDCBE48EEFE58E409, 0x1A9F3EC897EB404C);

        // 1. Map to the Weierstraß curve.
        //   u = (a - d)*(1 + y)/(1 - y)
        //   w = 2/x
        // We switch to the isomorphic curve with e = X*(Z - Y), so that
        // we get division-less expressions:
        //   u = (a - d)*(Z + Y)*(Z - Y)*X^2
        //   w = 2*Z*(Z - Y)
        let mut e = self.X * (self.Z - self.Y);
        let mut u = A_MINUS_D * (self.Z + self.Y) * self.X * e;
        let mut w = (self.Z * (self.Z - self.Y)).mul2();

        // We try to halve the point twice.
        let mut ok = 0xFFFFFFFFu32;
        for _ in 0..2 {
            // Loop invariant (if ok != 0 and the point is not low order):
            //   u*w^2 = u^2 + u*A0*e^2 + B0*e^4

            // Inverse iso().
            // We get (us, ws) in curve(As, Bs).
            let us = u.mul4();
            let ws = w.mul2();

            // Inverse psi2(). This works if and only if us is a square.
            let (mut wp, cc) = us.sqrt();
            ok &= cc;
            let mut up = (us - AP0 * e.square() - wp * ws).half();

            // We now have (if ok != 0 and the point is not low order):
            //   up*wp^2 = up^2 + up*Ap0*e^2 + Bp0*e^4

            // Inverse psi1().
            // If up is not a square, then we should have used Bp/up
            // instead. The sqrt_ext() function returns a square root of
            // 2*up or -2*up in that case; we can use that value to get
            // the proper result without needing another square root
            // extraction. We then multiply by sqrt(-1) the value if needed,
            // so that we have a square root of 2*up. Then, we move to a
            // new isomorphic curve with:
            //   e' = e*sqrt(2*up)
            //   up' = up*(2*up) = 2*up^2
            //   wp' = wp*sqrt(2*up)
            //   Bp' = Bp0*e'^4 = Bp0*4*up^2*e^4
            //   w = -sqrt(Bp'/up') = -sqrt(2*Bp0)*e^2
            // sqrt(2*Bp0) is a constant, which we can precompute.
            // Mind the sign on w! We use (Bp'/up', -wp') as source, which
            // implies a negation of w.
            //
            // If up was a square, then formulas are:
            //   w = sqrt(up)
            //   u = (w^2 - A - w*wp)/2
            // If not, then we use:
            //   w = sqrt(Bp/up)
            //   u = (w^2 - A + w*wp)/2

            // Tentative square root extraction.
            let (mut tt, cc) = up.sqrt_ext();
            w = tt;
            // If it failed (cc == 0), and we got a root of -2*up, convert
            // it into a square root of 2*up.
            tt.set_cond(&(tt * Self::SQRT_M1),
                !cc & tt.square().equals(-up.mul2()));
            // If cc == 0, update the isorphism and the computed w.
            up.set_cond(&(up.square().mul2()), !cc);
            wp.set_cond(&(wp * tt), !cc);
            w.set_cond(&(SQRT_2BP0 * e.square()), !cc);
            e.set_cond(&(e * tt), !cc);
            u = (w.square() - A0 * e.square() - w * wp).half();
            w.set_cond(&-w, !cc);
        }

        // We did two halvings; we can do a third one if and only if
        // the current u coordinate is a square.
        ok &= !((u.legendre() >> 1) as u32);

        // If the source point was a low order point then the computations
        // above might have failed; we fix that here. Among low-order
        // points, only the neutral is in the proper subgroup.
        let lop = self.has_low_order();
        let neu = self.isneutral();
        ok ^= lop & (ok ^ neu);

        ok
    }

    /*
     * Unused: old implementation of is_in_subgroup(), that multiplies the
     * source point by L with a custom addition chain.
     * Kept for reference only; the new is_in_subgroup() is about twice
     * faster.
     *
    pub fn old_is_in_subgroup(self) -> u32 {
        // L = 2^252 + 0x14DEF9DEA2F79CD65812631A5CF5D3ED
        // We use a wNAF chain with w = 5.
        let mut P = self.double();
        let mut win = [Self::NEUTRAL; 8];
        win[0] = self;
        for i in 1..8 {
            win[i] = win[i - 1] + P;
        }
        P.set_xdouble(129);

        // Pairs: number of doublings, next digit
        const DD: [i8; 44] = [
            0, 5, 5, 7, 5, -1, 5, -1, 6, 15, 8, -11, 5, 3, 5, -1, 6, -3,
            7, -13, 5, 11, 6, 11, 9, 5, 5, -13, 5, 3, 6, 7, 5, -13, 5, -3,
            7, -5, 7, -11, 5, -1, 5, 13,
        ];
        let mut i = 0;
        while i < DD.len() {
            P.set_xdouble(DD[i] as u32);
            let d = DD[i + 1];
            if d > 0 {
                P += win[(d >> 1) as usize];
            } else {
                P -= win[((-d) >> 1) as usize];
            }
            i += 2;
        }

        // The point P was in the proper subgroup if and only if L*P = 0.
        P.isneutral()
    }
    */

    /// Conditionally copies the provided point (`P`) into `self`.
    ///
    ///  - If `ctl` = 0xFFFFFFFF, then the value of `P` is copied into `self`.
    ///
    ///  - If `ctl` = 0x00000000, then the value of `self` is unchanged.
    ///
    /// `ctl` MUST be equal to 0x00000000 or 0xFFFFFFFF.
    #[inline]
    pub fn set_cond(&mut self, P: &Self, ctl: u32) {
        self.X.set_cond(&P.X, ctl);
        self.Y.set_cond(&P.Y, ctl);
        self.Z.set_cond(&P.Z, ctl);
        self.T.set_cond(&P.T, ctl);
    }

    /// Returns a point equal to `P0` (if `ctl` = 0x00000000) or `P1` (if
    /// `ctl` = 0xFFFFFFFF).
    ///
    /// Value `ctl` MUST be either 0x00000000 or 0xFFFFFFFF.
    #[inline(always)]
    pub fn select(P0: &Self, P1: &Self, ctl: u32) -> Self {
        let mut P = *P0;
        P.set_cond(P1, ctl);
        P
    }

    /// Conditionally negates this point.
    ///
    /// This point is negated if `ctl` = 0xFFFFFFFF, but kept unchanged
    /// if `ctl` = 0x00000000. `ctl` MUST be equal to 0x00000000 or
    /// 0xFFFFFFFF.
    #[inline]
    pub fn set_condneg(&mut self, ctl: u32) {
        self.X.set_cond(&-self.X, ctl);
        self.T.set_cond(&-self.T, ctl);
    }

    /// Maps this point to the corresponding Montgomery curve and returns
    /// the affine u coordinate of the resulting point.
    ///
    /// If this point is the neutral, then 0 is returned.
    pub fn to_montgomery_u(&self) -> GF25519 {
        (self.Z + self.Y) / (self.Z - self.Y)
    }

    /// Maps this point to the corresponding Montgomery curve and returns
    /// the projective u coordinate of the resulting point.
    ///
    /// The returned pair (X,Z) is such that the u coordinate is equal to
    /// X / Z. There are many possible pairs for a single point; which
    /// actual pair is obtained is unspecified. For the neutral point,
    /// Z = 0 (but X != 0); for all other points, Z != 0.
    pub fn to_montgomery_u_projective(&self) -> (GF25519, GF25519) {
        (self.Z + self.Y, self.Z - self.Y)
    }

    /// Recodes a scalar into 51 signed digits.
    ///
    /// Each digit is in -15..+16, top digit is in 0..+4.
    fn recode_scalar(n: &Scalar) -> [i8; 51] {
        let mut sd = [0i8; 51];
        let bb = n.encode();
        let mut cc: u32 = 0;       // carry from lower digits
        let mut i: usize = 0;      // index of next source byte
        let mut acc: u32 = 0;      // buffered bits
        let mut acc_len: i32 = 0;  // number of buffered bits
        for j in 0..51 {
            if acc_len < 5 {
                acc |= (bb[i] as u32) << acc_len;
                acc_len += 8;
                i += 1;
            }
            let d = (acc & 0x1F) + cc;
            acc >>= 5;
            acc_len -= 5;
            let m = 16u32.wrapping_sub(d) >> 8;
            sd[j] = (d.wrapping_sub(m & 32)) as i8;
            cc = m & 1;
        }
        sd
    }

    /// Lookups a point from a window, with sign handling (constant-time).
    fn lookup(win: &[Self; 16], k: i8) -> Self {
        // Split k into its sign s (0xFFFFFFFF for negative) and
        // absolute value (f).
        let s = ((k as i32) >> 8) as u32;
        let f = ((k as u32) ^ s).wrapping_sub(s);
        let mut P = Self::NEUTRAL;
        for i in 0..16 {
            // win[i] contains (i+1)*P; we want to keep it if (and only if)
            // i+1 == f.
            // Values a-b and b-a both have their high bit equal to 0 only
            // if a == b.
            let j = (i as u32) + 1;
            let w = !(f.wrapping_sub(j) | j.wrapping_sub(f));
            let w = ((w as i32) >> 31) as u32;

            P.X.set_cond(&win[i].X, w);
            P.Y.set_cond(&win[i].Y, w);
            P.Z.set_cond(&win[i].Z, w);
            P.T.set_cond(&win[i].T, w);
        }

        // Negate the returned value if needed.
        P.X.set_cond(&-P.X, s);
        P.T.set_cond(&-P.T, s);

        P
    }

    /// Multiplies this point by a scalar (in place).
    ///
    /// This operation is constant-time with regard to both the points
    /// and the scalar value.
    ///
    /// Note: a scalar is nominally an integer modulo the prime L, which
    /// is the order of a specific subgroup of the curve. If the source
    /// point is NOT in that subgroup, then what is computed is the
    /// product of the point by an integer in the 0 to L-1 range.
    pub fn set_mul(&mut self, n: &Scalar) {
        // Make a 5-bit window: win[i] contains (i+1)*P
        let mut win = [Self::NEUTRAL; 16];
        win[0] = *self;
        for i in 1..8 {
            let j = 2 * i;
            win[j - 1] = win[i - 1].double();
            win[j] = win[j - 1] + win[0];
        }
        win[15] = win[7].double();

        // Recode the scalar into 51 signed digits.
        let sd = Self::recode_scalar(n);

        // Process the digits in high-to-low order.
        *self = Self::lookup(&win, sd[50]);
        for i in (0..50).rev() {
            self.set_xdouble(5);
            self.set_add(&Self::lookup(&win, sd[i]));
        }
    }

    /// Lookups a point from a window of points in Duif coordinates, with
    /// sign handling (constant-time).
    fn lookup_duif(win: &[PointDuif; 16], k: i8) -> PointDuif {
        // Split k into its sign s (0xFFFFFFFF for negative) and
        // absolute value (f).
        let s = ((k as i32) >> 31) as u32;
        let f = ((k as u32) ^ s).wrapping_sub(s);
        let mut ypx = GF25519::ONE;
        let mut ymx = GF25519::ONE;
        let mut t2d = GF25519::ZERO;
        for i in 0..16 {
            // win[i] contains (i+1)*P; we want to keep it if (and only if)
            // i+1 == f.
            // Values a-b and b-a both have their high bit equal to 0 only
            // if a == b.
            let j = (i as u32) + 1;
            let w = !(f.wrapping_sub(j) | j.wrapping_sub(f));
            let w = ((w as i32) >> 31) as u32;

            ypx.set_cond(&win[i].ypx, w);
            ymx.set_cond(&win[i].ymx, w);
            t2d.set_cond(&win[i].t2d, w);
        }

        // Negate the returned value if needed. Negation in Duif coordinates
        // consists in exchanging the ypx and ymx values, and negating t2d.
        PointDuif {
            ypx: GF25519::select(&ypx, &ymx, s),
            ymx: GF25519::select(&ymx, &ypx, s),
            t2d: GF25519::select(&t2d, &-t2d, s),
        }
    }

    /// Sets this point by multiplying the conventional generator by the
    /// provided scalar.
    ///
    /// This operation is constant-time. It is faster than using the
    /// generic multiplication on `Self::BASE`.
    pub fn set_mulgen(&mut self, n: &Scalar) {
        // Recode the scalar into 51 signed digits.
        let sd = Self::recode_scalar(n);

        // We process four chunks in parallel. Each chunk is 13 digits,
        // except the top one which is 12 digits only.
        *self = Self::from_duif(&Self::lookup_duif(&PRECOMP_B, sd[12]));
        self.set_add_duif(&Self::lookup_duif(&PRECOMP_B65, sd[25]));
        self.set_add_duif(&Self::lookup_duif(&PRECOMP_B130, sd[38]));

        // Process the digits in high-to-low order.
        for i in (0..12).rev() {
            self.set_xdouble(5);
            self.set_add_duif(&Self::lookup_duif(&PRECOMP_B, sd[i]));
            self.set_add_duif(&Self::lookup_duif(&PRECOMP_B65, sd[i + 13]));
            self.set_add_duif(&Self::lookup_duif(&PRECOMP_B130, sd[i + 26]));
            self.set_add_duif(&Self::lookup_duif(&PRECOMP_B195, sd[i + 39]));
        }
    }

    /// Creates a point by multiplying the conventional generator by the
    /// provided scalar.
    ///
    /// This operation is constant-time.
    #[inline]
    pub fn mulgen(n: &Scalar) -> Self {
        let mut P = Self::NEUTRAL;
        P.set_mulgen(n);
        P
    }

    /// 5-bit wNAF recoding of a scalar; output is a sequence of 254
    /// digits.
    ///
    /// Non-zero digits have an odd value, between -15 and +15
    /// (inclusive). (The recoding is constant-time, but use of wNAF is
    /// inherently non-constant-time.)
    fn recode_scalar_NAF(n: &Scalar) -> [i8; 254] {
        // We use a branchless algorithm to avoid misprediction
        // penalties.
        //
        // Let x be the current (complete) integer:
        //  - If x is even, then the next digit is 0.
        //  - Otherwise, we produce a digit from the low five bits of
        //    x. If these low bits have value v (odd, 1..31 range):
        //     - If v <= 15, then the next digit is v.
        //     - Otherwise, the next digit is v - 32, and we add 32 to x.
        //    When then subtract v from x (i.e. we clear the low five bits).
        // Once the digit has been produced, we divide x by 2 and loop.
        //
        // Since L < 2^253, only 254 digits are necessary at most.

        let mut sd = [0i8; 254];
        let bb = n.encode();
        let mut x = bb[0] as u32;
        for i in 0..254 {
            if (i & 7) == 4 && i < 252 {
                x += (bb[(i + 4) >> 3] as u32) << 4;
            }
            let m = (x & 1).wrapping_neg();  // -1 if x is odd, 0 otherwise
            let v = x & m & 31;              // low 5 bits if x odd, or 0
            let c = (v & 16) << 1;           // carry (0 or 32)
            let d = v.wrapping_sub(c);       // next digit
            sd[i] = d as i8;
            x = x.wrapping_sub(d) >> 1;
        }
        sd
    }

    /// 5-bit wNAF recoding of a nonnegative integer.
    ///
    /// 129 digits are produced. Non-zero digits have an odd value,
    /// between -15 and +15 (inclusive). (The recoding is constant-time,
    /// but use of wNAF is inherently non-constant-time.)
    fn recode_u128_NAF(n: u128) -> [i8; 130] {
        // See recode_scalar_NAF() for details.
        let mut sd = [0i8; 130];
        let mut y = n;
        for i in 0..129 {
            let x = y as u32;
            let m = (x & 1).wrapping_neg();  // -1 if x is odd, 0 otherwise
            let v = x & m & 31;              // low 5 bits if x odd, or 0
            let c = (v & 16) << 1;           // carry (0 or 32)
            sd[i] = v.wrapping_sub(c) as i8;
            y = y.wrapping_sub(v as u128).wrapping_add(c as u128) >> 1;
        }
        sd
    }

    /// Given scalars `u` and `v`, sets this point to `u*self + v*B`
    /// (with `B` being the conventional generator of the prime order
    /// subgroup).
    ///
    /// This can be used to support EdDSA signature verification, though
    /// for that task `verify_helper_vartime()` is faster.
    ///
    /// THIS FUNCTION IS NOT CONSTANT-TIME; it shall be used only with
    /// public data.
    pub fn set_mul_add_mulgen_vartime(&mut self, u: &Scalar, v: &Scalar) {
        // Recode the scalars in 5-bit wNAF.
        let sdu = Self::recode_scalar_NAF(&u);
        let sdv = Self::recode_scalar_NAF(&v);

        // Compute the window for the current point:
        //   win[i] = (2*i+1)*self    (i = 0 to 7)
        let mut win = [Self::NEUTRAL; 8];
        let Q = self.double();
        win[0] = *self;
        for i in 1..8 {
            win[i] = win[i - 1] + Q;
        }

        let mut zz = true;
        let mut ndbl = 0u32;
        for i in (0..254).rev() {
            // We have one more doubling to perform.
            ndbl += 1;

            // Get next digits. If they are all zeros, then we can loop
            // immediately.
            let e1 = sdu[i];
            let e2 = sdv[i];
            if ((e1 as u32) | (e2 as u32)) == 0 {
                continue;
            }

            // Apply accumulated doubles.
            if zz {
                *self = Self::NEUTRAL;
                zz = false;
            } else {
                self.set_xdouble(ndbl);
            }
            ndbl = 0u32;

            // Process digits.
            if e1 != 0 {
                if e1 > 0 {
                    self.set_add(&win[e1 as usize >> 1]);
                } else {
                    self.set_sub(&win[(-e1) as usize >> 1]);
                }
            }
            if e2 != 0 {
                if e2 > 0 {
                    self.set_add_duif(&PRECOMP_B[e2 as usize - 1]);
                } else {
                    self.set_sub_duif(&PRECOMP_B[(-e2) as usize - 1]);
                }
            }
        }

        if zz {
            *self = Self::NEUTRAL;
        } else {
            if ndbl > 0 {
                self.set_xdouble(ndbl);
            }
        }
    }

    // Given scalars `u` and `v`, returns `u*self + v*B` (with `B` being
    // the conventional generator of the prime order subgroup).
    //
    // This can be used to support EdDSA signature verification, though
    // for that task `verify_helper_vartime()` is faster.
    //
    // THIS FUNCTION IS NOT CONSTANT-TIME; it shall be used only with
    // public data.
    #[inline(always)]
    pub fn mul_add_mulgen_vartime(self, u: &Scalar, v: &Scalar) -> Self {
        let mut R = self;
        R.set_mul_add_mulgen_vartime(u, v);
        R
    }

    /// Check whether `8*s*B = 8*R + 8*k*A`, for the provided scalars `s`
    /// and `k`, provided points `A` (`self`) and `R`, and conventional
    /// generator `B`.
    ///
    /// Returned value is true on match, false otherwise. This function
    /// is meant to support Ed25519 signature verification.
    ///
    /// THIS FUNCTION IS NOT CONSTANT-TIME; it shall be used only with
    /// public data.
    pub fn verify_helper_vartime(self,
        R: &Point, s: &Scalar, k: &Scalar) -> bool
    {
        // We want to compute:
        //   T = s*B - R - k*A
        // and then verify that 8*T = 0, or, equivalently, that T is one
        // of the known points of order 1, 2, 4 or 8. We split k into
        // a fraction of smaller integers: k = c0/c1; we can then
        // compute:
        //   c1*T = (s*c1)*B - c1*R - c0*A
        // Note that since c1 != 0 and is lower than L (it's a 128-bit
        // integer only), c1*T can be a low order point only if T is a
        // low order point (and vice versa). Thus, computing c1*T is
        // enough for our purposes.
        //
        // We can compute s*c1 as a scalar, then write the result as
        // a combination of two "halves":
        //    s*c1 = s0 + s1*2^130
        // with 0 <= s0 < 2^130, and 0 <= s1 < 2^123. We can thus make
        // the computation as a linear combination of four points, with
        // coefficients that fit on 130 bits each. (We choose '130'
        // because that's a multiple of 5 and we process bits by chunks
        // of 5).

        // Compute coefficients. We ensure we have only nonnegative
        // values by propagating signs to the points themselves if
        // necessary.
        let (c0, c1) = k.split_vartime();
        let (P1, d1) = if c1 >= 0 { (-R, c1) } else { (*R, -c1) };
        let (P2, d2) = if c0 >= 0 { (-self, c0) } else { (self, -c0) };
        let ss = s * Scalar::from_i128(c1);

        // Recode coefficients in 5-NAF. sd0 has 254 digits, sd1 and
        // sd2 have 129 digits each. Note: since coefficients d1 and d2
        // could fit on 127 bits, they really use only 128 digits, and
        // the top 129-th digit is always zero.
        let sd0 = Self::recode_scalar_NAF(&ss);
        let sd1 = Self::recode_u128_NAF(d1 as u128);
        let sd2 = Self::recode_u128_NAF(d2 as u128);

        // Compute windows for points P1 and P2:
        //   win1[i] = (2*i+1)*P1    (i = 0 to 7)
        //   win2[i] = (2*i+1)*P2    (i = 0 to 7)
        let Q1 = P1.double();
        let Q2 = P2.double();
        let mut win1 = [Self::NEUTRAL; 8];
        let mut win2 = [Self::NEUTRAL; 8];
        win1[0] = P1;
        win2[0] = P2;
        for i in 1..8 {
            win1[i] = win1[i - 1] + Q1;
            win2[i] = win2[i - 1] + Q2;
        }

        // Initialize the accumulator point with the top digits from
        // s0 (low half of sd0), which is the only of our four coefficients
        // that can use more than 128 digits.
        let mut T = if sd0[129] != 0 {
            let mut T2 = if sd0[129] > 0 {
                Self::from_duif(&PRECOMP_B[sd0[129] as usize - 1])
            } else {
                -Self::from_duif(&PRECOMP_B[(-sd0[129]) as usize - 1])
            };
            T2.set_double();
            if sd0[128] != 0 {
                if sd0[128] > 0 {
                    T2.set_add_duif(&PRECOMP_B[sd0[128] as usize - 1]);
                } else {
                    T2.set_sub_duif(&PRECOMP_B[(-sd0[128]) as usize - 1]);
                }
            }
            T2
        } else if sd0[128] != 0 {
            if sd0[128] > 0 {
                Self::from_duif(&PRECOMP_B[sd0[128] as usize - 1])
            } else {
                -Self::from_duif(&PRECOMP_B[(-sd0[128]) as usize - 1])
            }
        } else {
            Self::NEUTRAL
        };

        // Process all other digits. We coalesce long sequences of
        // doublings to leverage the optimizations of xdouble().
        let mut ndbl = 0u32;
        for i in (0..128).rev() {
            // We have one more doubling to perform.
            ndbl += 1;

            // Get next digits. If they are all zeros, then we can loop
            // immediately.
            let e1 = sd0[i];
            let e2 = if i < 124 { sd0[i + 130] } else { 0 };
            let f1 = sd1[i];
            let f2 = sd2[i];
            if ((e1 as u32) | (e2 as u32) | (f1 as u32) | (f2 as u32)) == 0 {
                continue;
            }

            // Apply accumulated doubles.
            T.set_xdouble(ndbl);
            ndbl = 0u32;

            // Process digits.
            if e1 != 0 {
                if e1 > 0 {
                    T.set_add_duif(&PRECOMP_B[e1 as usize - 1]);
                } else {
                    T.set_sub_duif(&PRECOMP_B[(-e1) as usize - 1]);
                }
            }
            if e2 != 0 {
                if e2 > 0 {
                    T.set_add_duif(&PRECOMP_B130[e2 as usize - 1]);
                } else {
                    T.set_sub_duif(&PRECOMP_B130[(-e2) as usize - 1]);
                }
            }
            if f1 != 0 {
                if f1 > 0 {
                    T.set_add(&win1[f1 as usize >> 1]);
                } else {
                    T.set_sub(&win1[(-f1) as usize >> 1]);
                }
            }
            if f2 != 0 {
                if f2 > 0 {
                    T.set_add(&win2[f2 as usize >> 1]);
                } else {
                    T.set_sub(&win2[(-f2) as usize >> 1]);
                }
            }
        }

        // We can skip the still accumulated doubles (if any) because they
        // won't change the status of T as a low order point.

        T.has_low_order() == 0xFFFFFFFF
    }
}

impl Add<Point> for Point {
    type Output = Point;

    #[inline(always)]
    fn add(self, other: Point) -> Point {
        let mut r = self;
        r.set_add(&other);
        r
    }
}

impl Add<&Point> for Point {
    type Output = Point;

    #[inline(always)]
    fn add(self, other: &Point) -> Point {
        let mut r = self;
        r.set_add(other);
        r
    }
}

impl Add<Point> for &Point {
    type Output = Point;

    #[inline(always)]
    fn add(self, other: Point) -> Point {
        let mut r = *self;
        r.set_add(&other);
        r
    }
}

impl Add<&Point> for &Point {
    type Output = Point;

    #[inline(always)]
    fn add(self, other: &Point) -> Point {
        let mut r = *self;
        r.set_add(other);
        r
    }
}

impl AddAssign<Point> for Point {
    #[inline(always)]
    fn add_assign(&mut self, other: Point) {
        self.set_add(&other);
    }
}

impl AddAssign<&Point> for Point {
    #[inline(always)]
    fn add_assign(&mut self, other: &Point) {
        self.set_add(other);
    }
}

impl Mul<Scalar> for Point {
    type Output = Point;

    #[inline(always)]
    fn mul(self, other: Scalar) -> Point {
        let mut r = self;
        r.set_mul(&other);
        r
    }
}

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

    #[inline(always)]
    fn mul(self, other: &Scalar) -> Point {
        let mut r = self;
        r.set_mul(other);
        r
    }
}

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

    #[inline(always)]
    fn mul(self, other: Scalar) -> Point {
        let mut r = *self;
        r.set_mul(&other);
        r
    }
}

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

    #[inline(always)]
    fn mul(self, other: &Scalar) -> Point {
        let mut r = *self;
        r.set_mul(other);
        r
    }
}

impl MulAssign<Scalar> for Point {
    #[inline(always)]
    fn mul_assign(&mut self, other: Scalar) {
        self.set_mul(&other);
    }
}

impl MulAssign<&Scalar> for Point {
    #[inline(always)]
    fn mul_assign(&mut self, other: &Scalar) {
        self.set_mul(other);
    }
}

impl Mul<Point> for Scalar {
    type Output = Point;

    #[inline(always)]
    fn mul(self, other: Point) -> Point {
        let mut r = other;
        r.set_mul(&self);
        r
    }
}

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

    #[inline(always)]
    fn mul(self, other: &Point) -> Point {
        let mut r = *other;
        r.set_mul(&self);
        r
    }
}

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

    #[inline(always)]
    fn mul(self, other: Point) -> Point {
        let mut r = other;
        r.set_mul(self);
        r
    }
}

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

    #[inline(always)]
    fn mul(self, other: &Point) -> Point {
        let mut r = *other;
        r.set_mul(self);
        r
    }
}

impl Mul<u64> for Point {
    type Output = Point;

    #[inline(always)]
    fn mul(self, other: u64) -> Point {
        let mut r = self;
        r.set_mul_small(other);
        r
    }
}

impl Mul<u64> for &Point {
    type Output = Point;

    #[inline(always)]
    fn mul(self, other: u64) -> Point {
        let mut r = *self;
        r.set_mul_small(other);
        r
    }
}

impl MulAssign<u64> for Point {
    #[inline(always)]
    fn mul_assign(&mut self, other: u64) {
        self.set_mul_small(other);
    }
}

impl Mul<Point> for u64 {
    type Output = Point;

    #[inline(always)]
    fn mul(self, other: Point) -> Point {
        let mut r = other;
        r.set_mul_small(self);
        r
    }
}

impl Mul<&Point> for u64 {
    type Output = Point;

    #[inline(always)]
    fn mul(self, other: &Point) -> Point {
        let mut r = *other;
        r.set_mul_small(self);
        r
    }
}

impl Neg for Point {
    type Output = Point;

    #[inline(always)]
    fn neg(self) -> Point {
        let mut r = self;
        r.set_neg();
        r
    }
}

impl Neg for &Point {
    type Output = Point;

    #[inline(always)]
    fn neg(self) -> Point {
        let mut r = *self;
        r.set_neg();
        r
    }
}

impl Sub<Point> for Point {
    type Output = Point;

    #[inline(always)]
    fn sub(self, other: Point) -> Point {
        let mut r = self;
        r.set_sub(&other);
        r
    }
}

impl Sub<&Point> for Point {
    type Output = Point;

    #[inline(always)]
    fn sub(self, other: &Point) -> Point {
        let mut r = self;
        r.set_sub(other);
        r
    }
}

impl Sub<Point> for &Point {
    type Output = Point;

    #[inline(always)]
    fn sub(self, other: Point) -> Point {
        let mut r = *self;
        r.set_sub(&other);
        r
    }
}

impl Sub<&Point> for &Point {
    type Output = Point;

    #[inline(always)]
    fn sub(self, other: &Point) -> Point {
        let mut r = *self;
        r.set_sub(other);
        r
    }
}

impl SubAssign<Point> for Point {
    #[inline(always)]
    fn sub_assign(&mut self, other: Point) {
        self.set_sub(&other);
    }
}

impl SubAssign<&Point> for Point {
    #[inline(always)]
    fn sub_assign(&mut self, other: &Point) {
        self.set_sub(other);
    }
}

// ========================================================================

/// An Ed25519 private key.
///
/// It is built from a 32-byte seed (which should be generated from a
/// cryptographically secure random source with at least 128 bits of
/// entropy, preferably 256 bits). From the seed are derived the secret
/// scalar and the public key. The public key is a curve point, that can
/// be encoded as such.
#[derive(Clone, Copy, Debug)]
pub struct PrivateKey {
    s: Scalar,                  // secret scalar
    seed: [u8; 32],             // source seed
    h: [u8; 32],                // derived seed (second half of SHA-512(seed))
    pub public_key: PublicKey,  // public key
}

/// An Ed25519 public key.
///
/// It wraps around the curve point, but also includes a copy of the
/// encoded point. The point and its encoded version can be accessed
/// directly; if modified, then the two values MUST match.
#[derive(Clone, Copy, Debug)]
pub struct PublicKey {
    pub point: Point,
    pub encoded: [u8; 32],
}

/// Constant string "SigEd25519 no Ed25519 collisions".
const HASH_HEAD: [u8; 32] = [
    0x53, 0x69, 0x67, 0x45, 0x64, 0x32, 0x35, 0x35,
    0x31, 0x39, 0x20, 0x6E, 0x6F, 0x20, 0x45, 0x64,
    0x32, 0x35, 0x35, 0x31, 0x39, 0x20, 0x63, 0x6F,
    0x6C, 0x6C, 0x69, 0x73, 0x69, 0x6F, 0x6E, 0x73,
];

impl PrivateKey {

    /// Generates a new private key from a cryptographically secure RNG.
    pub fn generate<T: CryptoRng + RngCore>(rng: &mut T) -> Self {
        let mut seed = [0u8; 32];
        rng.fill_bytes(&mut seed);
        Self::from_seed(&seed)
    }

    /// Instantiates a private key from the provided seed.
    ///
    /// The seed length MUST be exactly 32 bytes (a panic is triggered
    /// otherwise).
    pub fn from_seed(seed: &[u8]) -> Self {
        // We follow RFC 8032, section 5.1.5.

        // The seed MUST have length 32 bytes.
        assert!(seed.len() == 32);
        let mut bseed = [0u8; 32];
        bseed[..].copy_from_slice(seed);

        // Hash the seed with SHA-512.
        let mut sh = Sha512::new();
        sh.update(seed);
        let mut hh = sh.finalize();

        // Prune the first half and decode it as a scalar (with
        // reduction).
        hh[0] &= 0xF8;
        hh[31] &= 0x7F;
        hh[31] |= 0x40;
        let s = Scalar::decode_reduce(&hh[..32]);

        // Save second half of the hashed seed for signing operations.
        let mut h = [0u8; 32];
        h[..].copy_from_slice(&hh[32..]);

        // Public key is obtained from the secret scalar.
        let public_key = PublicKey::from_point(&Point::mulgen(&s));

        Self { s, seed: bseed, h, public_key }
    }

    /// Decodes a private key from bytes.
    ///
    /// If the source slice has length exactly 32 bytes, then these bytes
    /// are interpreted as a seed, and the private key is built on that
    /// seed (see `from_seed()`). Otherwise, `None` is returned.
    pub fn decode(buf: &[u8]) -> Option<Self> {
        if buf.len() == 32 {
            Some(Self::from_seed(<&[u8; 32]>::try_from(buf).unwrap()))
        } else {
            None
        }
    }

    /// Encodes a private key into 32 bytes.
    ///
    /// This actually returns a copy of the seed.
    pub fn encode(self) -> [u8; 32] {
        self.seed
    }

    /// Signs a message.
    ///
    /// This is the "Ed25519" mode of RFC 8032 (no pre-hashing, no
    /// context), also known as "PureEdDSA on Curve25519".
    pub fn sign_raw(self, m: &[u8]) -> [u8; 64] {
        self.sign_inner(false, 0, &[0u8; 0], m)
    }

    /// Signs a message with a context.
    ///
    /// This is the "Ed25519cx" mode of RFC 8032 (no pre-hashing, a
    /// context is provided). The context string MUST have length at most
    /// 255 bytes; it SHOULD NOT be of length zero.
    pub fn sign_ctx(self, ctx: &[u8], m: &[u8]) -> [u8; 64] {
        self.sign_inner(true, 0, ctx, m)
    }

    /// Signs a pre-hashed message.
    ///
    /// This is the "Ed25519ph" mode of RFC 8032 (message is pre-hashed),
    /// also known as "HashEdDSA on Curve25519". The hashed message `hm`
    /// is provided (presumably, that hash value was obtained with
    /// SHA-512; the caller does the hashing itself). A context string is
    /// also provided; it MUST have length at most 255 bytes.
    pub fn sign_ph(self, ctx: &[u8], hm: &[u8]) -> [u8; 64] {
        self.sign_inner(true, 1, ctx, hm)
    }

    /// Inner signature generation function.
    fn sign_inner(self, dom: bool, phflag: u8, ctx: &[u8],
                  m: &[u8]) -> [u8; 64]
    {
        // SHA-512(dom2(F, C) || prefix || PH(M)) -> scalar r
        let mut sh = Sha512::new();
        if dom {
            assert!(ctx.len() <= 255);
            let clen = ctx.len() as u8;
            sh.update(&HASH_HEAD);
            sh.update(&[phflag]);
            sh.update(&[clen]);
            sh.update(ctx);
        }
        sh.update(&self.h);
        sh.update(m);
        let hv1 = sh.finalize_reset();
        let r = Scalar::decode_reduce(&hv1);

        // R = r*B
        let R = Point::mulgen(&r);
        let R_enc = R.encode();

        // SHA-512(dom2(F, C) || R || A || PH(M)) -> scalar k
        if dom {
            assert!(ctx.len() <= 255);
            let clen = ctx.len() as u8;
            sh.update(&HASH_HEAD);
            sh.update(&[phflag]);
            sh.update(&[clen]);
            sh.update(ctx);
        }
        sh.update(&R_enc);
        sh.update(&self.public_key.encoded);
        sh.update(m);
        let hv2 = sh.finalize();
        let k = Scalar::decode_reduce(&hv2);

        // Signature is (R, S) with S = r + k*s mod L
        let mut sig = [0u8; 64];
        sig[0..32].copy_from_slice(&R_enc);
        sig[32..64].copy_from_slice(&(r + k * self.s).encode());

        sig
    }
}

impl PublicKey {

    /// Creates an instance from a curve point.
    pub fn from_point(point: &Point) -> Self {
        Self { point: *point, encoded: point.encode() }
    }

    // Decodes the provided bytes as a public key.
    //
    // This process may fail if the source slice does not have length
    // exactly 32 bytes, or if it has length 32 bytes but these bytes
    // are not the valid encoding of a curve point.
    //
    // Note: decoding success does not guarantee that the point is in
    // the proper subgroup of prime order L. The point may be outside of
    // the subgroup. The point may also be the curve neutral point, or a
    // low order point.
    pub fn decode(buf: &[u8]) -> Option<PublicKey> {
        let point = Point::decode(buf)?;
        let mut encoded = [0u8; 32];
        encoded[..].copy_from_slice(&buf[0..32]);
        Some(Self { point, encoded })
    }

    /// Encodes the key into exactly 32 bytes.
    ///
    /// This simply returns the contents of the `encoded` field.
    pub fn encode(self) -> [u8; 32] {
        self.encoded
    }

    /// Verifies a signature on a message.
    ///
    /// This is the "Ed25519" mode of RFC 8032 (no pre-hashing, no
    /// context), also known as "PureEdDSA on Curve25519". Return value
    /// is `true` on a valid signature, `false` otherwise.
    ///
    /// Note: this function is not constant-time; it assumes that the
    /// public key and signature value are public data.
    pub fn verify_raw(self, sig: &[u8], m: &[u8]) -> bool {
        self.verify_inner(sig, false, 0, &[0u8; 0], m)
    }

    /// Verifies a signature on a message.
    ///
    /// This is the "Ed25519cx" mode of RFC 8032 (no pre-hashing, a
    /// context is provided). The context string MUST have length at most
    /// 255 bytes; it SHOULD NOT be of length zero. Return value is
    /// `true` on a valid signature, `false` otherwise.
    ///
    /// Note: this function is not constant-time; it assumes that the
    /// public key and signature value are public data.
    pub fn verify_ctx(self, sig: &[u8], ctx: &[u8], m: &[u8]) -> bool {
        self.verify_inner(sig, true, 0, ctx, m)
    }

    /// Verifies a signature on a hashed message.
    ///
    /// This is the "Ed25519ph" mode of RFC 8032 (message is pre-hashed),
    /// also known as "HashEdDSA on Curve25519". The hashed message `hm`
    /// is provided (presumably, that hash value was obtained with
    /// SHA-512; the caller does the hashing itself). A context string is
    /// also provided; it MUST have length at most 255 bytes. Return
    /// value is `true` on a valid signature, `false` otherwise.
    ///
    /// Note: this function is not constant-time; it assumes that the
    /// public key and signature value are public data.
    pub fn verify_ph(self, sig: &[u8], ctx: &[u8], hm: &[u8]) -> bool {
        self.verify_inner(sig, true, 1, ctx, hm)
    }

    /// Inner signature verification function.
    fn verify_inner(self, sig: &[u8], dom: bool, phflag: u8, ctx: &[u8],
                    m: &[u8]) -> bool
    {
        /*
         * Old verification code which does not use verify_helper_vartime().
         * This code also checks the alternate verification equation,
         * without the cofactor, which is allowed by RFC 8032, but
         * slightly deviates from the behaviour we wish to achieve.

        if sig.len() != 64 {
            return false;
        }
        let R_enc = &sig[0..32];
        let (S, ok) = Scalar::decode32(&sig[32..64]);
        if ok == 0 {
            return false;
        }
        let mut sh = Sha512::new();
        if dom {
            assert!(ctx.len() <= 255);
            let clen = ctx.len() as u8;
            sh.update(&HASH_HEAD);
            sh.update(&[phflag]);
            sh.update(&[clen]);
            sh.update(ctx);
        }
        sh.update(R_enc);
        sh.update(&self.encoded);
        sh.update(m);
        let hv2 = sh.finalize();
        let k = Scalar::decode_reduce(&hv2);
        let R = self.point.mul_add_mulgen_vartime(&-k, &S);
        &R.encode()[..] == R_enc

         */

        // Signature must have length 64 bytes exactly.
        if sig.len() != 64 {
            return false;
        }

        // First half of the signature is the encoded point R;
        // second half is the scalar S. Both must decode successfully.
        // The decoding functions enforce canonicality (but point R
        // may be outside of the order-L subgroup).
        let R_enc = &sig[0..32];
        let R = match Point::decode(R_enc) {
            Some(R) => R,
            None    => { return false; }
        };
        let (S, ok) = Scalar::decode32(&sig[32..64]);
        if ok == 0 {
            return false;
        }

        // SHA-512(dom2(F, C) || R || A || PH(M)) -> scalar k
        // R is encoded over the first 32 bytes of the signature.
        let mut sh = Sha512::new();
        if dom {
            assert!(ctx.len() <= 255);
            let clen = ctx.len() as u8;
            sh.update(&HASH_HEAD);
            sh.update(&[phflag]);
            sh.update(&[clen]);
            sh.update(ctx);
        }
        sh.update(R_enc);
        sh.update(&self.encoded);
        sh.update(m);
        let hv2 = sh.finalize();
        let k = Scalar::decode_reduce(&hv2);

        // Check the verification equation 8*S*B = 8*R + 8*k*A.
        self.point.verify_helper_vartime(&R, &S, &k)
    }

    /// Verifies a truncated signature on a message.
    ///
    /// This is the "Ed25519" mode of RFC 8032 (no pre-hashing, no
    /// context), also known as "PureEdDSA on Curve25519". The signature
    /// slice (`sig`) MUST have length exactly 64 bytes; however, this
    /// function assumes that the last `rm` bits of the signature have
    /// been reused to store other data, and thus it ignores these bits.
    /// `rm` MUST be in the 8 to 32 range (inclusive); ignored elements
    /// are the last `floor(rm/8)` bytes, and the top (most significant)
    /// `rm%8` bits of the last non-ignored byte.
    ///
    /// If the original, untruncated signature was valid, then this
    /// function rebuilds it and returns it; otherwise, it returns `None`
    /// (if a rebuilt signature value is returned, then it has been
    /// verified to be valid and there is no need to validate it again).
    ///
    /// Note: this function is not constant-time; it assumes that the
    /// public key and signature value are public data.
    pub fn verify_trunc_raw(self,
        sig: &[u8], rm: usize, m: &[u8]) -> Option<[u8; 64]>
    {
        self.verify_trunc_inner(sig, rm, false, 0, &[0u8; 0], m)
    }

    /// Verifies a truncated signature on a message.
    ///
    /// This is the "Ed25519cx" mode of RFC 8032 (no pre-hashing, a
    /// context is provided). The context string MUST have length at most
    /// 255 bytes; it SHOULD NOT be of length zero. The signature slice
    /// (`sig`) MUST have length exactly 64 bytes; however, this function
    /// assumes that the last `rm` bits of the signature have been reused
    /// to store other data, and thus it ignores these bits. `rm` MUST be
    /// in the 8 to 32 range (inclusive); ignored elements are the last
    /// `floor(rm/8)` bytes, and the top (most significant) `rm%8` bits
    /// of the last non-ignored byte.
    ///
    /// If the original, untruncated signature was valid, then this
    /// function rebuilds it and returns it; otherwise, it returns `None`
    /// (if a rebuilt signature value is returned, then it has been
    /// verified to be valid and there is no need to validate it again).
    ///
    /// Note: this function is not constant-time; it assumes that the
    /// public key and signature value are public data.
    pub fn verify_trunc_ctx(self,
        sig: &[u8], rm: usize, ctx: &[u8], m: &[u8]) -> Option<[u8; 64]>
    {
        self.verify_trunc_inner(sig, rm, true, 0, ctx, m)
    }

    /// Verifies a truncated signature on a message.
    ///
    /// This is the "Ed25519ph" mode of RFC 8032 (message is pre-hashed),
    /// also known as "HashEdDSA on Curve25519". The hashed message `hm`
    /// is provided (presumably, that hash value was obtained with
    /// SHA-512; the caller does the hash itself). A context string is
    /// also provided; it MUST have length at most 255 bytes. The
    /// signature slice (`sig`) MUST have length exactly 64 bytes;
    /// however, this function assumes that the last `rm` bits of the
    /// signature have been reused to store other data, and thus it
    /// ignores these bits. `rm` MUST be in the 8 to 32 range
    /// (inclusive); ignored elements are the last `floor(rm/8)` bytes,
    /// and the top (most significant) `rm%8` bits of the last non-ignored
    /// byte.
    ///
    /// If the original, untruncated signature was valid, then this
    /// function rebuilds it and returns it; otherwise, it returns `None`
    /// (if a rebuilt signature value is returned, then it has been
    /// verified to be valid and there is no need to validate it again).
    ///
    /// Note: this function is not constant-time; it assumes that the
    /// public key and signature value are public data.
    pub fn verify_trunc_ph(self,
        sig: &[u8], rm: usize, ctx: &[u8], hm: &[u8]) -> Option<[u8; 64]>
    {
        self.verify_trunc_inner(sig, rm, true, 1, ctx, hm)
    }

    /// (2^227)*B
    const B227: Point = Point {
        X: GF25519::w64be(0x66E632DB15CC9B76, 0x95E73E2B94033B8D,
                          0x25FCA89B99EF5E24, 0x436B804BB683BE22),
        Y: GF25519::w64be(0x534E3C9C194CCCC1, 0x90FFEA06F048E7A6,
                          0xF65E6D5C7EF60746, 0x93FAD6A6134AF084),
        Z: GF25519::ONE,
        T: GF25519::w64be(0x7A4DEF7F4F0CCDC3, 0xC4D562F88E7B9652,
                          0xA1E950443BABDCA6, 0x25384DDF8800DB95),
    };

    /// Inner truncated signature verification function.
    fn verify_trunc_inner(self, sig: &[u8], rm: usize,
        dom: bool, phflag: u8, ctx: &[u8], msg: &[u8]) -> Option<[u8; 64]>
    {
        // Code is meant for removing between 8 and 32 bits.
        assert!(rm >= 8 && rm <= 32);

        // Signature array must have length 64 bytes exactly; but we ignore
        // the last rm bits. We copy the non-ignored bits to sig2, and clear
        // the rest.
        if sig.len() != 64 {
            return None;
        }
        let n = (519 - rm) >> 3;
        let mut sig2 = [0u8; 64];
        sig2[0..n].copy_from_slice(&sig[0..n]);
        if (rm & 7) != 0 {
            sig2[n - 1] &= 0xFFu8 >> (rm & 7);
        }

        // First half of the signature is the encoded point R; second
        // half is the scalar s (here truncated, we decode it as s0).
        // R must decode successfully (the decoding function enforces
        // canonicality). Scalar s was truncated by clearing the high
        // bits so it will always decode successfully.
        let R_enc = &sig2[0..32];
        let R = Point::decode(R_enc)?;
        let (s0, _) = Scalar::decode32(&sig2[32..64]);

        // SHA-512(dom2(F, C) || R || A || PH(M)) -> scalar k
        // R is encoded over the first 32 bytes of the signature.
        let mut sh = Sha512::new();
        if dom {
            assert!(ctx.len() <= 255);
            let clen = ctx.len() as u8;
            sh.update(&HASH_HEAD);
            sh.update(&[phflag]);
            sh.update(&[clen]);
            sh.update(ctx);
        }
        sh.update(R_enc);
        sh.update(&self.encoded);
        sh.update(msg);
        let hv2 = sh.finalize();
        let k = Scalar::decode_reduce(&hv2);

        // The verification equation is 8*s*B = 8*R + 8*k*A, but we
        // do not have the complete s, only a truncated version s0. We
        // write:
        //   n = 256 - rm
        //   m = 251 - n = rm - 5
        //   s = s0 + 2^251 + s1*2^n
        // with -2^m <= s1 <= +2^m (value s1 = +2^m is possible because
        // L is slightly above 2^252, hence s does not necessarily fit
        // in 252 bits).
        // We rewrite the equation as:
        //    (s0 + 2^251 + s1*2^n)*(8*B) = 8*R + 8*k*A
        // hence:
        //    s1*(8*2^n*B) = 8*R + 8*k*A - 8*(s0 + 2^251)*B
        // i.e. an equation:
        //    s1*U = V
        // for two known points U and V. V can be computed from the
        // equation elements; point U is fixed (it depends only on rm).
        //
        // We use the fact that for all points P on the curve, P and -P
        // have the same y coordinate; thus, we can get two tests for the
        // cost of one.
        //
        // Let I and J be two positive integers such that I*J >= 2^m;
        // We choose J = 2^(min(14,m)) (because we have a precomputed table
        // that can handle up to J = 2^14), and I = (2^m)/J. We write:
        //    s1 = a + I*b
        // for some 0 <= a < I and -J <= b <= J. The principle of the
        // algorithm is the following:
        //
        //   1. Compute all points U_j = j*I*U for 0 <= j <= J.
        //   2. Extract the y coordinates of the U_j and sort them
        //      appropriately to allow easy searches.
        //   3. For 0 <= i < +I, compute V_i = V - i*U.
        //   4. Lookup the y coordinate of V_i in the values extracted
        //      at step 2. If V_i has the same y coordinate as U_j,
        //      then s1 = i + I*j and s2 = i - I*j are possible solutions.
        //
        // Since (a + I*b)*U = V, we have:
        //    b*I*U = V - a*U
        // hence, V_a = V - a*U has the same y coordinate as U_{|b|}. Thus,
        // our search procedure must find the solution (if it exists).
        //
        // The U_j values depend only on the number of bits we ignored
        // in the signature, but not on the signature value; we can thus
        // precompute them, i.e. steps 1 and 2 above are done only once,
        // and the sorted y coordinates are a precomputed table included
        // in the code. To keep that table small enough, we only keep the
        // low 48 bits of each y coordinate (this is enough to make
        // matches, provided that we verify them).

        let m = rm - 5;
        let nJ = if m <= 14 { m } else { 14 };
        let nI = m - nJ;
        let I = 1usize << nI;
        // let J = 1usize << nJ;

        // Tn = 2^n = 2^(256 - n)  (scalar)
        let Tn = Scalar::from_w64le(0, 0, 0, 1u64 << (64 - rm));

        // U = 8*(2^n)*B = 2^(256 - rm + 3)*B = 2^(32 - rm)*(2^227)*B
        let U = Self::B227.xdouble((32 - rm) as u32);

        // V = 8*(R + k*A - (s0 + 2^251)*B)
        let T251 = Scalar::w64le(0, 0, 0, 0x0800000000000000);
        let t = -(s0 + T251);
        let V = (R + self.point.mul_add_mulgen_vartime(&k, &t)).xdouble(3);

        // I*U = 2^(m-nJ)*2^(32 - (m + 5))*(2^227)*B
        //     = 2^(227 + 32 - 5 - nJ)*B
        //     = 2^(254 - nJ)*B
        // Since we set nJ to a maximum of 14, we know that all values j*I*U
        // are necessarily in the set of j*2^240*B, for j = 0 to 2^14; these
        // are the values that were used to generate array UX_COMP[].

        // We compute V_i = V - i*U for all i such that 0 <= i < I. We
        // then compare these points with the points j*I*U. To optimize
        // this computation, we switch to the Montgomery curve x-only
        // representation, which allows computing the x coordinate of
        // V_(i+1) from the x coordinates of V_i and V_(i+1) with cost
        // 4M+2S. We also perform the computation by batches of 200 so
        // that we may normalize them with GF25519::batch_invert() (which
        // also works by batches of 200).
        //
        // Montgomery x-coordinate is equal to (1+y)/(1-y) for the
        // Edwards y coordinate. For the pseudo-addition formulas, we
        // just need from the point U the values X(U) + Z(U) and X(U) - Z(U)
        // (for fractional representation X(U)/Z(U) of the Montgomery
        // x-coordinate of point U), which simplify to U.Z and U.Y
        // (twisted Edwards projective coordinates), respectively.
        let mut X0 = V.Z + V.Y;
        let mut Z0 = V.Z - V.Y;
        let V1 = V - U;
        let mut X1 = V1.Z + V1.Y;
        let mut Z1 = V1.Z - V1.Y;
        let XpZu = U.Z;
        let XmZu = U.Y;

        let mut ii = 0usize;
        while ii < I {
            let ii_start = ii;

            // Generate next batch of V_i.
            let mut Xi = [GF25519::ZERO; 200];
            let mut Zi = [GF25519::ZERO; 200];
            let blen = if (I - ii) < 200 { I - ii } else { 200 };
            for i in 0..blen {
                Xi[i] = X0;
                Zi[i] = Z0;

                // Pseudo-addition formulas.
                // X0/Z0 is the Montgomery x-coordinate of V_i,
                // and X1/Z1 is the Montgomery x-coordinate of V_(i+1).
                // Since V_(i+1) - V_i = U, we get the Montgomery x-coordinate
                // of V_(i+2) as:
                //
                //   X2 = Z0*((X1 - Z1)*(Xu + Zu) + (X1 + Z1)*(Xu - Zu))^2
                //   Z2 = X0*((X1 - Z1)*(Xu + Zu) - (X1 + Z1)*(Xu - Zu))^2
                //
                // see: https://eprint.iacr.org/2017/212 (section 3.2)
                //
                // These formulas are valid for all points, except when
                // X0 = 0 or Z0 = 0, i.e. when V_i is either the neutral,
                // or the point of order 2. We can easily detect these two
                // cases and handle them separately.
                let (X2, Z2) =
                if Z0.iszero() != 0 {
                    // V_i is the neutral; hence, V_(i+1) = U and
                    // V_(i+2) = 2*U
                    let U2 = U.double();
                    (U2.Z + U2.Y, U2.Z - U2.Y)
                } else if X0.iszero() != 0 {
                    // V_i is the point N of order 2; hence V_(i+1) = U + N
                    // and V_(i+2) = 2*U + N
                    let U2 = U.double() + Point::ORDER2;
                    (U2.Z + U2.Y, U2.Z - U2.Y)
                } else {
                    let A = (X1 - Z1) * XpZu;
                    let B = (X1 + Z1) * XmZu;
                    (Z0 * (A + B).square(), X0 * (A - B).square())
                };
                (X0, Z0) = (X1, Z1);
                (X1, Z1) = (X2, Z2);
            }
            ii += blen;

            // We normalize these coordinates to get affine x-coordinates.
            // Note: if we got the point-at-infinity then we will obtain
            // 0 as its "x-coordinate" (on the Montgomery curve, that
            // point really has no x-coordinate); since we only use these
            // values for matching, and we verify the solution afterwards,
            // then it does not matter if we get a spurious zero here.
            GF25519::batch_invert(&mut Zi[0..blen]);
            for i in 0..blen {
                Xi[i] *= Zi[i];
            }

            // Perform the search.
            for i0 in 0..blen {
                // Extract the low 48 bits of the y coordinate of the
                // point. We make a searchable value with the 16-bit index
                // set to 0xFFFF.
                let mut tmp = [0u8; 8];
                tmp[0] = 0xFF;
                tmp[1] = 0xFF;
                tmp[2..8].copy_from_slice(&Xi[i0].encode()[0..6]);
                let x = u64::from_le_bytes(tmp);

                // Look for y in the array: we look for the largest value
                // which is lower than the search key; if the found value
                // has the same high 48 bits as y, then this is a match.
                if x < UX_COMP[0] {
                    continue;
                }
                let mut j1 = 0;
                let mut j2 = UX_COMP.len();
                while (j2 - j1) > 1 {
                    let jm = (j1 + j2) >> 1;
                    if x < UX_COMP[jm] {
                        j2 = jm;
                    } else {
                        j1 = jm;
                    }
                }
                let xf = UX_COMP[j1];
                if (x >> 16) != (xf >> 16) {
                    continue;
                }

                // We use UX_COMP, which contains 16385 values, but this
                // can be some overkill if we ignored fewer than 14 bits.
                // We skip spurious matches that would not correspond to
                // points j*I*U.
                let jf = xf & 0xFFFF;
                let j = (jf >> (14 - nJ)) as u64;
                if jf != ((j as u64) << (14 - nJ)) {
                    // Low bits are not 0, this is a spurious match.
                    continue;
                }

                // We got a match between some V_i and U_j.
                // Candidates for s1 are i + j*I and i - j*I. The signature
                // is valid if and only if s1*U = V. We already have the
                // points U and V, so we just need to compute s1*U for the
                // two candidates.
                let i = (ii_start + i0) as u64;
                let iU = U * i;
                let jIU = U.xdouble(nI as u32) * j;
                if V.equals(iU + jIU) != 0 {
                    let s1 = (i as i64) + ((j as i64) << nI);
                    return Some(Self::make_sig(R_enc,
                        &(s0 + T251 + Tn * Scalar::from_i64(s1))));
                }
                if V.equals(iU - jIU) != 0 {
                    let s1 = (i as i64) - ((j as i64) << nI);
                    return Some(Self::make_sig(R_enc,
                        &(s0 + T251 + Tn * Scalar::from_i64(s1))));
                }
            }
        }

        // We got no match; the signature is invalid.
        return None;
    }

    /// Rebuilds a signature value from the encoded R point (exactly
    /// 32 bytes) and a given scalar.
    fn make_sig(R_enc: &[u8], s: &Scalar) -> [u8; 64] {
        let mut sig = [0u8; 64];
        sig[0..32].copy_from_slice(R_enc);
        sig[32..64].copy_from_slice(&s.encode()[..]);
        sig
    }
}

// ========================================================================

// We hardcode known multiples of the points B, (2^65)*B, (2^130)*B
// and (2^195)*B, with B being the conventional base point. These are
// used to speed mulgen() operations up. The points are moreover stored
// in a three-coordinate format (which was attributed to Niels Duif in
// the original Ed25519 source code): we normalize the point to ensure
// that Z = 1, and we store X+Y, X-Y and 2d*T (with T = X*Y, since Z = 1).
// It can easily be seen that such values are in fact intermediaries used
// in the plain addition formulas.

/// A point in Duif format (y + x, y - x, 2*d*x*y).
#[derive(Clone, Copy, Debug)]
struct PointDuif {
    ypx: GF25519,
    ymx: GF25519,
    t2d: GF25519,
}

impl PointDuif {

    /* unused
    const NEUTRAL: Self = Self {
        ypx: GF25519::ONE,
        ymx: GF25519::ONE,
        t2d: GF25519::ZERO,
    };
    */
}

// Points i*B for i = 1 to 16, in Duif format.
static PRECOMP_B: [PointDuif; 16] = [
    // B * 1
    PointDuif { ypx: GF25519::w64be(0x07CF9D3A33D4BA65, 0x270B4898643D42C2,
                                    0xCF932DC6FB8C0E19, 0x2FBC93C6F58C3B85),
                ymx: GF25519::w64be(0x44FD2F9298F81267, 0xA5C18434688F8A09,
                                    0xFD399F05D140BEB3, 0x9D103905D740913E),
                t2d: GF25519::w64be(0x6F117B689F0C65A8, 0x5A1B7DCBDD43598C,
                                    0x26D9E823CCAAC49E, 0xABC91205877AAA68) },
    // B * 2
    PointDuif { ypx: GF25519::w64be(0x590C063FA87D2E2E, 0x5AA69A65E1D60702,
                                    0x9F469D967A0FF5B5, 0x9224E7FC933C71D7),
                ymx: GF25519::w64be(0x6BB595A669C92555, 0xE09E236BB16E37AA,
                                    0x8F2B810C4E60ACF6, 0x8A99A56042B4D5A8),
                t2d: GF25519::w64be(0x701AF5B13EA50B73, 0x500FA0840B3D6A31,
                                    0x36C16BDD5D9ACF78, 0x43FAA8B3A59B7A5F) },
    // B * 3
    PointDuif { ypx: GF25519::w64be(0x7A164E1B9A80F8F4, 0xC11B50029F016732,
                                    0x025A8430E8864B8A, 0xAF25B0A84CEE9730),
                ymx: GF25519::w64be(0x2AB91587555BDA62, 0x8131F31A214BD6BD,
                                    0x3BD353FDE5C1BA7D, 0x56611FE8A4FCD265),
                t2d: GF25519::w64be(0x5A2826AF12B9B4C6, 0xD170E5458CF2DB4C,
                                    0x589423221C35DA62, 0x14AE933F0DD0D889) },
    // B * 4
    PointDuif { ypx: GF25519::w64be(0x680E910321E58727, 0xCA348D3DFB0A9265,
                                    0x6765C6F47DFD2538, 0x287351B98EFC099F),
                ymx: GF25519::w64be(0x27933F4C7445A49A, 0xC3E8E3CD06A05073,
                                    0x327E89715660FAA9, 0x95FE050A056818BF),
                t2d: GF25519::w64be(0x7F9D0CBF63553E2B, 0x5DDBDCF9102B4494,
                                    0x6E9E39457B5CC172, 0x5A13FBE9C476FF09) },
    // B * 5
    PointDuif { ypx: GF25519::w64be(0x2945CCF146E206EB, 0xDD1BEB0C5ABFEC44,
                                    0x8D5048C3C75EED02, 0xA212BC4408A5BB33),
                ymx: GF25519::w64be(0x154A7E73EB1B55F3, 0xE33CF11CB864A087,
                                    0xD50014D14B2729B7, 0x7F9182C3A447D6BA),
                t2d: GF25519::w64be(0x43AABE696B3BB69A, 0xB41B670B1BBDA72D,
                                    0x270E0807D0BDD1FC, 0xBCBBDBF1812A8285) },
    // B * 6
    PointDuif { ypx: GF25519::w64be(0x51E57BB6A2CC38BD, 0x8065B668DA59A736,
                                    0x9B27158900C8AF88, 0x3A0CEEEB77157131),
                ymx: GF25519::w64be(0x38B64C41AE417884, 0xBB085CE7204553B9,
                                    0x575BE28427D22739, 0x499806B67B7D8CA4),
                t2d: GF25519::w64be(0x10B8E91A9F0D61E3, 0x53E4A24B083BC144,
                                    0xBE70E00341A1BB01, 0x85AC326702EA4B71) },
    // B * 7
    PointDuif { ypx: GF25519::w64be(0x461BEA69283C927E, 0x71B2528228542E49,
                                    0x7470353AB39DC0D2, 0x6B1A5CD0944EA3BF),
                ymx: GF25519::w64be(0x1D6EDD5D2E5317E0, 0x9DEA764F92192C3A,
                                    0x6CA021533BBA23A7, 0xBA6F2C9AAA3221B1),
                t2d: GF25519::w64be(0x7A9FBB1C6A0F90A7, 0x529C41BA5877ADF3,
                                    0xB3035F47053EA49A, 0xF1836DC801B8B3A2) },
    // B * 8
    PointDuif { ypx: GF25519::w64be(0x0915E76061BCE52F, 0xB1339C665ED9C323,
                                    0x6CB30377E288702C, 0x59B7596604DD3E8F),
                ymx: GF25519::w64be(0x3A9024A1320E01C3, 0x2C2741AC6E3C23FB,
                                    0x963D7680E1B558F9, 0xE2A75DEDF39234D9),
                t2d: GF25519::w64be(0x26907C5C2ECC4E95, 0x636412190EB62A32,
                                    0xB8A371788BCCA7D7, 0xE7C1F5D9C9A2911A) },
    // B * 9
    PointDuif { ypx: GF25519::w64be(0x34B9ED338ADD7F59, 0xCEB233C9C686F5B5,
                                    0xA6509E6F51BC46C5, 0x9B2E678AA6A8632F),
                ymx: GF25519::w64be(0x49C05A51FADC9C8F, 0x96CBC608E75EB044,
                                    0x98A081B6F520419B, 0xF36E217E039D8064),
                t2d: GF25519::w64be(0x73C172021B008B06, 0xAAF6FC2993D4CF16,
                                    0xE2FF83E8A719D22F, 0x06B4E8BF9045AF1B) },
    // B * 10
    PointDuif { ypx: GF25519::w64be(0x43AC7628AAE591ED, 0x0D5503639B554646,
                                    0x45F534D41617E057, 0xFF1D93D2B360748E),
                ymx: GF25519::w64be(0x0353832C4950B702, 0x84739745F5DC3958,
                                    0x04F8183665A9F02F, 0x75F3558E227081DD),
                t2d: GF25519::w64be(0x0EC62AF470BF4CE7, 0xFF169F0F6731B509,
                                    0x1D0C1CCBD3F06340, 0xD03D2AE403D0F8D8) },
    // B * 11
    PointDuif { ypx: GF25519::w64be(0x4275AAE2546D8FAF, 0x113E847117703406,
                                    0xE5D9FECF02302E27, 0x2FBF00848A802ADE),
                ymx: GF25519::w64be(0x18AB598029D5C77F, 0xA3A075556A8DEB95,
                                    0x3ED6B36977088381, 0x315F5B0249864348),
                t2d: GF25519::w64be(0x3DC65522B53DF948, 0x44311199B51A8622,
                                    0x031EB4A13282E4A4, 0xD82B2CC5FD6089E9) },
    // B * 12
    PointDuif { ypx: GF25519::w64be(0x078AAFDE8D3CDD58, 0x45ECDD2E701A6F93,
                                    0x88DE3DD7D834D1A9, 0xE2358042A71E7539),
                ymx: GF25519::w64be(0x7956ECE28A6022ED, 0x884DFB6E56D5DBDD,
                                    0x23B2BF90CCB25B24, 0x856F8375B53D54B9),
                t2d: GF25519::w64be(0x37C6A5151C83D0C6, 0xFFCB589AF4976461,
                                    0xF66CDA23A24E180B, 0xEEA594D87F944553) },
    // B * 13
    PointDuif { ypx: GF25519::w64be(0x234FD7EEC346F241, 0x537A0E12FB07BA07,
                                    0xBF84B39AB5BCDEDB, 0xBF70C222A2007F6D),
                ymx: GF25519::w64be(0x0267882D176024A7, 0x9D12B232AAAD5968,
                                    0xAEFCEBC99B776F6B, 0x506F013B327FBF93),
                t2d: GF25519::w64be(0x497BA6FDAA097863, 0xA2EF37F891A7E533,
                                    0x2437E6B1DF8DD471, 0x5360A119732EA378) },
    // B * 14
    PointDuif { ypx: GF25519::w64be(0x6EAF60B2464D1630, 0x1A474C042881BDD5,
                                    0x80277FC057EFA987, 0x26F870EC3F213DF2),
                ymx: GF25519::w64be(0x2DF0EA2C5B3C80D2, 0x112E56F16EEC47A9,
                                    0xCE69B20FDB7CA331, 0xDFDB8A44D4171280),
                t2d: GF25519::w64be(0x24BF7E3CD8BA9C93, 0x9C1FDF703ECB1BAA,
                                    0xF02397EDA2A9BF54, 0x96A1C5877A1E1B82) },
    // B * 15
    PointDuif { ypx: GF25519::w64be(0x61E22917F12DE72B, 0x2DBDBDFAC1F2D4D0,
                                    0x8648C28D189C246D, 0x24CECC0313CFEAA0),
                ymx: GF25519::w64be(0x43B5CD4218D05EBF, 0x7508300807B25192,
                                    0xD3829BA42A9910D6, 0x040BCD86468CCF0B),
                t2d: GF25519::w64be(0x511D61210AE4D842, 0x032E5A7D93D64270,
                                    0xEB38AF4E373FDEEE, 0x5D9A762F9BD0B516) },
    // B * 16
    PointDuif { ypx: GF25519::w64be(0x143B1CF8AA64FE61, 0x587A3A4342D20B09,
                                    0xB9C19F3375C6BF9C, 0x322D04A52D9021F6),
                ymx: GF25519::w64be(0x4CF210EC5A9A8883, 0xE6B5E4193288D1E7,
                                    0xA71284CBA64878B3, 0x7EC851CA553E2DF3),
                t2d: GF25519::w64be(0x21B546A3374126E1, 0xD0A7D34BEA180975,
                                    0x5F54258E27092729, 0x9F867C7D968ACAAB) },
];

// Points i*(2^65)*B for i = 1 to 16, in Duif format.
static PRECOMP_B65: [PointDuif; 16] = [
    // (2^65)*B * 1
    PointDuif { ypx: GF25519::w64be(0x4F675F5302399FD9, 0x77AFC6624312AEFA,
                                    0x537D5268E7F5FFD7, 0xFEC7BC0C9B056F85),
                ymx: GF25519::w64be(0x4AEFCFFB71A03650, 0x1AF07A0BF7D15ED7,
                                    0xB67544B570CE1BC5, 0xDC4267B1834E2457),
                t2d: GF25519::w64be(0x0BCCBB72A2A86561, 0x870A6EADD0945110,
                                    0xCD2BEF118998483B, 0xC32D36360415171E) },
    // (2^65)*B * 2
    PointDuif { ypx: GF25519::w64be(0x420BF3A79B423C6E, 0x1BB687AE752AE09F,
                                    0x914E690B131E064C, 0x21717B0D0F537593),
                ymx: GF25519::w64be(0x676B2608B8D2D322, 0xEBA13F0708455010,
                                    0x6155985D313F4C6A, 0x97F5131594DFD29B),
                t2d: GF25519::w64be(0x745D2FFA9C0CF1E0, 0x7BFF0CB1BC3135B0,
                                    0x8671B6EC311B1B80, 0x8138BA651C5B2B47) },
    // (2^65)*B * 3
    PointDuif { ypx: GF25519::w64be(0x08BA696B531D5BD8, 0x33809107F12D1573,
                                    0x48EEEF8EF52C598C, 0x2DFB5BA8B6C2C9A8),
                ymx: GF25519::w64be(0x2FF39DE85485F6F9, 0x5CE382F8BC26C3A8,
                                    0xC8C976C5CC454E49, 0xD8173793F266C55C),
                t2d: GF25519::w64be(0x120633B4947CFE54, 0xEA3D7A3FF1A671CB,
                                    0x04E05517D4FF4811, 0x77ED3EEEC3EFC57A) },
    // (2^65)*B * 4
    PointDuif { ypx: GF25519::w64be(0x0543618A01600253, 0x18A275D3BAE21D6C,
                                    0xA20D59175015E1F5, 0xAA6202E14E5DF981),
                ymx: GF25519::w64be(0x2E773654707FA7B6, 0xB093FEE6F5A64806,
                                    0xFAB8B7EEF4AA81D9, 0x157A316443373409),
                t2d: GF25519::w64be(0x4B1443362D07960D, 0x04202CB8A29ABA2C,
                                    0xAA6F0A259DCE4693, 0x0DEABDF4974C23C1) },
    // (2^65)*B * 5
    PointDuif { ypx: GF25519::w64be(0x3719BFDBC4548BF0, 0x4E3136E35C4C2D29,
                                    0x15055EA375641765, 0x5F49672DDC5E732A),
                ymx: GF25519::w64be(0x02F578EC9A5A71F5, 0x879AA4C76E5EEEE2,
                                    0x8EABFB006BB1BF82, 0x8B23A241C9162B28),
                t2d: GF25519::w64be(0x4280AE77A8228B15, 0xD689EF7D743C587A,
                                    0x00C8F5B68CF78E2C, 0xCCBDAB0F2F605518) },
    // (2^65)*B * 6
    PointDuif { ypx: GF25519::w64be(0x53CADF8D52E963EE, 0x45E7EAA6A658744F,
                                    0x5C00D1A9DD4A3811, 0xE4C10FF606F09880),
                ymx: GF25519::w64be(0x75C6AC1496383BAE, 0x06EBF5F5809055D0,
                                    0xCCDEBEB816C23084, 0x5D263EB31CC113C0),
                t2d: GF25519::w64be(0x036D4DFF214CCB4D, 0x09B12E297B984BDA,
                                    0x767AD824F8CB6165, 0x825B343F186E4A37) },
    // (2^65)*B * 7
    PointDuif { ypx: GF25519::w64be(0x2BCE4610C35A9C37, 0x3E774CC82FA35D5C,
                                    0x6B0E47FE980A8643, 0xFD03EAD6374DFD02),
                ymx: GF25519::w64be(0x3101C3210F55BB04, 0xFFB22982B46268F9,
                                    0xB3F10DBC73C79C79, 0x6432CF46CD9E38D0),
                t2d: GF25519::w64be(0x510EEF823E1B979E, 0xD5F7C521AE297E47,
                                    0x36E235F77A73F12D, 0xCC743CB43533E1D5) },
    // (2^65)*B * 8
    PointDuif { ypx: GF25519::w64be(0x2C435C24A44D9FE1, 0x3004806447235AB3,
                                    0x96CB929E6B686D90, 0x299B1C3F57C5715E),
                ymx: GF25519::w64be(0x48EA295BAD8A2C07, 0xE222FBFBE1D928C5,
                                    0x256DC48CC04212F2, 0x47B837F753242CEC),
                t2d: GF25519::w64be(0x7BCB4792A0DEF80E, 0x54F7450B161EBB6F,
                                    0x0E851578CA25EC5B, 0x0607C97C80F8833F) },
    // (2^65)*B * 9
    PointDuif { ypx: GF25519::w64be(0x496CF090064A0608, 0x2512555AEE0272D0,
                                    0xE28F38188A024866, 0x05B96B999A7EE0E4),
                ymx: GF25519::w64be(0x66F62082076347C1, 0x9C82D851A29B389A,
                                    0xAA201139C6D4AC1D, 0xB2777136EE675E2C),
                t2d: GF25519::w64be(0x1D70275FEB768CC5, 0x9EA6F6E4108105EA,
                                    0xE5E298562ED9B280, 0x4B5F958941A47600) },
    // (2^65)*B * 10
    PointDuif { ypx: GF25519::w64be(0x53085B03822DE6A8, 0x3EEE9B1D3EB1636A,
                                    0x5E6580ABD5EABC74, 0x47A1B04BC9818FE1),
                ymx: GF25519::w64be(0x0E11CDAC543A3B00, 0xCE92CE8444F1CD5B,
                                    0x6235A6330DE03DBA, 0xA0C043A82A7A9A32),
                t2d: GF25519::w64be(0x46A2D245D05FE473, 0x3CC38E6C29C792C2,
                                    0xBC07B226D267B65A, 0x25DE3437214201CA) },
    // (2^65)*B * 11
    PointDuif { ypx: GF25519::w64be(0x6670734032FAD700, 0x84A1506892F28482,
                                    0x767139A3EAC355BC, 0x40E8858AC3DEFE7C),
                ymx: GF25519::w64be(0x5181C8C96978562B, 0x6C3220475F12F8B1,
                                    0x596ADF7722460F2A, 0xD278466006EEB5E6),
                t2d: GF25519::w64be(0x7AE6F9429B50739E, 0x055112A4F99F9740,
                                    0xB311DCC9EF413331, 0xB8346F7319F27259) },
    // (2^65)*B * 12
    PointDuif { ypx: GF25519::w64be(0x79C0E0FEC46C5B71, 0xD9929904D2DF7648,
                                    0x131691364E6964CC, 0x5B4614EC6116F3CA),
                ymx: GF25519::w64be(0x1B1B2C535FC8EA31, 0xF6BF72415992B150,
                                    0x149E595106F7931F, 0x13795198693756E6),
                t2d: GF25519::w64be(0x37D76EC8A91B9EA0, 0x921372848F0B2688,
                                    0xF83FB13E814CB1A1, 0x710593B24AED47CF) },
    // (2^65)*B * 13
    PointDuif { ypx: GF25519::w64be(0x1962668863F82C69, 0x666618C99A176A9F,
                                    0x7B4DFA8DC38624D8, 0x0FAA62D322656FE5),
                ymx: GF25519::w64be(0x181C797B6AFDE525, 0x8EB7B6AFA5F2B2DA,
                                    0x4D8A6186B2089372, 0x6505E445215F2E7C),
                t2d: GF25519::w64be(0x1EA31DCC327F524C, 0x6BC24CAAD615C878,
                                    0x6F7DCF828B707287, 0xC37208F578DFF6A3) },
    // (2^65)*B * 14
    PointDuif { ypx: GF25519::w64be(0x5DFF3528725B783A, 0x87532EC4A1830DB8,
                                    0x138EBE5CBEC187C9, 0x6CD60FBAC78DA244),
                ymx: GF25519::w64be(0x259CA3DD93A96DEF, 0x8103A85F9C598984,
                                    0xAD54A807E03E23A6, 0xB594D80A106BFA62),
                t2d: GF25519::w64be(0x1D744B3212031DB8, 0xB8DF09E458923E05,
                                    0x4AABAFCA0D461F00, 0xDBDBBB96AE414FD9) },
    // (2^65)*B * 15
    PointDuif { ypx: GF25519::w64be(0x79D51D0A3165A2E2, 0x6000B30B7B56C2CB,
                                    0xEBE27A2F70B4006A, 0x4F30118837E1C0C6),
                ymx: GF25519::w64be(0x52BBFE9FC78E5A02, 0xB796012A59639018,
                                    0xD7639FDBEF14837F, 0x9A05A5497C41ECEB),
                t2d: GF25519::w64be(0x7C352C3F74C60034, 0x25BB40710E1A8E24,
                                    0xF5C9F06C59947F4B, 0x9CBCF813651CB4AE) },
    // (2^65)*B * 16
    PointDuif { ypx: GF25519::w64be(0x1B6CC62016736148, 0x775B7A925289F681,
                                    0x757F1B1B69E53952, 0x1CECD0A0045224C2),
                ymx: GF25519::w64be(0x57369F0BDEFC96B6, 0xD17C975ADCAD6FBF,
                                    0x4BAF8445059979DF, 0x8487E3D02BC73659),
                t2d: GF25519::w64be(0x63FA6E6843ADE311, 0x849471334C9BA488,
                                    0x353DD1BEEEAA60D3, 0xF1A9990175638698) },
];

// Points i*(2^130)*B for i = 1 to 16, in Duif format.
static PRECOMP_B130: [PointDuif; 16] = [
    // (2^130)*B * 1
    PointDuif { ypx: GF25519::w64be(0x720A5BC050955E51, 0x20F5B522AC4E60D6,
                                    0x844A06E674BFDBE4, 0x9D18F6D97CBEC113),
                ymx: GF25519::w64be(0x2E0C92BFBDC40BE9, 0x61E3061FF4BCA59C,
                                    0xF700660E9E25A87D, 0xC3A8B0F8E4616CED),
                t2d: GF25519::w64be(0x0E9B9CBB144EF0EC, 0x691417F35C229346,
                                    0xE84E8B376242ABFC, 0x0C3F09439B805A35) },
    // (2^130)*B * 2
    PointDuif { ypx: GF25519::w64be(0x3451995F2944EE81, 0x6F619B39F3B61689,
                                    0xE60577AAFC129C08, 0x962CD91DB73BB638),
                ymx: GF25519::w64be(0x445484A4972EF7AB, 0xA61E6B2D368D0498,
                                    0x5F541C511857EF6C, 0x44BEB24194AE4E54),
                t2d: GF25519::w64be(0x10B89CA6042893B7, 0x258E9AAA47285C40,
                                    0x4A816C94B0935CF6, 0x9152FCD09FEA7D7C) },
    // (2^130)*B * 3
    PointDuif { ypx: GF25519::w64be(0x346F01ACD2869617, 0x97D27CFEC07F8BDA,
                                    0x481C5D4F3B9FF9AE, 0xA9EADFBE60E27CD7),
                ymx: GF25519::w64be(0x0AC2129186A49D9B, 0x80DA4DB2A1C7710C,
                                    0xE6E815182154EF4E, 0x532A2120BF310F6E),
                t2d: GF25519::w64be(0x0970C3F002E3FD69, 0x4768AC983FE2336C,
                                    0x42307975A28106B2, 0xD88700075CB01065) },
    // (2^130)*B * 4
    PointDuif { ypx: GF25519::w64be(0x2E05D9EAF61F6FEF, 0xA535A456E35D190F,
                                    0xCC0B9EC0CC4DB39F, 0xD67CDED679D34AA0),
                ymx: GF25519::w64be(0x06409010BEA8DE75, 0xA25CFFC2DD6DEA45,
                                    0x32127190385CE4CF, 0x9B2A426E3B646025),
                t2d: GF25519::w64be(0x293C778CEFE07F26, 0x24685482B7CA6827,
                                    0x661F19BCE5DC880A, 0xC447901AD61BEB59) },
    // (2^130)*B * 5
    PointDuif { ypx: GF25519::w64be(0x5C6AC6F202184045, 0x20FED32C065A104F,
                                    0x9C019C2825B1F150, 0xEE763B43896F9346),
                ymx: GF25519::w64be(0x4F9B5A5B384DDE6D, 0xE09C2A13B3D5B1D3,
                                    0x527DC853036F67DE, 0xE0555077BBA9D1CF),
                t2d: GF25519::w64be(0x116ADE0F0C3AF50B, 0xD70BA52C3114662F,
                                    0xD34AD18F6B75FBEE, 0x03D4FB81703B3A97) },
    // (2^130)*B * 6
    PointDuif { ypx: GF25519::w64be(0x6CB38C89E457BE03, 0x2F371874E2368013,
                                    0x009634D955BCA989, 0x1C91324BFF709850),
                ymx: GF25519::w64be(0x46785F6E9E09137D, 0x937159234A83201F,
                                    0x1D1F7A3DA1BE148B, 0x795EEE0FE838AB22),
                t2d: GF25519::w64be(0x7BBA5431BCA76FF6, 0x311C59D360912E37,
                                    0xDF18333709F08B4A, 0xAF4E83ECA129234C) },
    // (2^130)*B * 7
    PointDuif { ypx: GF25519::w64be(0x786C11F422B6004B, 0x565E0C35171FDEF1,
                                    0x3CAC00D6BFA0A80A, 0x397354FDF67DBB72),
                ymx: GF25519::w64be(0x082B3D95B81D503C, 0x4C39E0A61ADD99B4,
                                    0x012AC6090631C0CB, 0x2950952982E7BDAB),
                t2d: GF25519::w64be(0x79593A61EFC2EC67, 0x545B6109D88516F1,
                                    0xDF0501CC3CCC3A31, 0xEAB54BC1C835B149) },
    // (2^130)*B * 8
    PointDuif { ypx: GF25519::w64be(0x50B8C2D031E47B4F, 0x89F293209B5395B5,
                                    0xCB70D0E2B15815C9, 0x16C795D6A11FF200),
                ymx: GF25519::w64be(0x0487F3F112815D5E, 0x07F35715A21A0147,
                                    0xAAD75B15E4E50189, 0x86809E7007069096),
                t2d: GF25519::w64be(0x4B0553B53CDBA58B, 0x17AF4F4AAF6FC8DD,
                                    0x6FFDD05351092C9A, 0x48350C08068A4962) },
    // (2^130)*B * 9
    PointDuif { ypx: GF25519::w64be(0x59DE00448918589B, 0x7993E00119B5096B,
                                    0xC3B6E67668F9C475, 0x9A5DAB65F3F624FE),
                ymx: GF25519::w64be(0x502DDB71BB9AA9ED, 0x43EBF62E8D0D44FF,
                                    0xFC4F1C1C6F153C79, 0x85CE2F2F2593E409),
                t2d: GF25519::w64be(0x4560794D369EDBA6, 0x1E845B0665A0587D,
                                    0xA4550A54513060EE, 0xFDFE1EB21320283C) },
    // (2^130)*B * 10
    PointDuif { ypx: GF25519::w64be(0x024D127C0B1CCBAA, 0x9F10EFC5D398BED0,
                                    0xEF06CE8D3EB359D6, 0xE89E457F0B1A90DA),
                ymx: GF25519::w64be(0x2071B5F318A12E8E, 0x15F6272AE65152CB,
                                    0x0D21836087E3D5C1, 0xB1F43460EAA08DE2),
                t2d: GF25519::w64be(0x1D140135E65254CE, 0xE2A53C02D8931553,
                                    0xE33633B297C70357, 0xA504B50084927524) },
    // (2^130)*B * 11
    PointDuif { ypx: GF25519::w64be(0x35982724CABFA938, 0xD8D589F9ADDF6878,
                                    0xB252E3230DA55379, 0x5662E35E5B95F14D),
                ymx: GF25519::w64be(0x7180771C829F7074, 0x4D217B04B888AFB2,
                                    0xE31B531083BBA50C, 0x6752C35DFE727026),
                t2d: GF25519::w64be(0x6F6EFDCBE76EFF9E, 0xB55CEB78FCA5DB50,
                                    0x04EF65043B30CD77, 0x4ADB64F2DDC97340) },
    // (2^130)*B * 12
    PointDuif { ypx: GF25519::w64be(0x054C8BDD50BD0840, 0x5E0B2CAA8E6FAB98,
                                    0x5EC26849BD1AF639, 0xBF05211B27C152D4),
                ymx: GF25519::w64be(0x35106CD551717908, 0xFCED2A6C6C07E606,
                                    0xEB75EA9F03B50F9B, 0x9C65FCBE1B32FF79),
                t2d: GF25519::w64be(0x72E82D5E5505C229, 0xFED5AC25D3404F9A,
                                    0x4B60A8A3B7F6A276, 0x38A0B12F1DCF073D) },
    // (2^130)*B * 13
    PointDuif { ypx: GF25519::w64be(0x176E80FEAFD65ED2, 0x88E0B82A1B342085,
                                    0x8E4C8E3879BD17D8, 0x73D765296C75B946),
                ymx: GF25519::w64be(0x199CC3EB95298DFC, 0x67779DA5AB282061,
                                    0xC81FCC7F02AD2E2E, 0x9FE1CBB8AB6BEFCF),
                t2d: GF25519::w64be(0x07A91D9445F84939, 0x4AE9B1B877D1439E,
                                    0x0BEA9479144A07FB, 0xDC17D38DC8893B1A) },
    // (2^130)*B * 14
    PointDuif { ypx: GF25519::w64be(0x3AD2C7546B769BE0, 0xB9B3A7101986D3A9,
                                    0x30ACB6C129F01E5C, 0xDE5A3A631C0053B9),
                ymx: GF25519::w64be(0x597EA4C5CB2E0C3A, 0x0676E61728E90987,
                                    0x080222C92B5BF106, 0xEFC754E1F471F7DD),
                t2d: GF25519::w64be(0x2F35BDDA16D3BBA1, 0x4A057C65378F51CE,
                                    0xADE79B45B5C50642, 0x2345AA028B43F0EF) },
    // (2^130)*B * 15
    PointDuif { ypx: GF25519::w64be(0x0385DA9F416DB3AA, 0x8AB069671B274D45,
                                    0xDC6B1D45A5B02F86, 0x618F3104FD45F3C8),
                ymx: GF25519::w64be(0x2D68905794A8F9C4, 0xA66F8060DDD6B730,
                                    0x404FC1F72CEE8614, 0xFD826C0C0F266C17),
                t2d: GF25519::w64be(0x1469BC10E9AFCA61, 0xF3AC04BAE33A17A5,
                                    0x2BB04136D062E37A, 0x9007EBBE0B574C75) },
    // (2^130)*B * 16
    PointDuif { ypx: GF25519::w64be(0x2857BF1627500861, 0x4C45306C1CB12EC7,
                                    0x410276CD6CFBF17E, 0x00D9CDFD69771D02),
                ymx: GF25519::w64be(0x7B7C242958CE7211, 0xD2A541C6C1DA0F1F,
                                    0xBB12F85CD979CB49, 0x6B0B697FF0D844C8),
                t2d: GF25519::w64be(0x510DF84B485A00D4, 0xA122EE5F3DEB0F1B,
                                    0xD779DFD3BF861005, 0x9F21903F0101689E) },
];

// Points i*(2^195)*B for i = 1 to 16, in Duif format.
static PRECOMP_B195: [PointDuif; 16] = [
    // (2^195)*B * 1
    PointDuif { ypx: GF25519::w64be(0x1DEFC6AD32B587A6, 0x29A17FD797373292,
                                    0x8F72EB2A2A8C41AA, 0x671FEAF300F42772),
                ymx: GF25519::w64be(0x0EB28BF671928CE4, 0x38AC15510A4811B8,
                                    0x388DDECF1C7F4D06, 0x0AE28545089AE7BC),
                t2d: GF25519::w64be(0x467D201BF8DD2867, 0x2991F7FB7AE5DA2E,
                                    0x148C1277917B15ED, 0xAF5BBE1AEF5195A7) },
    // (2^195)*B * 2
    PointDuif { ypx: GF25519::w64be(0x51FC2B28D43921C0, 0xB0E5B13F5839E9CE,
                                    0x993580D4D8152E65, 0x745F9D56296BC318),
                ymx: GF25519::w64be(0x44C218671C974287, 0x8D5CFE45B941A8A4,
                                    0x05D270D6109ABF4E, 0x7906EE72F7BD2E6B),
                t2d: GF25519::w64be(0x6E6B9DE84C4F4AC6, 0x5B30E7107424B572,
                                    0x1C4E5EE12B6B6291, 0x1B8FD11795E2A98C) },
    // (2^195)*B * 3
    PointDuif { ypx: GF25519::w64be(0x1E6C5FF840BF2091, 0x75E8270254F351AD,
                                    0xBB735DE8E48757AA, 0xFC5E11F5CA6957BF),
                ymx: GF25519::w64be(0x71E99AB52400BC69, 0x580FD74A91193BBC,
                                    0x1CB03668863E1BE3, 0xB1BEA535D198EB58),
                t2d: GF25519::w64be(0x7756230E8AC60507, 0x268B2EF2B9FBF01B,
                                    0x4575091331453155, 0xB16656A080DE210D) },
    // (2^195)*B * 4
    PointDuif { ypx: GF25519::w64be(0x5F4C802CC3A06F42, 0xC2B620A5C6EF99C4,
                                    0x736B54DC56E42151, 0x6B7C5F10F80CB088),
                ymx: GF25519::w64be(0x2D292459908E0DF9, 0x2554B3C854749C87,
                                    0xD841C0C7E11C4025, 0xDFF25FCE4B1DE151),
                t2d: GF25519::w64be(0x66ED5DD5BEC10D48, 0xC3B514F05B62F9E3,
                                    0x881CE338C77EE800, 0x9B65C8F17D0752DA) },
    // (2^195)*B * 5
    PointDuif { ypx: GF25519::w64be(0x650E0BCEB4CB7051, 0xC9EF50B4EB01C714,
                                    0xA131131F0525EF3F, 0x1DF8E1F64A821880),
                ymx: GF25519::w64be(0x288A9AB8CFC8E850, 0x5A49CB0C9B689275,
                                    0x624C17111B1529AB, 0x6B78A6BE8D33AAA7),
                t2d: GF25519::w64be(0x51FEF585F8870F17, 0x0D7A118618982E44,
                                    0x410821C7BD75C5C9, 0xBB4A244319BD9EAB) },
    // (2^195)*B * 6
    PointDuif { ypx: GF25519::w64be(0x1F23A0C77E20048C, 0xCFDA112D44735F93,
                                    0x81C3B2CBF4552F6B, 0xF0ADF3C9CBCA047D),
                ymx: GF25519::w64be(0x2EACF8BC03007F20, 0xC4A70B8C6C97D313,
                                    0x808334E196CCD412, 0x7D38A1C20BB2089D),
                t2d: GF25519::w64be(0x0840BEF29D34BC50, 0x27529AA2FCF9E09E,
                                    0x03D2D9020DBAB38C, 0xF235467BE5BC1570) },
    // (2^195)*B * 7
    PointDuif { ypx: GF25519::w64be(0x707A98EE755F8778, 0x7DABE4EB93F9027C,
                                    0x17657B1D4F344DE3, 0xE85DA7971916B50F),
                ymx: GF25519::w64be(0x158C0EAAFAD87769, 0x6199DB0D592C9C8D,
                                    0x2D0EACF13DBF360F, 0xC46CF746B9DADBB0),
                t2d: GF25519::w64be(0x456055ACF5854046, 0xAB8E4284ADE8E127,
                                    0x421C216A64D7CFC3, 0xA25D104C57D9B9E6) },
    // (2^195)*B * 8
    PointDuif { ypx: GF25519::w64be(0x246AFFA06074400C, 0xB8248BB0D3597DCE,
                                    0x8CC15F87F5E96CCA, 0xCD54E06B7F37E4EB),
                ymx: GF25519::w64be(0x0304F5A191C54276, 0x7F3D43E8C7B24905,
                                    0x27176BCD5C7FF29D, 0x796DFB35DC10B287),
                t2d: GF25519::w64be(0x25A83CAC5753D325, 0x4E9B13EF894A0D35,
                                    0x86097548C0D75032, 0x37D88E68FBE45321) },
    // (2^195)*B * 9
    PointDuif { ypx: GF25519::w64be(0x477CB667E65969FE, 0xBC0D547FDB7C5829,
                                    0x784F776F728F9AF4, 0x31358659E065D14B),
                ymx: GF25519::w64be(0x401F1A1301E87D05, 0xE4BDA0BCBD529628,
                                    0xD460D63857FA2C64, 0x9999E9AED084E426),
                t2d: GF25519::w64be(0x7CA09B26D3FF0F8E, 0xF239EF64F011E4AD,
                                    0xA4A388118D0D965A, 0xC2060E9C27139D1F) },
    // (2^195)*B * 10
    PointDuif { ypx: GF25519::w64be(0x06BE10F5C506E0C9, 0xFF45252BD609FEDC,
                                    0x33DB5E0E0934267B, 0x9F0F66293952B6E2),
                ymx: GF25519::w64be(0x7CCFA59FCA782630, 0x1E145C09C221E8F0,
                                    0x623FC1234B8BCF3A, 0x10222F48EED8165E),
                t2d: GF25519::w64be(0x5E82770A1A1EE71D, 0xA7A2788528BC0DFE,
                                    0x22050C564A52FECC, 0x1A9615A9B62A345F) },
    // (2^195)*B * 11
    PointDuif { ypx: GF25519::w64be(0x58BF4063F679C9EE, 0xB615F749019BCCE0,
                                    0x54B31FD699070785, 0x417DF7554D12AAF0),
                ymx: GF25519::w64be(0x618F7132438ED474, 0x5E4753DCC22E53D0,
                                    0x765FD7271E6B7B02, 0x6293B15A69FDE737),
                t2d: GF25519::w64be(0x768287DFDA4A954D, 0xA527A1A552A50BDC,
                                    0x76C519467C9718E0, 0x850EA7591A673BEA) },
    // (2^195)*B * 12
    PointDuif { ypx: GF25519::w64be(0x2CCA982C605BC5EE, 0x34865D1F1C408CAE,
                                    0x34175166A7FFFAE5, 0xE802E80A42339C74),
                ymx: GF25519::w64be(0x09D04F3B3B86B102, 0x2C66F25F92A35F64,
                                    0xE8673AFBE78D52F6, 0x35425183AD896A5C),
                t2d: GF25519::w64be(0x7A325D1727741D3E, 0x2613D8DB325AE918,
                                    0x207C2EEA8BE4FFA3, 0xFD2D5D35197DBE6E) },
    // (2^195)*B * 13
    PointDuif { ypx: GF25519::w64be(0x2F9ED53170073087, 0x60ADA9DA4CF73B50,
                                    0x33B4C6F2E4046E9D, 0x1067A6BF0E98C031),
                ymx: GF25519::w64be(0x57435F4B4C60F5C1, 0xFC963922C6BC5F7C,
                                    0xD40589473F153DFD, 0x0D8CC27A7EB7F9AE),
                t2d: GF25519::w64be(0x5ED5CEA05FD33707, 0x3E6F740EA65DD90C,
                                    0x67C163BF1C6EB44F, 0xC8655BBFD0D7649B) },
    // (2^195)*B * 14
    PointDuif { ypx: GF25519::w64be(0x2A479DF17BB1AE64, 0x52A61AF0919233E5,
                                    0xD788689F1636495E, 0xECD27D017E2A076A),
                ymx: GF25519::w64be(0x4D3B1A791239C180, 0x8E6CC966A7F12667,
                                    0xA2055757C497A829, 0xD036B9BBD16DFDE2),
                t2d: GF25519::w64be(0x27AD5538A43A5E9B, 0xA41C22C592718138,
                                    0x189854DED6C43CA5, 0x9E5EEE8E33DB2710) },
    // (2^195)*B * 15
    PointDuif { ypx: GF25519::w64be(0x53A59A9B3E95E758, 0x7073DBC9EA999F40,
                                    0xE8A5066258C81CD2, 0xA19050992EFB92D5),
                ymx: GF25519::w64be(0x0F7014C65C32DF89, 0x77C5FCE2C6AAB79C,
                                    0x63B14D2BC9E50F57, 0x5E38A720E54B655C),
                t2d: GF25519::w64be(0x739DDBE845A7512D, 0x161BCB08B61D7A6D,
                                    0x794562FCDADDAC29, 0x2082A0615E1607EA) },
    // (2^195)*B * 16
    PointDuif { ypx: GF25519::w64be(0x080153B7503B179D, 0x549E1E4D8BEDFDCC,
                                    0x8DB7536120A1C059, 0xCB5A7D638E47077C),
                ymx: GF25519::w64be(0x510E987F7E7D89E2, 0xE86E365A138672CA,
                                    0xD03FCBC8EE9521B7, 0x2746DD4B15350D61),
                t2d: GF25519::w64be(0x23BE8D554FE7372A, 0xC817AD58BDAA4EE6,
                                    0x3D386EF1CD60A722, 0xDDA69D930A3ED3E3) },
];

// ========================================================================

/// Let U_i = i*(2^240)*B  (i = 0 to 16384, B = generator).
/// Let M_i = mapping of U_i into the Montgomery curve; if the y-coordinate
/// of U_i is y_i, then the x-coordinate of M_i (in the Montgomery curve)
/// is x_i = (1 + y)/(1 - y).
/// Let z_i = (x % 2^48)*2^16 + i  (low 48 bits of x_i, shifted, and index i)
/// `UX_COMP[]` contains the z_i in ascending numerical order. All values are
/// distinct in their top 48 bits.
static UX_COMP: [u64; 16385] = [
    0x0000000000000000, 0x000750609E2A158D, 0x000B7B1CDE3A3C32,
    0x000FF312538F0BFF, 0x0020DCECE10D122F, 0x00257F14A0433BAC,
    0x003193397FE03677, 0x00326FC1D9DC23D9, 0x0033E27B8BEA33F3,
    0x0036A7C5403F1C30, 0x0038AF69C4FD3E56, 0x0042802517E01C5B,
    0x0043CB6667802080, 0x00467186B6433A75, 0x00538130E38E2251,
    0x005596E73DDE2597, 0x0057C0A0C5FF2FDB, 0x005863520368327D,
    0x0058648A3D892DA3, 0x005F28A921031C6E, 0x00614CFCB5300C80,
    0x00662FA340FF2169, 0x0067DE903A6F15A0, 0x0067E4B7AA092F2C,
    0x0069409FA632190F, 0x0069A08A9B482E48, 0x0077C66954A12A9E,
    0x00783A49D6122129, 0x0078517060CF3CB4, 0x00797169528B0099,
    0x007D566933AF3605, 0x007F3CD9B3D62C87, 0x0081205FCCA6397E,
    0x0082EEC7A4752852, 0x0087F2173DF1096F, 0x008861AA5EF103C4,
    0x0088A73240CD2A68, 0x008CE526F90A3838, 0x008F2C0B44B20A7A,
    0x00972BEF8E973840, 0x009B5C5D877933A1, 0x009B66A1D1410F13,
    0x009BA5E597803A68, 0x009EB406B4B034EF, 0x009F65D73E660180,
    0x00A084B0B44D31FC, 0x00A4B0BA53573AA6, 0x00A5A223EFFE3852,
    0x00B0AF7A3ABB0A41, 0x00B444B8B7C23304, 0x00B5358A678D10D3,
    0x00B7C0B086B40B0F, 0x00BB94297194208B, 0x00BC36F6B46F0922,
    0x00BC53316AC00C66, 0x00C425B6B8D5285C, 0x00DA9E64D0BC27B7,
    0x00F2A44D1F2E1309, 0x00F60E7140FE23A2, 0x00FFE141E6870F29,
    0x00FFF95146603E72, 0x0101EDF81C9E3314, 0x01109812FEEE2D3C,
    0x01122F7BFCF62AFB, 0x0113B2D8B7762A04, 0x01143A1264D83A35,
    0x011575C42F4A2665, 0x0119921A800B128A, 0x011CF86BED5F1989,
    0x011F87687DBE1E69, 0x01250BF662530E9D, 0x012585D9D6713F95,
    0x0126E7F22E631A0C, 0x012BE6F55BFB3086, 0x0133A84D1F853C12,
    0x01355F2FFB522CF6, 0x01365A8858A80249, 0x0138E92CBE6C03B7,
    0x0140BA20568911F0, 0x0141657BF4A52520, 0x014254CEB7DA1A5A,
    0x014CB79739B21858, 0x0158B2B1B1E6150F, 0x015C1E505F4D075A,
    0x015D34846E5F065B, 0x015EB9A28EEA0D08, 0x0161202664FD2D27,
    0x0164836E57D80B79, 0x016F7724146214B3, 0x0173B26820E01701,
    0x0179717FDF6C3E08, 0x0179B29CCE8F076E, 0x0187177AEA4C2A78,
    0x018BD1186F3C3E5F, 0x018C3BA622BC180D, 0x018D6FFF3FFF1855,
    0x018E4BC769AF254A, 0x0192896D5E7019B5, 0x0194734899F63925,
    0x0194AAE9DCEF2475, 0x01977ED036B92A38, 0x01982493FE1831C9,
    0x01B3D10B64EC2510, 0x01B509936A0935AA, 0x01B5F75254C837F0,
    0x01C047C9B2B210EF, 0x01C15E87F9202BAA, 0x01C20A394AFA2FEB,
    0x01C532CF3A28067B, 0x01C9F3D711B01505, 0x01CADA066D96213D,
    0x01D080ACBBD425BA, 0x01D85B7A518D005F, 0x01E239E3F436277C,
    0x01E254B93B5D0B6C, 0x01E60831DBDA34FF, 0x01E8031E36770415,
    0x01E83176C7442075, 0x01EA04DD8E6F0839, 0x01EA6CDEC3AF111E,
    0x01EF2451648030EF, 0x01EFF80D9DE1210E, 0x01F4E4381D4F3CE7,
    0x01F5EE02732D2EAA, 0x01F66DCB4CCA2374, 0x020564E6566A0BB0,
    0x02058FC59D6F35DF, 0x0207CC99A4032209, 0x0215AAD353D9378F,
    0x021687C2CF020B7D, 0x0219A18D32210D1A, 0x021B2D3C89941F41,
    0x02217D6C363404B1, 0x0221995E7EAD3F0C, 0x02258AB2B670320C,
    0x022918A9796523BA, 0x0231422FD9CE110C, 0x02364B9708ED1EB7,
    0x023C4879A7132189, 0x02432423CCE235A7, 0x024613218AF728DF,
    0x024D94D883823237, 0x0253C013E89A360B, 0x025E329D8AFA3C5A,
    0x025EEE9C1DD63E62, 0x02694A07B7CE0AF3, 0x026EBA6B849F08B2,
    0x02769FFDE66E2CA0, 0x0277AC2EB8F60419, 0x0283F95D785020C5,
    0x028B1930158A065A, 0x029D31B478EB3CC7, 0x02A00FD59FE412DE,
    0x02A2439381240A91, 0x02A5435D9EA8224A, 0x02AB3478F6A63BA2,
    0x02B4598F08DF17E7, 0x02B665EBAFEF30C9, 0x02B6B0025CCF0057,
    0x02B7ECBF615F3089, 0x02B7F6033B730793, 0x02C25F475E5F2584,
    0x02CF531556181012, 0x02D220D7F4AB39A8, 0x02D358743B910278,
    0x02D4BCFAAB061573, 0x02D4DCDFB7A70E27, 0x02DD12B500C61B2C,
    0x02E17C08539C0151, 0x02E225467C942541, 0x02E8EF44CA082B84,
    0x02E9499527D43E23, 0x02EE67CEDF141EDC, 0x02F167955D2A2DA4,
    0x02F3702C2EAE32A5, 0x02F57C9D87762E93, 0x02F7332965522F4F,
    0x02F77DB1F53F2458, 0x02F98AF5D38106F4, 0x02FBB3EA114A0ED4,
    0x02FF67E7148D252F, 0x03019DD4A2EC1C9F, 0x0305CEA0676B0511,
    0x0308DA0A6AA8029E, 0x030A9C432D2228F7, 0x031781CD8C920C13,
    0x032819F5639625D2, 0x032FEA275025263E, 0x03363246574F2DF6,
    0x0336C847AFAA359F, 0x0339145CAE340361, 0x0339B09F8E4B2032,
    0x033F5317769E2E84, 0x03412FCE0BAD2E19, 0x034952C7178B2285,
    0x034A843818181BBB, 0x034CE3A54209073A, 0x034E2F01038F1734,
    0x035297937A8B092E, 0x035C5BDA50B5011D, 0x03614445EAF01FA6,
    0x0362DA721B0B3A81, 0x0363887BD0410C9E, 0x0363FD5A26C42680,
    0x0364EC1F7CD81AA9, 0x03660BB827BD1D20, 0x0368B3A33FC62983,
    0x037559CFAC7A1C8A, 0x037C37CA68D417F5, 0x0386774E3E8323D0,
    0x03872D998DC039BB, 0x038C81BC268B1B55, 0x038D234E025B2F33,
    0x038D25B8DC8A15F1, 0x038EAFDC4B5C1536, 0x0391E308F50F169E,
    0x0394D019755A04F0, 0x03953D659A7A0F95, 0x039FB2A107910A16,
    0x03A01C1527AC2AD5, 0x03A284007A781D8C, 0x03A599BA97AB33B1,
    0x03A6B7A5384020CD, 0x03A74AA3DFE31623, 0x03A7F0D40A1F3525,
    0x03A9342D6B8C330F, 0x03ACD25BD4033B87, 0x03AD3E27622A2D48,
    0x03B29556EEF52AC7, 0x03B35A5669C12FAA, 0x03BAC069F16B2307,
    0x03BC9A11A38438EC, 0x03C41C33781C219E, 0x03D2174CC69F1414,
    0x03D3CA7DD8D731DF, 0x03D3DED63F0F291E, 0x03E45EC5661E2C25,
    0x03EF0B9C696511DF, 0x03F4E9082F480255, 0x03F6A99B528F0098,
    0x03FADDF7322319BC, 0x0400CA30DC87213A, 0x040F8B0569DB3252,
    0x040FADCFCA150B27, 0x0411E81FF2351E16, 0x0418BFBFC5430808,
    0x041FBE0DCBB632BB, 0x0424894E9DAD0171, 0x0425A1F3341D3BF3,
    0x042E704B141904C3, 0x0433E8A0997C3BAB, 0x043A7C87238F0FF3,
    0x043BD27604420F8B, 0x043D0C963AE623A6, 0x04424696E17911E0,
    0x0442EC796A172EF8, 0x0444BB36F43F1F2D, 0x0444C1CBE9B507DD,
    0x044DD76313101AFE, 0x04512E738B072FD3, 0x0453561922CA269B,
    0x0455846FD95E0027, 0x04595D740D1D209E, 0x0460E3C444CE0861,
    0x0463391E3B8113DF, 0x0463C2262CA938F9, 0x04649FB004BC3A45,
    0x046C5756DADC2015, 0x047771CC71CE2025, 0x04799F0DD7C4135D,
    0x047EE59027412362, 0x04821A9743A01A21, 0x048689F5E9751A03,
    0x048FA84A32F5119D, 0x049590441C7C0008, 0x0496865F552E17FD,
    0x049EBBBD0C0F1729, 0x04A0DDC9D38500F0, 0x04A42DB112E40ADF,
    0x04A526AFF07C05C5, 0x04A93004D5251F74, 0x04AABA187D8B0E4B,
    0x04AD90D84A603B4B, 0x04BCB51F38312BFF, 0x04BE9175A4162FF5,
    0x04C00A68F90D3BFC, 0x04C65F4B1CC71B75, 0x04CCD47994551528,
    0x04D11E475F670037, 0x04D15306488D15AC, 0x04D4980EAE5435F2,
    0x04D87BA345931EAC, 0x04E0D883321905E3, 0x04ED317D794A048C,
    0x04F017D01B5A075B, 0x04F3923FD69B1E25, 0x04F5A1C781A409C1,
    0x04F5C27714372BC2, 0x04FA75695EFD13B9, 0x04FBB8466D7E24C1,
    0x04FF29785B0D39D3, 0x050165E88BDC30A3, 0x0502FF6D631F302E,
    0x050AFEE4DD970829, 0x050B0DA03B342AF5, 0x050E20C956812659,
    0x0512A51ED82E3574, 0x05154B637C69399B, 0x0519DF17B6561A31,
    0x0520BA0A5943299C, 0x05293C29DA461E26, 0x052BB9AD053B2888,
    0x05303CD9DEE72305, 0x053375D32382001C, 0x0534AE4B90DE3A6D,
    0x053AA562C3133C2F, 0x053D307E462632A9, 0x053F2148988C3E10,
    0x05452CFBC2AD2E2B, 0x054B2A6FED7D03CA, 0x054C9CF8592138B6,
    0x0551357FEF8A1479, 0x055C129EAF043EF6, 0x055F891C38243718,
    0x0564A8C9232E1719, 0x05654BC2AC351AF6, 0x0565D6B8291D2123,
    0x056CF914E64230EB, 0x056DF1C8AB8D2DF9, 0x056E225AFEBE203B,
    0x056EC018A9770402, 0x0572FA6DB8C80E21, 0x05781550873C2440,
    0x0578AE6937B73E48, 0x05796DD866D82821, 0x057D92117EF70BDF,
    0x057EF23A4F542634, 0x057F820731B71154, 0x058069AA48861071,
    0x0584E834DD5F367B, 0x05888239542911C1, 0x05955F5529B832D9,
    0x0598BA432A7A3642, 0x0598DE6488200681, 0x059923D46D481174,
    0x059AC84F2B0612A7, 0x059D2546D8FA2263, 0x059D827A2E7E1A8C,
    0x05A06025E4442CC9, 0x05A2DE9272B22C6A, 0x05A3F851B3841D5B,
    0x05A52900EAD137D7, 0x05B495BAE9A71A37, 0x05B7F47E55D812E8,
    0x05B963640E651B41, 0x05C125FE13C92C58, 0x05C5A3A56BE011E8,
    0x05C995177EE51EC0, 0x05CDF4B10B3533BC, 0x05D283919E16342E,
    0x05D733E207573339, 0x05D94D1B2A5E25E0, 0x05DB43F46BF202A3,
    0x05F14A1003FD1FAE, 0x05F4B1015F6F32D0, 0x05FB6F3809A5309D,
    0x05FD3A0EA24D04CF, 0x0601FBD12E5F1151, 0x06066222922E305E,
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    0x486EBD3F56DA3B80, 0x487435A03F801D5D, 0x4875558FCA420401,
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    0xAC308C41C8CD30F6, 0xAC3316E126982BAD, 0xAC331BA6FFB60E7F,
    0xAC3DB15502770E37, 0xAC447E2343291974, 0xAC54025039FD3109,
    0xAC5E833D450C2F12, 0xAC65462794BF0E7C, 0xAC6B2B3F3AED0AEA,
    0xAC6C2F8DFA8E1570, 0xAC6EC8BCBCDC0BE0, 0xAC70A02A6AF52C52,
    0xAC743BDE3B2B1DBF, 0xAC7CB04597A11FDF, 0xAC82295E2AAF3FA9,
    0xAC88B5644256005B, 0xAC90D461C2AC319D, 0xAC91E60B48EA15AD,
    0xAC9924A402B53487, 0xAC99E15FB1970502, 0xACA1473B45A03E4A,
    0xACA373CE5C10054E, 0xACA546112D2A24A5, 0xACA71912B7512673,
    0xACAA99ABE43D1877, 0xACAEA7CC646A24E1, 0xACB1B16A332F3FA8,
    0xACB6490315B21F14, 0xACBE750EDDFB1EB3, 0xACC0B86081ED3FC5,
    0xACC17883C6F31112, 0xACC19C300F9F34FE, 0xACC2FFFE95CB0BBF,
    0xACC3C76F300E004F, 0xACC9B0FB2DC02DF0, 0xACCA6276C60D0BE5,
    0xACCADEB775FF10AA, 0xACD8224B7DA316C3, 0xACDC1CCE7A4A324B,
    0xACE3D3A36C2C2E79, 0xACF167A6E2E61D31, 0xACF28C46CF113FAA,
    0xACF36FC362921875, 0xACFB908E0E0B3287, 0xACFF7D29278C3C75,
    0xAD01DE3BBF403F69, 0xAD07F6E27AF10F5A, 0xAD0FA6CE77F91859,
    0xAD10DD8B890521F8, 0xAD17A99F839B2EE3, 0xAD1D9FE88A723100,
    0xAD33EC5CDFEE212F, 0xAD34B3899BDB3B5B, 0xAD35C6BB4308103F,
    0xAD3AC906FE91119B, 0xAD50257A977C3D16, 0xAD53A6A19DC03181,
    0xAD55016F85EE2580, 0xAD59559236D93927, 0xAD602CD72D2F1773,
    0xAD68F754A5BD3BCF, 0xAD6EB18314220FD7, 0xAD70988810891E13,
    0xAD749F8B25E80AAA, 0xAD761340F9841DC5, 0xAD7748C7A4B5089B,
    0xAD78436C4EDD3F34, 0xAD79FF434F4F1D2F, 0xAD7D727944153CCB,
    0xAD7D778FB5A60039, 0xAD7D969D7B920470, 0xAD8840AF83BA1267,
    0xAD88AD0A640D2155, 0xAD8E72A492533FF7, 0xAD903EA21ED51899,
    0xAD9BF416A96D158E, 0xAD9D9DB3D0920A92, 0xADA2990B01CD0955,
    0xADA4418FAD1A2046, 0xADB1E6815C11091F, 0xADB2597E1B220C15,
    0xADB34E45B2B70883, 0xADB45563FF3A2DEE, 0xADBB819348A3074A,
    0xADC0D8BC46521C6C, 0xADCAF2E8D53017E2, 0xADCBC3D421AA07C1,
    0xADCEAE230BB50956, 0xADD2D0F1975A2C45, 0xADD5E9BC044509C5,
    0xADDB62C3A35A3CCF, 0xADDBDCC306240E5E, 0xADDF82E33ACB10A6,
    0xADE07DCE5C333EEC, 0xADE6E3C9E82E331E, 0xADEE1860C7CC17C7,
    0xADEF13BAC3873759, 0xADF73EB71C4C17D9, 0xADF8FF8358031B50,
    0xAE0545C7AF5A3B26, 0xAE09B0F0DE2A3FB0, 0xAE0CB1A8964A1216,
    0xAE0E2968A4A43504, 0xAE16C472D36A26AC, 0xAE197D1D406F36A3,
    0xAE1AC9EA7ECD3DBF, 0xAE1B4BC49DBC30A0, 0xAE1C6693E96E18E5,
    0xAE1FD8AA472B2AB4, 0xAE2010737DBE2F4E, 0xAE21678A53061D01,
    0xAE28A4A63D2E3E0F, 0xAE2C90CE39100646, 0xAE2D4734CB880B19,
    0xAE315750E2B3045F, 0xAE36FF7FB2EB2685, 0xAE3A1A95D53535DC,
    0xAE4DE0F62FA52658, 0xAE55BBB8DEDE1762, 0xAE56E91D66F215B8,
    0xAE605FE0015201B8, 0xAE61E735537B3B3F, 0xAE6602C0E76D1287,
    0xAE662399D8472DB7, 0xAE68D99B0A5A27F5, 0xAE6D1F8202C912D2,
    0xAE6D5064475331F8, 0xAE6D98E8C783125F, 0xAE772313A322283F,
    0xAE77CBFACCCB0480, 0xAE799027C6492548, 0xAE7A31DA7AFE3B5E,
    0xAE7E439D2A52068C, 0xAE83F4BB50FB1945, 0xAE8689083D3A144B,
    0xAE88E3A2255323A7, 0xAE8A90C7033D092A, 0xAE8B965B44C03DCB,
    0xAE8E4A1AFDCB2F78, 0xAE93E637137B05F1, 0xAE9694FDA32D01A2,
    0xAE9FD18CE8CB147A, 0xAEA5BB5A06F128B3, 0xAEA63BAD14FD0DE4,
    0xAEA9461601A82B31, 0xAEAAB9444D63150C, 0xAEAE822EBF651CF4,
    0xAEB637B286F825E4, 0xAEB7DD0C813A2EC9, 0xAEBA497E47B61A7F,
    0xAEBBE8A8DD9E0D19, 0xAEBD56F1A7F82C0E, 0xAEC1F9EA44050603,
    0xAEC6DDC6E88F0AE0, 0xAEC71A4149C9103C, 0xAEC944F4E1660F9E,
    0xAECD44421BBC0781, 0xAED3697D1E1A00DC, 0xAED4E00FF9471323,
    0xAEE4E02719470E14, 0xAEE53DDEEB273CCE, 0xAEE8545C9E1C151A,
    0xAEEB608C32092C54, 0xAEED194BE4A63139, 0xAEF4AA6380D402EC,
    0xAEFAD5D3A9273030, 0xAEFD56C6EAC30D98, 0xAEFD637842A30DCA,
    0xAEFE532561C01E36, 0xAF038B71F2CB21C4, 0xAF06B090D43427B1,
    0xAF07BE260ECC0453, 0xAF08D51AEC2B3708, 0xAF08EC9F91711424,
    0xAF10E86229CC0499, 0xAF15B37EE25525CD, 0xAF175B00311E1C75,
    0xAF175E698BE52AC1, 0xAF1A6B640BFA17A0, 0xAF1C49997B9A2733,
    0xAF1D2EF0C2083B8A, 0xAF230973BE34334A, 0xAF27327A9132201F,
    0xAF275DBB00E92D6E, 0xAF2C8F8A76163FBE, 0xAF32601FB3800C6C,
    0xAF3427C3EC2E3070, 0xAF36E05B8C4E368D, 0xAF462EFB33AE0C04,
    0xAF4C85AC2FC22335, 0xAF52B7849BF5268F, 0xAF56113A82B73C0E,
    0xAF564A7E5C780A7C, 0xAF56D285AB01127C, 0xAF57D74BE6ED11DD,
    0xAF594537C35604D7, 0xAF5A607A0B110B56, 0xAF6011F52C7F0B78,
    0xAF609AFEE5F92A10, 0xAF62D803FEEF0D26, 0xAF68732AE15233F0,
    0xAF68E93A62DA09A4, 0xAF6E1C48BCF022B9, 0xAF7123F17E3E0949,
    0xAF76C7F0E9F12620, 0xAF86F3D08CF33688, 0xAF8A07FA058724C9,
    0xAF8C09AB6CF53094, 0xAF8E2A0F2A7505A5, 0xAF921F93FE133378,
    0xAFA6CDF2FCB528D2, 0xAFAE29A2D0F02996, 0xAFB0EC7F93C020DB,
    0xAFB468F0188C1382, 0xAFB4D071AAE02DC7, 0xAFB67AD369F034B6,
    0xAFB6B55B172B26DF, 0xAFCAF316FB2A2E4E, 0xAFCB19353E901108,
    0xAFCE7BA6CF462CB9, 0xAFCEAD12353135DD, 0xAFDA3D44F6020FBC,
    0xAFDCA130717A2336, 0xAFE6887B8C462418, 0xAFE9273E3E1E0305,
    0xAFEA01BCD4EA1CBE, 0xAFEAFD5B26180CD3, 0xAFF0D39D31671447,
    0xAFF5671B9D9D3627, 0xB00BB3462692071D, 0xB00EB0146EEE0DB2,
    0xB0115D39B2411CFB, 0xB017EF00C8D717BB, 0xB0199965AB460668,
    0xB01B46C2239C056D, 0xB01D37B9203F1054, 0xB031EFD7A9B811B0,
    0xB032B72402AE2039, 0xB0407B53FF5A2A20, 0xB0474BA075D50BFA,
    0xB0480DD61F1F2D0B, 0xB048509CCB8E12AE, 0xB048C243C6A0398E,
    0xB049DAEE7F651EAB, 0xB04CEFF7391B3F68, 0xB05A677EEF841440,
    0xB05EBDB2DC1B3FDA, 0xB06348028B353226, 0xB066D9BCA03E336A,
    0xB06BCA3649CE15E8, 0xB06EB9ACC4173B2F, 0xB070AB8814273A58,
    0xB073AEB6BA0A2908, 0xB07493AF4F463037, 0xB07602578F8B3B17,
    0xB07779FBE2353C55, 0xB078DE252F7625F0, 0xB07960BC781F295C,
    0xB08506574F430703, 0xB086FDC99CE83E75, 0xB0884B14320D36E3,
    0xB08AA2C6F40418F4, 0xB08BB85F839E3C2C, 0xB09071F62B84263C,
    0xB097A080B5AE2B8D, 0xB097B2CB9C0A3703, 0xB097C26A2DEC279F,
    0xB09887D9D4BC17A7, 0xB0A1FC8A52E907A2, 0xB0A9699010C418E9,
    0xB0ABF59DD0F02232, 0xB0AD5394E0DD1AB4, 0xB0B5D60872970EEC,
    0xB0B83A8F47832F6E, 0xB0C0A2FDB4490D69, 0xB0C0FEBD83AB3C34,
    0xB0C1C5D3B32A3D37, 0xB0C3DD85B55F0742, 0xB0D70A3B0E630929,
    0xB0D83BBFA725036A, 0xB0DC1387342F0337, 0xB0E2BF8DB2E41081,
    0xB0ECC98641781D41, 0xB0F42CC9143336A6, 0xB0F7D96840B029B3,
    0xB0FAA7203DFB15D4, 0xB106023E1C723067, 0xB1061436EC620748,
    0xB10ED5F4BDB20B32, 0xB11256FA91AF06B1, 0xB1127BCC9B7D3C6D,
    0xB11922DE2BD937B7, 0xB11A03B19B1722EB, 0xB11CFCD7718F0B5C,
    0xB11ED16D239F2051, 0xB1203A78E23E19A4, 0xB1259D09AB8528F3,
    0xB1265BF99C6F032F, 0xB12EEDF0887F23F7, 0xB13071187DAA1DBB,
    0xB130CA188F7013BD, 0xB1337DF5445923FA, 0xB13DCEE895D62E9C,
    0xB14304EEB6363C0C, 0xB1463538268D0EB9, 0xB1468D04E704350E,
    0xB147CB51AB1A095F, 0xB14850AF755D0498, 0xB148AC0E283B0408,
    0xB156A2C00BA10265, 0xB160CBB6C5DF2436, 0xB164FD47378B00A4,
    0xB1653D861CDB11A4, 0xB165C76C36110FD8, 0xB16828DB8A141D3B,
    0xB16AC8E7830B3D17, 0xB16F2ED8E52326FE, 0xB173FBD086FE1660,
    0xB17516E9138F3595, 0xB17871EDE0FC1B74, 0xB1848C8C564A174E,
    0xB1875BABFB8D24A0, 0xB18B6FB9EC9019F1, 0xB18BAEF6ECB51F4E,
    0xB18D2B0C48EB28C0, 0xB1911EDD88572705, 0xB192CD8B1E2F0BC4,
    0xB1938A188CAC326A, 0xB1991492ACDF3C6A, 0xB199FC18C1501437,
    0xB19E617B9B5805B8, 0xB1A2842D990F1F93, 0xB1A7A9E2CD462B58,
    0xB1A8573F01922B05, 0xB1B5CC9471B33723, 0xB1BCAE961B6E3C22,
    0xB1C6708EF3320202, 0xB1C6AD64ED701024, 0xB1C6BAA4EF292178,
    0xB1C6DA28CA1F01B4, 0xB1CA43143BAE0B90, 0xB1D57B94F2B72C97,
    0xB1DB614A13581CC8, 0xB1DF139B26432349, 0xB1E22A0DEB2F343D,
    0xB1E2745950690CCE, 0xB1E80C56C1A03CE6, 0xB1EFB6B203DD1409,
    0xB1F218C1D332149A, 0xB200B0AA6F7C3F2A, 0xB209EEEE5E2208A6,
    0xB20A1E67115E10E2, 0xB20E37663E393C39, 0xB211162FAEF225EB,
    0xB2114B11E44D3D58, 0xB219521BDC71375A, 0xB21B70BD3E543B7D,
    0xB21BA776CFC12C8E, 0xB22DF26EA9F4260A, 0xB23083C75352183B,
    0xB232C1BA30191F9B, 0xB232E2B329860DF5, 0xB236DDCF46C1316C,
    0xB23AD075A72F0132, 0xB23CB32D1D281E19, 0xB241D324EA30272E,
    0xB24313D4D141033F, 0xB25B2357195A1B40, 0xB264F2674BBC2A8C,
    0xB269CAF9C0F3350F, 0xB26A05BFB2D40041, 0xB26A0D8D2EA13761,
    0xB26C7C0AB3FF0A13, 0xB26EA2F72A52228E, 0xB2720670C2F904E7,
    0xB273395F2D370A96, 0xB279EFB0AF810DD0, 0xB2807F05D9322963,
    0xB2880C8125682843, 0xB288C342AE1E3E06, 0xB2899DEA8CE71F5C,
    0xB28FAE546D5B1577, 0xB296D52C1B5E3E2A, 0xB2A07BD187681BB1,
    0xB2A0837296210267, 0xB2A113AAC0780C24, 0xB2A520E632432840,
    0xB2ACD7A7851A303F, 0xB2B80A2ECF513D6F, 0xB2B8B23214EC249E,
    0xB2C4EE498EF82C19, 0xB2C55ACE25053C16, 0xB2C803E34DCC1742,
    0xB2CAEFF2BFFA10DD, 0xB2CEE760427C0894, 0xB2CF0607DF201DA4,
    0xB2CF4FAD59C01AD5, 0xB2CFDB7E283C35B5, 0xB2CFF1F583163B5A,
    0xB2D02B41EAF62EFB, 0xB2D1CA877B5F20E0, 0xB2D68DCC928A2677,
    0xB2DD268D897200CA, 0xB2DEB9C0F69335FA, 0xB2DFF3E45CB2230C,
    0xB2E0C7ABAA21007F, 0xB2E4CC8D617317BD, 0xB2E4D2B00CB511A1,
    0xB2EDAEB04AD93E0A, 0xB2F4432E9FA6365F, 0xB2FAC71E753E1BEE,
    0xB2FC2D8202780AF0, 0xB3076EC45ACE24CC, 0xB30BD45F212F28A9,
    0xB30C1894C14319CE, 0xB3155D7A711A3FC0, 0xB324F281D07408EA,
    0xB3267CDEF52528A1, 0xB3279DF839503A3E, 0xB329E083971531C7,
    0xB329E4468DA31C3F, 0xB329FF440BCF1A74, 0xB32F3AD9DD0D3F1F,
    0xB331DE0514D13C42, 0xB3377107CCAA2AAA, 0xB33924ADA1973E99,
    0xB340A78899032E9E, 0xB347A23330B233AC, 0xB349E1382D973F5E,
    0xB34F197AB1C038C5, 0xB351F3BE2BC03FE3, 0xB352350535903BA7,
    0xB355EFE425B02BBD, 0xB35717E6EF5E0AF1, 0xB35E11F448791D12,
    0xB3606130BD69242E, 0xB3629E72281005B4, 0xB365033BC50A293B,
    0xB36D6181B68C27E5, 0xB36DE8FEA3C230DF, 0xB374D893200E3CAC,
    0xB37AE6E9AEF03A60, 0xB37B964ABE4436C9, 0xB37E1B5167633648,
    0xB37F4A758D95348C, 0xB381E6317D2C228F, 0xB388D858CCD3396B,
    0xB38BA72F5A593D07, 0xB3905A2739B4251D, 0xB396C43E1BDB3CCD,
    0xB396F35811DF34DA, 0xB39BEF074E5F2D46, 0xB3A04CB86B5418FA,
    0xB3A11783098916F4, 0xB3A160A10DDB13A7, 0xB3A16C061899332A,
    0xB3A29810FE3D0B61, 0xB3A3985620D72C46, 0xB3A46C85A77419B4,
    0xB3A4D0B9C3EF1674, 0xB3A849F60EA81187, 0xB3A8CC8B83F41D9E,
    0xB3ACB9D4D46A3B6A, 0xB3B7399CDC112D7B, 0xB3B80105C92D2FC0,
    0xB3C7E34FD2D12650, 0xB3D054AF7A970D0E, 0xB3D29A59AE633ACD,
    0xB3D87BBCF7EF1FF0, 0xB3D8CE14790C2924, 0xB3DDF7FCB48D35B9,
    0xB3E0413D62BB188C, 0xB3E148ADD5392EAF, 0xB3E68E199FE81F11,
    0xB3E9AF963C13322D, 0xB3ED6D19BC7618A6, 0xB3EF97D4CDC6094E,
    0xB3F71AC00989190B, 0xB3F7D37907610C5D, 0xB3FFAA56C4CD2E4F,
    0xB4010EF3E6E30D04, 0xB402CD27880A0587, 0xB405CD88619D13CA,
    0xB4080D0AB08E3D47, 0xB40BA4E8AF220EB3, 0xB40BD87F55372040,
    0xB41828CA38D103B6, 0xB41B3AB47F232A75, 0xB41C908AE8D10322,
    0xB4218C8A3E4B2AE5, 0xB42279B2CC51076B, 0xB42DEE2E3A7A3954,
    0xB435B1DD526701EC, 0xB4366FAB1C0C1971, 0xB43A768930683D08,
    0xB4426C4BA61F3587, 0xB4472DC21C640A26, 0xB448C1EA02932019,
    0xB44E2C371F950B50, 0xB451EE3A970818AC, 0xB4550C9521CC30DB,
    0xB456EBC36E962BD7, 0xB458071574A40CA9, 0xB45E45C36BDB137A,
    0xB462EF21DFE90A1C, 0xB4686317A0FA3FA6, 0xB46864B20C3D3AA7,
    0xB46D1A856D8A2203, 0xB475B60671BE1607, 0xB47A28550B332688,
    0xB47A650F3DC537B0, 0xB47A762B93652710, 0xB47B2BE2B2AC3769,
    0xB47D579DC44D2F92, 0xB4815067BF0200E6, 0xB4853767CA542D88,
    0xB48708D8EE162DB6, 0xB489C9257FE11EBF, 0xB48DAE4615C118C0,
    0xB4A42CFCA7B938DD, 0xB4A4BF84B3B02DB0, 0xB4B0FE789DC92BF2,
    0xB4B384BC88162F02, 0xB4B98190E14017FE, 0xB4C8B037D1BF069C,
    0xB4CA36656F8E12D5, 0xB4D14500A7852B33, 0xB4D4C363981A0866,
    0xB4DAE00A6DA2185B, 0xB4DB7873CF373D3E, 0xB4DD32447B2B36EB,
    0xB4E0985CAD85080F, 0xB4E4749C5543070A, 0xB4E9B1C7E53623A9,
    0xB4EA2E1644CF1C9B, 0xB4EC6601E6280F52, 0xB4EC6EADA02329EC,
    0xB4EEB0E5C0DF2DA6, 0xB4F0A07113E622F7, 0xB4F22FBD24932DCF,
    0xB4F4EC1C682023A4, 0xB4F8264E647A3371, 0xB4FCC3242ED6351B,
    0xB4FCD52C3A3125DA, 0xB4FE22A08FEC3FAE, 0xB5065921E90201C2,
    0xB5084062B5F70302, 0xB508706E5F050002, 0xB508D642ABAD012E,
    0xB50AC6D1455F1B98, 0xB513F3BD20470DCB, 0xB51742A5C20919C8,
    0xB51CE80EE0790F65, 0xB52842013C640D13, 0xB5289E8BC80C26F1,
    0xB52A66B47DB12220, 0xB534697AE893085D, 0xB536619D6FE20269,
    0xB5379BA8B7023B59, 0xB53D2B11D3A5079A, 0xB540EAE4E24F2BAF,
    0xB5411DE85FEA0F0B, 0xB543B87D30DB1D9A, 0xB543E2C7F7332AEB,
    0xB5442D33C50904C7, 0xB5447AE2784A09BE, 0xB54684E337AC0E5C,
    0xB54B76CBADA313F2, 0xB54C0FE7F2EA2995, 0xB54FDB99B8CA1FBF,
    0xB5537888BB841D34, 0xB55396B741530CE7, 0xB554E8C41DF71ECF,
    0xB55B1EFC91AD014A, 0xB55D514EB4EC080A, 0xB562C6E35633395F,
    0xB568798CBD802A88, 0xB56B313CA5BB3A17, 0xB56CC56B991812F0,
    0xB5713DF0412E3A0D, 0xB57418BCD97A3DDB, 0xB577E929DA8C111D,
    0xB5820A25F0770E60, 0xB589146528B6301D, 0xB589355CB97728BB,
    0xB589E7E25D6E0DA4, 0xB58ED54B36673381, 0xB598C35C170A34E0,
    0xB59E51E785141439, 0xB59F8A46214A3F62, 0xB5A56F4FC3A81122,
    0xB5A78DEE6A741E5C, 0xB5A7F21C996A239E, 0xB5B2C38BBCA62FA8,
    0xB5B7D528F1A936AB, 0xB5C4FB7434C13A53, 0xB5CDA1A62EF23785,
    0xB5CEF157B98018EB, 0xB5D183E77A1A174A, 0xB5DAFE326C012EEE,
    0xB5E1324A1CB02E0A, 0xB5F06D840A7F1BEB, 0xB5F1420D366B3CA5,
    0xB5F2A9A7EED807B1, 0xB601297DFC092B40, 0xB603BE337B4B120E,
    0xB604426DB88E1520, 0xB604FD5E60752E47, 0xB607CDC775FD0D85,
    0xB60B5CE075A419C6, 0xB61348C954703051, 0xB61612D2937E0B6D,
    0xB6177DCB792F3534, 0xB61842D4D0F40692, 0xB61C76F4B1C70EC9,
    0xB61F7423928337FC, 0xB627D223F9D9167F, 0xB6303009900007BE,
    0xB632E161C39327F1, 0xB6384DCB53842FA2, 0xB63A4133B01736FD,
    0xB63CA3178456337C, 0xB63F2FEF06662746, 0xB63FEFED1F2D2243,
    0xB6420C46056D0E8F, 0xB643822D6B033F9C, 0xB64B65BEEFEC0122,
    0xB651FCC2FFB51558, 0xB6556D2A6F930663, 0xB661F921450B2272,
    0xB66311D71B253E50, 0xB664EF098C4C0493, 0xB6671031F53C17DF,
    0xB673B579AC44128D, 0xB6740EF01D8B1F05, 0xB6765E0B2E6F24B3,
    0xB679999DA5BE382E, 0xB67A1C99100C0088, 0xB67B00EEB6DC0007,
    0xB67D9F416BD62F04, 0xB67DAF1D46281843, 0xB67EB90891FF2CF8,
    0xB683128427743602, 0xB683B96585541E10, 0xB685088F998810DE,
    0xB687EF2EAE71317B, 0xB688D613D2F2057A, 0xB68C8B818E0A0F48,
    0xB692B603D860321C, 0xB693F12D06F63D64, 0xB6A05FCFC9C3108C,
    0xB6A7572870C221E4, 0xB6ADB735B8741FB4, 0xB6B10C2689E52ECC,
    0xB6B385B9413323B7, 0xB6B6066A8FD50C64, 0xB6BE025A6FE7229A,
    0xB6BE389CFBDD16A2, 0xB6C3445CE3FA2460, 0xB6C6C26DE85A0164,
    0xB6C71A627D352FB0, 0xB6C82BFDEB103C04, 0xB6CCCE3333DA3C8C,
    0xB6CDAC64AE813268, 0xB6D0EA14C61719AB, 0xB6D10C8462AF3439,
    0xB6D47E5EBA1F2770, 0xB6D778D5DF500636, 0xB6D7D6B3D53434BA,
    0xB6DB1DE3C5EA11C4, 0xB6DB884C16091AC9, 0xB6DF198286B20AA4,
    0xB6E0A4D6FBEC37A7, 0xB6F202ECDBE815BA, 0xB6F39AEFC80C1F02,
    0xB6F647E815E603E5, 0xB6F7119909C2171D, 0xB705CEE98EE4041D,
    0xB70F1A78A78F1841, 0xB711B13117573F36, 0xB7123C00B2893F20,
    0xB71301A74B852623, 0xB713DC3CAA8F1AFC, 0xB71CCD1CC2491104,
    0xB71F7672F5D61726, 0xB724C998425C1ED1, 0xB726F2EC21371050,
    0xB72915D5FAA733B5, 0xB729F157CD96382D, 0xB73A32CD1C6E1768,
    0xB73F5EF2D87305AF, 0xB7441E29D8502EE8, 0xB746A93E863A2D31,
    0xB747466E174C2AAD, 0xB749C02847E91595, 0xB74EE526CB581B39,
    0xB753A215EAE80B21, 0xB756A8C6CDF60CDB, 0xB758F4D82BC01778,
    0xB75A318B4AB42CFA, 0xB75B52EFD1602BFE, 0xB75C9E0EC90D0137,
    0xB760D2D747F52A14, 0xB76204C9AA2430B4, 0xB7626AE94EAD31E8,
    0xB76F0E1A95CA2EE1, 0xB76FB2BF81022AAB, 0xB7781B0C05841DF2,
    0xB77E786B41BC38ED, 0xB77F53EA3C7A275A, 0xB7829410246509E7,
    0xB783EE69EB122D77, 0xB788A23544B42EA4, 0xB78925E1E6BD06C8,
    0xB78B0B5557A6142C, 0xB78ED7AE5C36206C, 0xB792A76C51AF18D4,
    0xB7934CF525F53ACA, 0xB795EF4227F30350, 0xB7978B942A5107D6,
    0xB798F7D30FC126E1, 0xB79EE38604750A4D, 0xB7B262F20A272432,
    0xB7B3FBB1BC062E2C, 0xB7B4888DEB293FDF, 0xB7BEE06B5FA92085,
    0xB7C56733D9B9234C, 0xB7C76D45E5560C78, 0xB7C7945D0C6E2B87,
    0xB7CCDF88566F2FF4, 0xB7D334D2BF433437, 0xB7D590FA58281D82,
    0xB7DB4034BAF0308E, 0xB7DEEC232C6C2084, 0xB7E380AF0B632BCA,
    0xB7E6FC657EBB0ACF, 0xB7F1F98862CF0A37, 0xB7F30FAE01963FD4,
    0xB7F740DA45BE0387, 0xB7FEAB05825E3667, 0xB7FF761B3A0638DA,
    0xB802229B2B882100, 0xB803B3F16DAD089C, 0xB804BB45BBA215B7,
    0xB805E445AAD43B6E, 0xB806E38ED69A04EE, 0xB80966A5A0742BBE,
    0xB80A2E687EAD275D, 0xB80CC5BB18AD168E, 0xB80E6FB0B4892B7C,
    0xB81A97C7C31E14CA, 0xB81DF0004B4C3A66, 0xB822F14462DB2F88,
    0xB824C05834C83173, 0xB824C8D833313E21, 0xB8253B1C5FAC3DF5,
    0xB8261D1DAD223163, 0xB83BBCBEF16A329B, 0xB845A071A9FD30F9,
    0xB8466C49EE0D3A62, 0xB848145DEA703399, 0xB84DAE4A7F982325,
    0xB84E6DF68BC63ADC, 0xB84F94FB0BD03DCC, 0xB85131C2B1D3080D,
    0xB862E92FB40519AF, 0xB867D7F1A7350AC4, 0xB868BB86842224CA,
    0xB86C789C294A3CD0, 0xB86D473EEA772FC8, 0xB872EB92F63202F6,
    0xB874764748BC2E9B, 0xB874BF8DF77339B0, 0xB876DC768BF52D18,
    0xB87B105BAFD9382F, 0xB88139A3B0FA01F6, 0xB8828E143A13383B,
    0xB88521BD5DFF1D5A, 0xB8878EF121143A11, 0xB8929A84EE2F3BFB,
    0xB895617313AE2AD8, 0xB89EB0D80D8E130E, 0xB8A2DBC8DCBE099A,
    0xB8A369E639E413E8, 0xB8A50A4079CE197C, 0xB8AABD96B135172E,
    0xB8B6564355C309A3, 0xB8BD3D193CD306A4, 0xB8C3AD1FEB3323B3,
    0xB8CC8086297F0DF4, 0xB8D2E68BDE7534FD, 0xB8DA2F15A5421AD2,
    0xB8E1D65D4CA93CAB, 0xB8E448402ABA2C67, 0xB8E7CC33F8932F7D,
    0xB8EAE7FEAC503049, 0xB8ED37D2D0B30C44, 0xB8EDC5EC0F5736A0,
    0xB8F353318A6B2778, 0xB8F38EB175930C75, 0xB8F3C34EBA252C4A,
    0xB903899B3F83239F, 0xB905930DA0F5328D, 0xB911B91962D31374,
    0xB91A318BDB8C0CC5, 0xB921243A55B631A4, 0xB9417BA9262B1E86,
    0xB9447377C92A255A, 0xB94542659DD43FA1, 0xB946216399CD17ED,
    0xB94697EF1C9335CF, 0xB94D7B71A3D02C28, 0xB94ED7E1B37B04F4,
    0xB94F391C07423023, 0xB9508CB74FE21AF1, 0xB9537754F5012BFA,
    0xB958ABB7595812A1, 0xB95B8F578D040995, 0xB9602B4C1FB83FFF,
    0xB961E6C40E5F3C3D, 0xB963BA7CEE0007F9, 0xB963BF23F20A3A57,
    0xB963E09F35580A20, 0xB968BB56CA1920C8, 0xB968E497DC3405A4,
    0xB96A1924A35F1FDD, 0xB96DFB4DADDB3BAF, 0xB96EB56490F71331,
    0xB96FC255711C1AEC, 0xB974A3FF3BF12FDD, 0xB97729210B5A284B,
    0xB979426DDCA03376, 0xB97BBD215D5C09B1, 0xB97D56A593322C1C,
    0xB97EB222D3AB1876, 0xB97EB236EE9800E7, 0xB97EBEB8E5E215D5,
    0xB97F40E642BB2937, 0xB9853B205AB026E2, 0xB98C3E76137E0B49,
    0xB99192D3808F2A33, 0xB9950B303EE30EDF, 0xB997C7EF98AB39EA,
    0xB99DE3C92F8C0DA3, 0xB99EA10959F43C6C, 0xB99FE0CFD0511A2E,
    0xB9A0855839283B3B, 0xB9A5A23097AA3455, 0xB9A6A78EF1C6008C,
    0xB9B0DBCB2FE02B2C, 0xB9B4A9D1470531CA, 0xB9B579D6D2D310E8,
    0xB9B9799F248D18F9, 0xB9BBE1E51B433B73, 0xB9BFFFE67E412777,
    0xB9C2FB691A730536, 0xB9C4ACD825061EB0, 0xB9C5ABD9DF52044C,
    0xB9C5B32DDA651B88, 0xB9C8BCE059561E53, 0xB9C93394AAF13E1A,
    0xB9CD83F1349822CD, 0xB9CEC2F9295F3B61, 0xB9D60F5EB4560902,
    0xB9D689EB2C6107FE, 0xB9DF95169EF225EE, 0xB9E0842541951097,
    0xB9E33F0290F6235C, 0xB9E420BF51D41303, 0xB9E9FACBAF550538,
    0xB9EDE5D5669B2C8C, 0xB9F0A0D118F536EA, 0xB9F1DED61173007D,
    0xB9F49C3517523970, 0xB9F4DC8CDE942ABE, 0xB9F537F1EB150870,
    0xB9F651AFF9CE1C94, 0xB9F8148D8A813E85, 0xB9FD4EF2DB8F0904,
    0xBA009CED3AC131AC, 0xBA08201E3AEA3A32, 0xBA09E1A25C463842,
    0xBA0B3702CECF2227, 0xBA0D274948D63429, 0xBA105EB991F13C7D,
    0xBA16F5058B6A0DF3, 0xBA1A4AE572E30043, 0xBA1C3A62C62314E3,
    0xBA1F504174572ABA, 0xBA28FF463B2B1A77, 0xBA2CA8890ED03F24,
    0xBA32304E396F0183, 0xBA361493D5D32DEF, 0xBA36CEF4EF78061D,
    0xBA373E2F685E1CF6, 0xBA39EFF0268B1C51, 0xBA3B77E8F9981B45,
    0xBA3D1DCEFA850941, 0xBA3DAF65FDE31533, 0xBA48E216B7CF3B31,
    0xBA4DFF2BB21D24BE, 0xBA4F1A44D4A13F91, 0xBA4F5A1F57BD08E3,
    0xBA55AAAA5CAB06DB, 0xBA570CFA89AC2A30, 0xBA683C781D8D2C49,
    0xBA6CCC8273D72064, 0xBA7682808F361335, 0xBA76A7BFB5691667,
    0xBA7BF89767B309AE, 0xBA833617EB430BA9, 0xBA84BEBEA66A0E04,
    0xBA8A946782EC38C1, 0xBA8B64C01BD2232F, 0xBA8BE5FE177F1934,
    0xBA8C6BC1AE900C0E, 0xBA91EA7C2ADA07A1, 0xBA94423176D907CC,
    0xBA97382721320632, 0xBA9B566598E701C3, 0xBA9DE6F29E043582,
    0xBAA377D43A64241E, 0xBAA5B26653532482, 0xBAAA077584EA02B6,
    0xBAAB70B42FF0163F, 0xBAB26375290C2E5D, 0xBABC293B05FD3669,
    0xBABE63861F0E1294, 0xBAC9E0278494234E, 0xBACDF3F2DFA137F2,
    0xBACF131D3BE51C70, 0xBAE309A6FD2C17E4, 0xBAE5EFFE9BCB0E78,
    0xBAE7A2565848280F, 0xBAE807C1D22128BE, 0xBAEC6F3E7F933475,
    0xBAF0A6BFBBC93BE4, 0xBAF322A0DC6A0C62, 0xBAF4384840F31C90,
    0xBAF6582497BA1A6B, 0xBAF96D3E53D71AA0, 0xBAFC75D0084C2B25,
    0xBAFD4268636F19CF, 0xBAFD7797365C3EA6, 0xBAFF1ECCE950392B,
    0xBB037EA4EAAA0B6F, 0xBB06032A717204E2, 0xBB08F5C733283DFA,
    0xBB0EB687FDAF312F, 0xBB15F67EBB572C68, 0xBB1D304298273258,
    0xBB1EA71F59382B49, 0xBB204D5127C70550, 0xBB21D940CCC20014,
    0xBB22A63A35042F6C, 0xBB23515A911E33ED, 0xBB25C10BDA0E343B,
    0xBB2FE63465252E59, 0xBB36CF5E447610C9, 0xBB3F6D49EC6D2679,
    0xBB41A4BC6C8E252B, 0xBB41E195A28F1CA2, 0xBB570BD5575235BC,
    0xBB5717E273A4394E, 0xBB649AF9F1493626, 0xBB6A720CF2D71CC1,
    0xBB7308E1179F1445, 0xBB7AA62B80EF1D1E, 0xBB7C6658D84824D9,
    0xBB7C881DFF741A5B, 0xBB80E2E22A821ACC, 0xBB8333DF56B22278,
    0xBB835658302E314A, 0xBB86DF154130216A, 0xBB897B080483113D,
    0xBB93AF2A85A92306, 0xBB9417A8BCF03D0B, 0xBB97EAE522271F2E,
    0xBB98E74BCD030B77, 0xBB9FB28BA246108E, 0xBBA675463C793877,
    0xBBACC7C9D30E29E6, 0xBBB374FB81652E55, 0xBBB3FEA70D5D20B6,
    0xBBB6548713A40311, 0xBBC1844DFC5F3299, 0xBBC2DBD43C25186B,
    0xBBC3C9089672191D, 0xBBD28C850E9D2C2E, 0xBBD54F5591F022E5,
    0xBBE27ED0A48F283A, 0xBBE3F3A4943224E4, 0xBBE4899CB243309F,
    0xBBE4C1E539880447, 0xBBE5F0DBB4681926, 0xBBE64C7192413BDB,
    0xBBEABC5D17431D90, 0xBBEF950FB84522E2, 0xBBF1A7AA27750F6F,
    0xBBF22150C8B2043C, 0xBBF84C1F77182428, 0xBBFA9B2F888B229D,
    0xBBFD07374FB9257A, 0xBC009F5BA54B27B6, 0xBC07E37571F237E7,
    0xBC08BCC952132715, 0xBC0BD01FCE9F1985, 0xBC0E5194DB542C2F,
    0xBC10D6254F972F35, 0xBC123A0F133D0B46, 0xBC14ABAE811F3985,
    0xBC185D50EB4739DD, 0xBC1D1294061D2AC8, 0xBC2339D9ED971826,
    0xBC2C30E3FCD702D6, 0xBC300787E9B40DC4, 0xBC3CF8F9616119FE,
    0xBC3E91D6EE781AC0, 0xBC47C62E42C32661, 0xBC4AF7ACC4EA0B18,
    0xBC4B4744BFED05BF, 0xBC4DB8588C32064D, 0xBC4E7132EFDE0E96,
    0xBC54E29CBD473D5A, 0xBC630D11BE6B1887, 0xBC653F1C3BC60B7B,
    0xBC690022C8441F5D, 0xBC6E47FC979E016D, 0xBC795C56C31E34C3,
    0xBC7C0DE815A72AAE, 0xBC7C71E2A6A52C9F, 0xBC805506B53C3792,
    0xBC85B7D9938800CF, 0xBC8A177C87A5061A, 0xBC911659997805DB,
    0xBC9637A7AA073382, 0xBC98B973CBA53D0E, 0xBC9A7ECCB59327FF,
    0xBC9EC6673781278C, 0xBCA6EBA1C2C60919, 0xBCB3DE22E38027B2,
    0xBCBC2C33CDF419CB, 0xBCC006EFCE70209B, 0xBCCC2DE9F1973836,
    0xBCCEDFBA887F0217, 0xBCD0227E3CFC0D0A, 0xBCD2A99262302B8F,
    0xBCD37A3E62043A4B, 0xBCE0BB52269B1997, 0xBCE918988EFE11B5,
    0xBCEBEB3F4E952E14, 0xBCEDCCB19DC1177B, 0xBCEF079C27171A7B,
    0xBCEF11A0C1AC14EF, 0xBCEF98550E110CE3, 0xBCFB88E50BEE3073,
    0xBCFC94851EF32ECB, 0xBCFCC6107ED3029F, 0xBD015BA69C191D4E,
    0xBD0615D1A48F37C8, 0xBD07C43354440031, 0xBD0BC5B795531F3E,
    0xBD0E66A14F672637, 0xBD10BD90A65F0699, 0xBD1B272FF84D1DE6,
    0xBD1D784D2B02329F, 0xBD1F8142FAFD3632, 0xBD24DC8D5B432C1F,
    0xBD2B23A712CE26ED, 0xBD2C04D1562E3EED, 0xBD304D0221AF257C,
    0xBD350C12E7F52D4C, 0xBD375EAF102407A4, 0xBD3ADA0C444B1195,
    0xBD45F31044CA1160, 0xBD478F6D56460359, 0xBD488CA2381134A7,
    0xBD4BD8A7B00709B4, 0xBD4C2487AAB1157D, 0xBD4E68E1F6621C5E,
    0xBD53DC4AB81608FC, 0xBD542284E5301E4F, 0xBD594E4ADFB20984,
    0xBD5FCA850F2F36B9, 0xBD640A8862B83E36, 0xBD6810157A272212,
    0xBD6BD658CD5430CE, 0xBD6FA93CAFD910B4, 0xBD7087711B092CA8,
    0xBD73B1F2104518E2, 0xBD7C84C7A0A23C9E, 0xBD7F0F1497AC2C30,
    0xBD7F380FABDC1D36, 0xBD828853F6CB2D20, 0xBD86E159992E339D,
    0xBD886E5BB6732FB1, 0xBD8899F9F4090BCE, 0xBD899F4DB8471F03,
    0xBD8E7A9CE6F70335, 0xBD90E8C111EE33D4, 0xBD96143114C62741,
    0xBD9DA5BB4F233720, 0xBDA2FBDB658312FF, 0xBDA590E96FCD3B60,
    0xBDA6C431BAB5157F, 0xBDAD7C597EB206C0, 0xBDB5A4C141A62FB6,
    0xBDB63F6B6F2D0813, 0xBDBB47F7D8053A2F, 0xBDC0512A39CE09B8,
    0xBDC4005FE351332C, 0xBDC5C4F723FA32BC, 0xBDCC229FB5491D1D,
    0xBDCD971E03241715, 0xBDD2F53E69CA0FF2, 0xBDDC04E8AD271AF5,
    0xBDDC51EBC3931998, 0xBDDCD87FE4D706D8, 0xBDE0B27DF2A82BEE,
    0xBDE1FDD69E083248, 0xBDE278D6DB0314A6, 0xBDE683B4E8571A40,
    0xBDF20126ADA00248, 0xBDF4BE9387540698, 0xBDF94BC5ED2E1A96,
    0xBDF9927A5C742822, 0xBDFCB284A2D23DB0, 0xBE006C1DD1043EF3,
    0xBE07502D95192EEB, 0xBE0CD2CDCBC719A0, 0xBE1B1A4B7F230B55,
    0xBE1C1B3EC03A16B5, 0xBE271E3854732392, 0xBE2EBB8993C23A47,
    0xBE33C87974222D61, 0xBE349943B2302D82, 0xBE35CA35306D3656,
    0xBE37865D3E232600, 0xBE3ADB9929861BD8, 0xBE40B0B4EC70004A,
    0xBE43F86A1A2B2353, 0xBE45BA5AB4A137CD, 0xBE4FCFCD85021410,
    0xBE6176805DF30C97, 0xBE6677AA9A4E2EC7, 0xBE68280E18701D1C,
    0xBE68C8A6CFF13BC0, 0xBE68D6B4B2B20B87, 0xBE6C6F6E5DD32494,
    0xBE6EF3DF5F312F23, 0xBE70D03530E218D8, 0xBE719FD6DC2F131D,
    0xBE72E3149CAE1C38, 0xBE7C75E968EF1A0F, 0xBE7F27DC29D512B9,
    0xBE7FDF8D98212041, 0xBE87D082078D33B8, 0xBE8CAC82301B2FBC,
    0xBE8E356C9B461845, 0xBE9199CF024C2BC6, 0xBE98D8AE165B2360,
    0xBE9BC7DBB5051A22, 0xBE9EDBCFA4332569, 0xBEA1BDE1E6D93C8A,
    0xBEA62C8BBECF3A87, 0xBEA99C5A9A7B307B, 0xBEAB11BE975D3746,
    0xBEB5E7AB1D82108A, 0xBEBBCE7FEC51332E, 0xBEC0D754185A08D8,
    0xBEC7705E33ED0717, 0xBEC7A89BFFE52608, 0xBED5267B113B14B8,
    0xBEE15B3C52AA21D2, 0xBEE37A0697BD2A3C, 0xBEE3DC921A9D05D1,
    0xBEEC7825B7D6121E, 0xBEEC86829B901F23, 0xBEF150D142703674,
    0xBEF32BBE49982314, 0xBEF3CED397CF25DC, 0xBEF91385D9502FE7,
    0xBEF9FA4FE3610E10, 0xBEFF5D503B1D1334, 0xBF039139D68637C6,
    0xBF03F07D91023A21, 0xBF058A20229C27BB, 0xBF05B18FC694003B,
    0xBF0BDB87D84D0D9B, 0xBF0C61EBE1D83FE2, 0xBF0FA26E75DF2347,
    0xBF13488C7CDE3FDE, 0xBF13C1FEA0BA3AF3, 0xBF16E845B2CD0A00,
    0xBF1ACFAAFC41367A, 0xBF20ED3B51FA3C46, 0xBF23832721541DCC,
    0xBF248829EFBB0A9C, 0xBF2503BD47B31363, 0xBF260D36E3E80C23,
    0xBF286CDC3F393AC0, 0xBF29AB0829D3392C, 0xBF31E765245C2B0F,
    0xBF32C7AC9B76029D, 0xBF3FC8F6DAD732DA, 0xBF418AEC19021B67,
    0xBF47B7F660300309, 0xBF49071EA9F509B6, 0xBF4940170AE215F3,
    0xBF4B4FB9A00708D3, 0xBF4C077E83BB11C2, 0xBF51B3A78AD6292F,
    0xBF593DD9A7A93729, 0xBF5A25F0510528A6, 0xBF623D457DEE2CE4,
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    0xCB2C6AE628F5277B, 0xCB2F30E64EC91E5B, 0xCB2F97D6B91131FB,
    0xCB36923667D02A03, 0xCB3B3AE7D5541BC7, 0xCB3D74EA44E937E3,
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    0xCBB311BBB27931D9, 0xCBB612D8309328DD, 0xCBB85F2EF84919C1,
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    0xD32616321F763414, 0xD326F3E0906F2294, 0xD32A6F104BE12671,
    0xD32B0D3269B21EED, 0xD335909D766E1921, 0xD33650152992107D,
    0xD33E2111817811FA, 0xD33F7D5D26351CB6, 0xD3429C71BB3A227C,
    0xD3467C5BFB5227A0, 0xD350F4F46E603D12, 0xD35323A4D4852EEC,
    0xD35A4253686126C0, 0xD360ABB87C592AF4, 0xD364F12EA9F42C47,
    0xD3663C319CE5280E, 0xD366B64764FB3FC8, 0xD36B182891C92823,
    0xD376F619563E0192, 0xD378FCC519060706, 0xD37C9C1929462ECD,
    0xD382848471780328, 0xD382F4E398813D6D, 0xD3843BFDD774010E,
    0xD38D9863C27237BA, 0xD38ECE3064BA33D1, 0xD3924C21801A1458,
    0xD3934E2E500510CC, 0xD394955C087A2A26, 0xD397092858751980,
    0xD397D7454AF000E0, 0xD398E2D928DF212C, 0xD3A0A4258B2A1E03,
    0xD3A296AD6EBC0D21, 0xD3AED9E239231E43, 0xD3AFDE5D6F8C1C98,
    0xD3B2FE241A1E0719, 0xD3B41C28B8503CFB, 0xD3BC19AA56DD00D1,
    0xD3BED17B18E521DE, 0xD3BF6EBA1D6C37C3, 0xD3C67F1D1A7809F8,
    0xD3C9BE99FF05093E, 0xD3CC3B169FF03938, 0xD3D0BCA9CD5C1DED,
    0xD3D15E6014890E5B, 0xD3D406BC6CE73373, 0xD3D6E447CCE22F13,
    0xD3D6FE0D2CFD1AC4, 0xD3DB7DB3D2733418, 0xD3ED12B08CA13E2E,
    0xD3EF9EF1F272333E, 0xD3F77516BE2419BF, 0xD3FC67049D2E3025,
    0xD402A577AE8F1B6A, 0xD403ED67F2B80C73, 0xD407B24DF12603AF,
    0xD408BF12205D060B, 0xD40DC3D71DFD0F6B, 0xD40F89B077F53F45,
    0xD41173D3DF2D18AF, 0xD417D758AC5026AE, 0xD418EED228B73D09,
    0xD41A087E56290E44, 0xD41BCC23FE0C29AA, 0xD421217015033FDD,
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    0xD439CEB8DD3915C0, 0xD43A36B97A3F103D, 0xD43B3E4CF3112644,
    0xD43F134084730DF8, 0xD440C3924CD53E2B, 0xD44E7C3456590C74,
    0xD4542C884C3108D6, 0xD45846CCF617315F, 0xD4598812EC9713C6,
    0xD459E73CC6511CD9, 0xD46329122B911C16, 0xD464E0B8BF0A2431,
    0xD46A13CD0CB60BC9, 0xD46A27106EDE2A72, 0xD46E259431B410F7,
    0xD47423A48FE522EF, 0xD475464CBF2C35D1, 0xD478727195AB096C,
    0xD478C21ED3331691, 0xD47D2AE3AF241A92, 0xD481EF7AD0573D52,
    0xD482158E50113F5B, 0xD48DAD06B0A201D8, 0xD4923327DA7720FF,
    0xD4981AE8B85F3D55, 0xD499864D12EA0532, 0xD49B7A9125FB3229,
    0xD49C04653E7534F5, 0xD49CA3AAF82E09E1, 0xD4A391F4742E0F54,
    0xD4A733AA14F43355, 0xD4AF3FA0F4130C0D, 0xD4B0E7E2ADF23FE0,
    0xD4B123CBCC6B1DE7, 0xD4BBF81423D937E2, 0xD4C0BA3E50BB2339,
    0xD4C35EA92D3B3829, 0xD4CA6F203FDF0EF4, 0xD4CD6A46EA493924,
    0xD4CF07833C562A0B, 0xD4D15EE30F1D218D, 0xD4D3CA0AEF3E2B3D,
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    0xD530AA735F163333, 0xD53485FAA1BB0848, 0xD53886D111880BC2,
    0xD53B426BD12D3098, 0xD540256A895A24B5, 0xD540A5CCDB4F07A8,
    0xD548600A44EC1EBC, 0xD54F51BB54062D0E, 0xD54F839509F61689,
    0xD5503E516ADC1A38, 0xD55212E20FA91233, 0xD55B4D4EE94D1B9D,
    0xD5628D3E2084210D, 0xD568B08698F21C01, 0xD569CB24072D29BC,
    0xD56B1CFE06A42577, 0xD56D64D333C4035B, 0xD573048940260FAE,
    0xD57671BE171116A7, 0xD57B1701BA9433D7, 0xD57B505EA591166A,
    0xD57C2FBCE47D0B00, 0xD57C30A88E3D3035, 0xD580F74EBF9F2D64,
    0xD58155522E3E0D8E, 0xD583EE5BDE1016A1, 0xD5873C0220E2386B,
    0xD5887A824D3C18DA, 0xD5891AF22F58339A, 0xD58B51DA74A42640,
    0xD58C6F7E5B212341, 0xD596373A798521D9, 0xD598C1BAEAD13188,
    0xD59A97FB96543FE7, 0xD59ABFD737B53E33, 0xD5AF29EF3B4B03E2,
    0xD5B37A6D57B70DCE, 0xD5B6A67532972C3D, 0xD5B90D9175E00257,
    0xD5BE39F7F684096D, 0xD5D2F98BA2E71C43, 0xD5D44292D8541552,
    0xD5D46202C782034A, 0xD5D5C366A4C71FB5, 0xD5DBA553FA582CE6,
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    0xD5E6A6FF3E503CB9, 0xD5E93C28E4D30AF2, 0xD5EDEB6A764419DF,
    0xD5F6CCCB1EAD2A8D, 0xD5FF3405109232E3, 0xD603389D546623F0,
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    0xD60EEE6B54EB09EF, 0xD61C8D36832D2D4A, 0xD61CF3D0FD0112F5,
    0xD61F3E7576CD1FD2, 0xD624F00107FF20FE, 0xD625DBB8F4F831F3,
    0xD626B3FF918C3E82, 0xD62DDB684E5A0B5E, 0xD62EE26D8B2B3E07,
    0xD62EFB688E2B0F82, 0xD630C5CFB2B02A37, 0xD633223E6F46231F,
    0xD6339AD3DC6804D0, 0xD63CD6F3A55113AF, 0xD63F92057E6A088D,
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    0xD65BC48FBF931588, 0xD66270160B0C2D08, 0xD6684EFCE5271448,
    0xD66E07178FE709CC, 0xD67116A64B4F03F0, 0xD6729F6CE4080D5B,
    0xD679D3EA23881825, 0xD67F218197341F01, 0xD67FCD37BDF7272A,
    0xD680BA8DB3172B38, 0xD681F59513B10FB1, 0xD68BAA910DC31D47,
    0xD68F2C8C61A3054B, 0xD69015A190EA2419, 0xD6969BFBA8DF2D7A,
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    0xF00F97C5EEEF0DB0, 0xF01035618C290E0F, 0xF011EB076CB62240,
    0xF012BFAAD94536C2, 0xF01987E142E22C84, 0xF01DE470EA793E79,
    0xF02308E7F61F238C, 0xF027C9BF697D16BC, 0xF029C12AA35E13F9,
    0xF02EDAF8C991234F, 0xF02F1862743A184A, 0xF0364E81A5C408CA,
    0xF037E92A5C121FBB, 0xF039F3BC930C056B, 0xF03F75F9594F1FED,
    0xF0415815F6F525C7, 0xF041F35B46552DC1, 0xF0440EB53DD83271,
    0xF044916C8B01016F, 0xF049108F786F0FBA, 0xF04B1C96487E3EF2,
    0xF0517A4ACE960952, 0xF055FD5F034605F6, 0xF05817B7EF4F05CA,
    0xF05AB99E7D620565, 0xF05D11CB4D971568, 0xF05D259CF2763230,
    0xF062FA35546738B5, 0xF064DBF6839F3AA0, 0xF064F77FD8853167,
    0xF065F0FC99BA2505, 0xF06867910A7026BC, 0xF0736639EE501E02,
    0xF0736CB9A5610A55, 0xF075998527C6389D, 0xF077CAE1F3C53332,
    0xF078AF24834529F1, 0xF07AD4470C6E0ACD, 0xF083791E05B31FE3,
    0xF083D81D3A492CF2, 0xF0849076FE6033B9, 0xF084F567A1B40340,
    0xF08692BDBF2529E9, 0xF089D5A5B40E0D45, 0xF08F2569D5EB1898,
    0xF09805732CEC0E4C, 0xF09C1CADA03628D5, 0xF09E9875D069188F,
    0xF0A893FF551A2F75, 0xF0AC1DC6D01C1D66, 0xF0AD4147E7B81800,
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    0xF0D489E48E000AA0, 0xF0D9AF4559E835A0, 0xF0DAB0A3B856256B,
    0xF0DFE2D9577E331A, 0xF0E099F9F6EE04B6, 0xF0E58F10D1713058,
    0xF0F10CEFCC0B06BB, 0xF0FB09E339723291, 0xF1047BC63875115A,
    0xF106ABE436D700FE, 0xF10C50BEDC723C1B, 0xF1128B996F6822C6,
    0xF112A32664230C7C, 0xF114059A9E2F1E46, 0xF117AC69373831EB,
    0xF1196F658B3F0001, 0xF128596D984026E4, 0xF12C218FA8AF2346,
    0xF1384A7709753F3C, 0xF1395A8AD33F0684, 0xF14510D8262E0A56,
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    0xF156B79B1D112A4D, 0xF1585327C0D031B8, 0xF15A0D3C186937D1,
    0xF15AE8A92F0036A8, 0xF15E58167EA73ADE, 0xF15FA23DE8DB2D01,
    0xF1643C85241C1365, 0xF167229482853A88, 0xF1696D33DF751929,
    0xF16AEA4105E4369A, 0xF16C05BA72B930E3, 0xF16C171323201D54,
    0xF16CA1F531C33F32, 0xF17136F17EE62A18, 0xF171EC2F43C0071E,
    0xF178075DA2133E8C, 0xF180EC2F0F5819DD, 0xF182CC6B51C52406,
    0xF1835A5BE35A16C8, 0xF183F2FA12C00D7B, 0xF1872A597F232DE8,
    0xF18893F861662C8A, 0xF18B4780317D0089, 0xF1982FEB31552921,
    0xF19ACD80EBBB377A, 0xF19DA4AF7BDF070B, 0xF19F9AAE8D94015D,
    0xF1A09881ADF707C9, 0xF1A21C2AA1F80968, 0xF1A2B5D296152AA5,
    0xF1A6ABB1C0483914, 0xF1AB511B68061DAA, 0xF1AC1F0A885B36E4,
    0xF1BA7C90DF8011D8, 0xF1BB7CBB8EAD25B9, 0xF1BF1C55045E1CEC,
    0xF1BFC6CBCF3F0023, 0xF1CCA1B8A8BA0459, 0xF1D0C01C467929F8,
    0xF1D368E2AF732845, 0xF1D4F7A798CD1B46, 0xF1D5340DA2080E9F,
    0xF1D990BBCC2D1584, 0xF1DF948865341666, 0xF1E3E828E6F312D9,
    0xF1E418C14F150EFB, 0xF1E697AD79A929C1, 0xF1E6FB8146B212C6,
    0xF1EB8111FCF41737, 0xF1ECF28CA44435D2, 0xF1F08372D96D250E,
    0xF1F3B334450D04DF, 0xF1F5517BB5FE1C8D, 0xF1F55E01F1633C10,
    0xF1F662D0DEC025DD, 0xF1FC344CD1803AEF, 0xF1FF46D8D4933E5A,
    0xF210E247470A0DC5, 0xF211F40C49F823FB, 0xF21B6DC2A1D32560,
    0xF21BACCFD86B32B3, 0xF21FBC62F8463342, 0xF223CF6A845E3E9F,
    0xF22546FD62FA383F, 0xF226A52DFB42349F, 0xF2298AAA2BBD0365,
    0xF22B8539BB0A14FC, 0xF22CE49D880B3EA2, 0xF234640C527D20F3,
    0xF234B6EDA23134A1, 0xF2366A8FFCBE13FC, 0xF241125C92B03047,
    0xF24582E79F2932D6, 0xF248F23BF506357B, 0xF24C1571D1E12CB1,
    0xF24D910E093911F6, 0xF24DF3281B5A3246, 0xF250128C109A252D,
    0xF2504880DDE3325C, 0xF250648107D8291F, 0xF25C430654713D71,
    0xF25E17AF5CA02A7C, 0xF25E1BB0D9370D70, 0xF26203ED99002337,
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    0xF26C7D70762B124C, 0xF2780818E95404EB, 0xF27DFA87CF6938F5,
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    0xF2B336EDDC8A3523, 0xF2BC20820F7805C1, 0xF2BEE54A66503B98,
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    0xF304EC275E171F4D, 0xF306EEE2647A2EF2, 0xF30CC8830388055E,
    0xF314075E629C21FE, 0xF317816F12220152, 0xF3195040543215FC,
    0xF31AC8ED4799192C, 0xF320727F876201A3, 0xF320AD57ACE812CE,
    0xF3226BD783AD2D8D, 0xF327C38FF8E733C0, 0xF330166412F71E21,
    0xF33C789BD29D172D, 0xF3405ADFB46F317A, 0xF34110D76EB63AF2,
    0xF345817293DF3E7F, 0xF3464CA989513DAB, 0xF347079DA53921B0,
    0xF348E9C9E9E51560, 0xF34A5AC5B27601F8, 0xF34A7666D76719A5,
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    0xF97DFCA9E5A323A3, 0xF9857A93741E20F7, 0xF989B73461303DB7,
    0xF98B9EEBBFDB04A7, 0xF9906BDB96843CF3, 0xF9921E50193214E8,
    0xF992F2F7996D0AC7, 0xF99593B4A5CF047D, 0xF996AC4AAF7327C3,
    0xF99705E10D331C00, 0xF99EF4444794385C, 0xF99FD5CFF46A0B70,
    0xF9A0EAE83F862DA1, 0xF9A497DF16F82FE8, 0xF9A7C252FEF30A50,
    0xF9AC2ACE1871380A, 0xF9AF1D7E863108A5, 0xF9B27AA33C153DFC,
    0xF9B3AD5D65B62892, 0xF9B98B9863C522FC, 0xF9BB4CB001592BDE,
    0xF9BBDC9C3A9D2FBF, 0xF9CAD5A41A81181D, 0xF9CC3A4523BF2EF5,
    0xF9CCC5E2C50A0C90, 0xF9D2325939C42C0F, 0xF9D2C37078140DE9,
    0xF9D75E533FF32B82, 0xF9D8B2E11B573325, 0xF9D8CC72C028394D,
    0xF9D8E16349E72B09, 0xF9DDB52DC1C6271C, 0xF9DEDB954D152EFD,
    0xF9E8698484AB1FF3, 0xF9EF3A783752269A, 0xF9F5EEC0AF000777,
    0xF9F9C2129E6C3008, 0xF9FA02599D641164, 0xF9FC6D2F37CF38A0,
    0xF9FDC277F7143F4C, 0xF9FDEDF6306A2164, 0xFA0178625CC20864,
    0xFA04D1AE3BDE3DFD, 0xFA06AC2517FD1F33, 0xFA09A2506C4D2985,
    0xFA109269216731A2, 0xFA118BDA24411AE1, 0xFA16F656AF5F3906,
    0xFA18A2CD4B3C2DD2, 0xFA1F396544280086, 0xFA1FA34D1B9C3B4C,
    0xFA24E88819130377, 0xFA322DA1CB262BDF, 0xFA34671061C03850,
    0xFA37BA40482029CD, 0xFA40D35728C808DC, 0xFA435E5B263D0B2E,
    0xFA43714922D43318, 0xFA5152E6D0D32020, 0xFA53611A60C90A09,
    0xFA53DEF3ACC902D8, 0xFA568A024F04335D, 0xFA5802E5F8381C96,
    0xFA599FF3A51B3C69, 0xFA5A26B5F3AF3DB6, 0xFA63FCBEA62A0A12,
    0xFA6771D9D61000D2, 0xFA6A51A5AE9E08FB, 0xFA72C95981801227,
    0xFA7567353DEF0354, 0xFA79AF8585232DAB, 0xFA8C414BFA0A04CC,
    0xFA901684296F1A88, 0xFA9031714BCB3930, 0xFA94D5F413C63953,
    0xFA9666A21802108D, 0xFA99999939042FB5, 0xFA9D34AEAD4C2732,
    0xFA9FCC68C28B3A01, 0xFAA4237C1CE830B3, 0xFAA5B3EEF5941E94,
    0xFAA65DCBAD893E8F, 0xFAA6B3D1070D1350, 0xFAA720D2D7001150,
    0xFAA7EC39A73D04A3, 0xFAAFBC73D318218C, 0xFAB69B00B5F3032B,
    0xFAB792E2C6BB0BD6, 0xFABC3AC501B2359E, 0xFABF451BECF62265,
    0xFAC00C30C20A331C, 0xFAC12FAC4B2B2290, 0xFAC3F49B05FA11E1,
    0xFAC72D01A1F43C8F, 0xFAC82F7ACC793993, 0xFACC867ADD623DA2,
    0xFAD5CA2F90F80524, 0xFAD6D2F4566B08CC, 0xFAE1496844700E66,
    0xFAED5FE99B2506E1, 0xFAF14F5D63D72E8C, 0xFAFA0881DBE025A7,
    0xFAFA256632FB2271, 0xFB013E01C04C2054, 0xFB06EECE587308E8,
    0xFB0C8830A1F93B2D, 0xFB0CD35FD28D3B3C, 0xFB0DC7BFB5BA0B0C,
    0xFB115B14A8CD0594, 0xFB12F484E43C2E3D, 0xFB19ACA83DCF2853,
    0xFB241401DFBE28AD, 0xFB26E42C2E4E3108, 0xFB289D100905125E,
    0xFB334CB9E46D0EEA, 0xFB3374A0B29F3352, 0xFB3958510E1B0C51,
    0xFB3D492F12303A6B, 0xFB427237358E30EC, 0xFB439702C69D3F8B,
    0xFB44AD8D54DA2810, 0xFB4562E9481112B8, 0xFB46784A6149138D,
    0xFB4702654BEA117A, 0xFB47C7A2F54C1F87, 0xFB50BBA7789C1AA6,
    0xFB519D3AB02F15DC, 0xFB519EF316B32E13, 0xFB523E83CE743B75,
    0xFB58CE0A609E242F, 0xFB5D1C76BACD3003, 0xFB613471AABB0301,
    0xFB6507083E202E53, 0xFB676923DC603ED8, 0xFB694EFB6F9C16A5,
    0xFB6F101B3FCA3573, 0xFB712F072CBF07F5, 0xFB744BD6748F2895,
    0xFB7F69BCB5CF07AE, 0xFB7FD2AE382B3095, 0xFB808796B95C0259,
    0xFB843EBDA86B16AB, 0xFB8C01F8EA571C14, 0xFB90677A498B0E70,
    0xFB998BB992B800BC, 0xFB9A4F1DA6F02E01, 0xFBA25A0C4DE404A0,
    0xFBA8C02CD98435B2, 0xFBAC380B93653E4B, 0xFBAD049CE1741770,
    0xFBB92D722F5C16E6, 0xFBBC5F442CF72B91, 0xFBBD9CB3D28A2626,
    0xFBC81CDC6679236A, 0xFBD21D7628342007, 0xFBD4C4C99572082A,
    0xFBDBBC84E05F3D40, 0xFBE12912AF102E88, 0xFBE36873B960331B,
    0xFBE90C26C2192E37, 0xFBF59C9F024D34E3, 0xFBF8A755889B2F8A,
    0xFBFF5D0EC24E3D91, 0xFC040618A3DA238A, 0xFC06D17F68EA1243,
    0xFC0888C94A69113A, 0xFC0982712B122390, 0xFC0A689F396921F0,
    0xFC0E96DA72760E4F, 0xFC0FB13CDF430489, 0xFC1D3D14F0712904,
    0xFC1F552A1F170B66, 0xFC20370D219928E8, 0xFC2813318A7C09A9,
    0xFC2CFDCBE3DB301B, 0xFC2CFF83A37707A6, 0xFC323DD5070F38C6,
    0xFC3349CB22632DDE, 0xFC34502212763198, 0xFC397C0FE7B72B76,
    0xFC39D47B11131D28, 0xFC3A3E218E233902, 0xFC3ACF73ACC2382B,
    0xFC3C42DD92420E58, 0xFC4024AC20F6055C, 0xFC404BB5CDFD2D5D,
    0xFC445CCF5F231DBA, 0xFC486C1D58CB3338, 0xFC4A81667693090D,
    0xFC4BB1CD821B2275, 0xFC4DB8ADE1AF0B5D, 0xFC4F18D3769C03F7,
    0xFC4F1F822EEE2C77, 0xFC50CD7428790DB7, 0xFC5154E001F627E0,
    0xFC5D0ED95E7E155C, 0xFC5E8DEF6D9123B8, 0xFC60824798260B26,
    0xFC634AEB33393A99, 0xFC64939A0F55199C, 0xFC6C00375D7726AF,
    0xFC6FDAB8BCC806A2, 0xFC70147340B12327, 0xFC76F36BEA5B2484,
    0xFC7FC3589D751D48, 0xFC899250ED3D2AB0, 0xFC8D0E930D7E0D4F,
    0xFC9725573D7C051E, 0xFC98AE97459E12B0, 0xFC9C010A81213CC2,
    0xFC9E427B89A2190E, 0xFCA84F0C33A311D4, 0xFCA8C004862F1AC8,
    0xFCAA45F0454A2D68, 0xFCAB0AB7307725A1, 0xFCAD79DE532116FC,
    0xFCB41411DF56398F, 0xFCBAFD798CFC2E63, 0xFCBB44ED1AA10D25,
    0xFCBB84D282012381, 0xFCBBFF1306DE3884, 0xFCBF44FE80250B3F,
    0xFCC679D51A4617EE, 0xFCC7F12E17C13E29, 0xFCCDB3AF92E613D1,
    0xFCD4CBA7C8472D04, 0xFCD54DE1607327FC, 0xFCD5753D98A32E02,
    0xFCDEE299FCF23E31, 0xFCDF3F9F1F760207, 0xFCECDD4263970787,
    0xFCEE0C462EE63040, 0xFCF1ADDA04D63347, 0xFCF21E50E15C1E9D,
    0xFCF3DB77328506B3, 0xFCF42C3F4F750A23, 0xFCFA869F395B038C,
    0xFCFC126DDEEA1251, 0xFCFEDA184EB82DC4, 0xFD06875C34533DB8,
    0xFD0A38BB341708DD, 0xFD20F6F2B84908A3, 0xFD30870E326D1A3F,
    0xFD365835F99C1B97, 0xFD3981D3ADC20A0B, 0xFD3DD531D15B05E1,
    0xFD4364DB0FDF0CC8, 0xFD440EA621620831, 0xFD499675A8DC1CFF,
    0xFD4C21653B6A3F04, 0xFD51590629302D6B, 0xFD5703620A693E59,
    0xFD571CFA86532114, 0xFD57B9D3EED222A4, 0xFD5A8D6614BD380B,
    0xFD5EC513CE891D87, 0xFD5FB5B7752F1D3D, 0xFD660B12D1461892,
    0xFD67D975597C24A6, 0xFD680E5EB161204D, 0xFD6C4FD7C02B160E,
    0xFD6E78D2DF8F1B24, 0xFD707E8300692A0D, 0xFD73F138B4EC2A31,
    0xFD7F0F40241E2DA5, 0xFD8A568FE3090A38, 0xFD92798AE8900DDA,
    0xFD92A3AE11393D62, 0xFD94100F6E991FAD, 0xFD95325B9E6A292E,
    0xFD96BBC2A965053E, 0xFD9CF573DE3233AF, 0xFDA0A72F00B821A8,
    0xFDA3E558AFEC2F85, 0xFDA574C48DE70967, 0xFDA6592D5D553052,
    0xFDA6CB15E5F73C5B, 0xFDA923562D260D88, 0xFDA9CFA77F86052A,
    0xFDB024309A4C08FE, 0xFDBD79DAFDEC2EBE, 0xFDBDA1E4DA602A70,
    0xFDC0579AA54507EC, 0xFDC15FE8B730154A, 0xFDC2A6FBB225023F,
    0xFDC3BCEB0DF91BEF, 0xFDCCEBD0EE8B0A18, 0xFDCD079D922419ED,
    0xFDCDBB14D81C0382, 0xFDD1815655922DA9, 0xFDD2171825BA38E7,
    0xFDE23595490004DE, 0xFDEC892646741198, 0xFDF6DCF757A0262A,
    0xFDFFA19C7D3E19A9, 0xFE01A202BBB23778, 0xFE035C424809080E,
    0xFE076A3F748C362B, 0xFE0810925B5D3283, 0xFE17E587CA70282D,
    0xFE1BB14F09CC2C59, 0xFE21E83E10330128, 0xFE23EC1238223392,
    0xFE36FCCFD0683C26, 0xFE391F751D45308F, 0xFE3ADBBB47E218DD,
    0xFE3E5284BDF500D8, 0xFE427DD04AB72292, 0xFE48573B0C0A1884,
    0xFE4F0908005A2683, 0xFE625DB2EDFF3F70, 0xFE66BEF01B65275C,
    0xFE68B09D73D11ACA, 0xFE6BA650C2BD07C3, 0xFE6DC50D6FE71E72,
    0xFE70183ABEF70F96, 0xFE71D7F17C28351F, 0xFE754643624A1BB8,
    0xFE76969E869A02F5, 0xFE787E0693130938, 0xFE7B5C0C27772EBC,
    0xFE7F81E9550D265C, 0xFE839FB32E3D19B8, 0xFE8AD31F30A514D2,
    0xFE8D000FE16705E0, 0xFE8DDBCD3B882802, 0xFE8ED0B5081C3B65,
    0xFE95917762D42D50, 0xFE9B12CD6E49198A, 0xFE9F7AF3416828FA,
    0xFEA10B0BC86D12AB, 0xFEA1FC426E9A114A, 0xFEAF5513659E083C,
    0xFEB015C1B69C230B, 0xFEB348BE9B90315A, 0xFEB9DDC4C4B61A43,
    0xFEB9F6569874091E, 0xFEBE770A414808EB, 0xFEC08575D1572369,
    0xFEC153A5BACC0CCB, 0xFEC2733553481736, 0xFEC32DAF613706AD,
    0xFEC43183D772077A, 0xFEC5BC9B4DDA2A23, 0xFED419762BBA1B9F,
    0xFED81345DCFF19DE, 0xFED838AE491B3814, 0xFEDCE385BC3617F2,
    0xFEDE36CAEDB30465, 0xFEDFC203FABE04D1, 0xFEE4E070ED483B99,
    0xFEE5DF73183211EC, 0xFEE6D70E9EA63C14, 0xFEEB0E6288DD39DA,
    0xFEEF15454AB83D98, 0xFEEF7412A9C22C2A, 0xFEF15C908FFC15ED,
    0xFEFAFBA121A630B8, 0xFEFC73D273623749, 0xFEFD4EBBCB7930D2,
    0xFEFEA5A5B8722C9C, 0xFF02C8C8CAAC2D37, 0xFF063F571E25220D,
    0xFF0D2A5C24542125, 0xFF0E038073B71D75, 0xFF118546E12C29E8,
    0xFF14B22020CB11A5, 0xFF18A10FB96428C7, 0xFF1914DB3DBA32C2,
    0xFF1A1A5047910E0E, 0xFF1CAFD462F602E8, 0xFF21EE853A663172,
    0xFF27C84A324F1019, 0xFF2E8600B84A0B31, 0xFF350EE42B922462,
    0xFF35B6C469310F30, 0xFF3D9551B00B0384, 0xFF450FDD68F73432,
    0xFF4B29D80F0F2573, 0xFF4EFF48F2723545, 0xFF5543B543E03E9D,
    0xFF569041EE1C3F92, 0xFF5739D103102EB1, 0xFF57CEE91DD82DD6,
    0xFF5EF4D72FE5129E, 0xFF6AA811C14E0124, 0xFF6B15171A8D317C,
    0xFF6DE38E61041C3D, 0xFF6F4AAB673924BD, 0xFF74BBFBFC0501BB,
    0xFF7509FF09052BB2, 0xFF80087ECFD51168, 0xFF88B4A157BC0660,
    0xFF8AEA878640188B, 0xFF8BD1021D3319A7, 0xFF8E61ACB4002E98,
    0xFF928914D1BE387D, 0xFF941932DDE93A4F, 0xFF9481B6185F1A2B,
    0xFF96C83660883AD1, 0xFF9735F4B6632909, 0xFF979A68A4980D2C,
    0xFF97BD2FA5802E32, 0xFF9994EB0BD03DB1, 0xFF9A4B2D16801070,
    0xFF9C5777D07008A8, 0xFFA9E20467E91EB2, 0xFFAAFA5EC83F0E06,
    0xFFAD3D1A77613EA7, 0xFFAF3F87F3610ED5, 0xFFBD2E4A7D351270,
    0xFFC0797967C03D8E, 0xFFC334A50EB124EA, 0xFFC86FC407730237,
    0xFFC9A07B8AC33A7B, 0xFFCA14E542B90FCC, 0xFFCADA902F9C2340,
    0xFFCB823997B00E2E, 0xFFCC804CD60C3D13, 0xFFD00F2259DE2521,
    0xFFD09C80835D0C86, 0xFFD1ADD32F5C3417, 0xFFD6FA00CD3A3303,
    0xFFDB46E4B7510B33, 0xFFDE91679C85283C, 0xFFDEC7009CEC01FC,
    0xFFE04FEA37EB3FEA, 0xFFE6AC574C103D8D, 0xFFEA04AF3C9C3EC2,
    0xFFEA82B37061367E, 0xFFEED4EA71102674, 0xFFF5C98450D92A2D,
    0xFFF89D61746A09FE, 0xFFF8A421B49A23CB,
];

// ========================================================================

#[cfg(test)]
mod tests {

    use super::{Point, Scalar, PrivateKey, PublicKey};
    use sha2::{Sha256, Sha512, Digest};

    /* unused
    use std::fmt;
    use crate::field::GF25519;

    fn print_gf(name: &str, x: GF25519) {
        print!("{} = 0x", name);
        let bb = x.encode();
        for i in (0..32).rev() {
            print!("{:02X}", bb[i]);
        }
        println!();
    }

    fn print(name: &str, P: Point) {
        println!("{}:", name);
        print_gf("  X", P.X);
        print_gf("  Y", P.Y);
        print_gf("  Z", P.Z);
        print_gf("  T", P.T);
    }
    */

    #[test]
    fn base_arith() {
        // For a point P (randomly generated on the curve with Sage),
        // points i*P for i = 0 to 6, encoded.
        const EPP: [[u8; 32]; 7] = [
            [
                0x01, 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
            ],
            [
                0x91, 0x7E, 0x2B, 0x2F, 0xF9, 0xC9, 0x66, 0x45,
                0x1D, 0x28, 0xC9, 0x3E, 0xD4, 0xDE, 0x9A, 0xE9,
                0xCE, 0x2D, 0x67, 0x2C, 0xD3, 0xCF, 0x74, 0x06,
                0xAE, 0x0D, 0x86, 0xC4, 0x21, 0xB9, 0x02, 0xE6
            ],
            [
                0x18, 0x68, 0xD3, 0xEA, 0xE6, 0x62, 0x1B, 0xAF,
                0xD0, 0x4C, 0x2C, 0xEC, 0x8B, 0xA9, 0xEE, 0xF3,
                0x28, 0xCE, 0xC3, 0x07, 0x5A, 0x57, 0xCE, 0x98,
                0x69, 0x83, 0x3C, 0x8A, 0x8E, 0xF2, 0x90, 0xD4
            ],
            [
                0x37, 0x31, 0xAB, 0x10, 0x85, 0x48, 0x38, 0xC7,
                0x2E, 0x2F, 0xCF, 0x29, 0xD9, 0xCD, 0xA0, 0xBD,
                0xC8, 0xAE, 0xD6, 0x70, 0x58, 0x56, 0x6F, 0xAE,
                0xCD, 0x6F, 0xFC, 0xB7, 0x0D, 0x6A, 0xC5, 0x60
            ],
            [
                0x47, 0x18, 0x8B, 0xDD, 0x31, 0xDE, 0x9E, 0x3E,
                0x29, 0x2B, 0x52, 0x6C, 0x50, 0x0A, 0x91, 0x29,
                0x96, 0x9D, 0xAD, 0xE2, 0x6B, 0x13, 0x3A, 0x8E,
                0xAB, 0x55, 0xED, 0xBA, 0xD9, 0x01, 0x34, 0x26
            ],
            [
                0xE4, 0xD0, 0x2A, 0x56, 0x90, 0xE1, 0x86, 0xCD,
                0xEC, 0x21, 0x41, 0xF0, 0x49, 0x4F, 0x19, 0x70,
                0x7C, 0x3B, 0x4D, 0xC0, 0x0E, 0x6B, 0x90, 0x1C,
                0x9D, 0x8E, 0xF0, 0xE2, 0xC6, 0x91, 0x11, 0xD2
            ],
            [
                0x3B, 0x7F, 0xFC, 0x19, 0x3B, 0x7C, 0xC4, 0x58,
                0xDF, 0x56, 0xE3, 0xD8, 0xBB, 0xEC, 0x71, 0x3F,
                0xE9, 0xE3, 0x97, 0xD8, 0x6E, 0x4A, 0x7E, 0x08,
                0x25, 0x1A, 0xCC, 0x29, 0xED, 0x17, 0x88, 0x32
            ],
        ];

        let mut PP = [Point::NEUTRAL; 7];
        for i in 0..7 {
            let P = Point::decode(&EPP[i][..]).unwrap();
            assert!(EPP[i] == P.encode());
            PP[i] = P;
            if i == 0 {
                assert!(P.isneutral() == 0xFFFFFFFF);
            } else {
                assert!(P.isneutral() == 0x00000000);
            }
        }

        let P0 = PP[0];
        let P1 = PP[1];
        let P2 = PP[2];
        let P3 = PP[3];
        let P4 = PP[4];
        let P5 = PP[5];
        let P6 = PP[6];

        for i in 1..7 {
            assert!(PP[i].equals(PP[i - 1]) == 0);
            let Q = PP[i - 1] + PP[1];
            assert!(PP[i].equals(Q) == 0xFFFFFFFF);
            assert!((Q + Point::NEUTRAL).equals(Q) == 0xFFFFFFFF);
            let R = Q + P0;
            assert!(PP[i].equals(R) == 0xFFFFFFFF);
        }

        let Q2 = P1 + P1;
        assert!(Q2.encode() == EPP[2]);
        assert!(Q2.equals(P2) == 0xFFFFFFFF);
        let R2 = P1.double();
        assert!(R2.encode() == EPP[2]);
        assert!(R2.equals(P2) == 0xFFFFFFFF);
        assert!(R2.equals(Q2) == 0xFFFFFFFF);

        let Q3 = P2 + P1;
        assert!(Q3.encode() == EPP[3]);
        assert!(Q3.equals(P3) == 0xFFFFFFFF);
        let R3 = Q2 + P1;
        assert!(R3.encode() == EPP[3]);
        assert!(R3.equals(P3) == 0xFFFFFFFF);
        assert!(R3.equals(Q3) == 0xFFFFFFFF);

        let Q4 = Q2.double();
        assert!(Q4.encode() == EPP[4]);
        assert!(Q4.equals(P4) == 0xFFFFFFFF);
        let R4 = P1.xdouble(2);
        assert!(R4.encode() == EPP[4]);
        assert!(R4.equals(P4) == 0xFFFFFFFF);
        assert!(R4.equals(Q4) == 0xFFFFFFFF);
        let R4 = P1 + Q3;
        assert!(R4.encode() == EPP[4]);
        assert!(R4.equals(P4) == 0xFFFFFFFF);
        assert!(R4.equals(Q4) == 0xFFFFFFFF);

        let Q5 = Q3 + R2;
        assert!(Q5.encode() == EPP[5]);
        assert!(Q5.equals(P5) == 0xFFFFFFFF);
        let R5 = R3 + Q2;
        assert!(R5.encode() == EPP[5]);
        assert!(R5.equals(P5) == 0xFFFFFFFF);
        assert!(R5.equals(Q5) == 0xFFFFFFFF);

        assert!((R5 - Q3).equals(Q2) == 0xFFFFFFFF);

        let Q6 = Q3.double();
        assert!(Q6.encode() == EPP[6]);
        assert!(Q6.equals(P6) == 0xFFFFFFFF);
        let R6 = Q2 + Q4;
        assert!(R6.encode() == EPP[6]);
        assert!(R6.equals(P6) == 0xFFFFFFFF);
        assert!(R6.equals(Q6) == 0xFFFFFFFF);
    }

    #[test]
    fn mulgen() {
        // Test vector generated randomly with Sage.
        let s = Scalar::w64be(0x0CD02221F61282F5, 0xA89A40FB39C73FE9,
                              0xFAC8A7DDC31F4E9B, 0x453E5B8615D29F82);
        let enc: [u8; 32] = [
            0xA3, 0xBB, 0x37, 0xCB, 0xA6, 0x00, 0x65, 0xA0,
            0x23, 0x1A, 0xB8, 0x72, 0x51, 0x78, 0x5E, 0x96,
            0x3D, 0x28, 0xCE, 0xDB, 0xA0, 0x09, 0xA6, 0x5A,
            0x4A, 0x9F, 0x6E, 0x5A, 0x64, 0x8F, 0xB4, 0xB0
        ];

        let R = Point::decode(&enc).unwrap();
        let P = Point::BASE * s;
        assert!(P.equals(R) == 0xFFFFFFFF);
        assert!(P.encode() == enc);
        let Q = Point::mulgen(&s);
        assert!(Q.equals(R) == 0xFFFFFFFF);
        assert!(Q.encode() == enc);
    }

    #[test]
    fn mul() {
        let mut sh = Sha256::new();
        for i in 0..20 {
            // Build pseudorandom s1 and s2
            sh.update(((2 * i + 0) as u64).to_le_bytes());
            let v1 = sh.finalize_reset();
            sh.update(((2 * i + 1) as u64).to_le_bytes());
            let v2 = sh.finalize_reset();

            let s1 = Scalar::decode_reduce(&v1);
            let s2 = Scalar::decode_reduce(&v2);
            let s3 = s1 * s2;
            let P1 = Point::mulgen(&s1);
            let Q1 = s1 * Point::BASE;
            assert!(P1.equals(Q1) == 0xFFFFFFFF);
            let P2 = Point::mulgen(&s3);
            let Q2 = s2 * Q1;
            assert!(P2.equals(Q2) == 0xFFFFFFFF);
        }
    }

    #[test]
    fn mul_add_mulgen() {
        let mut sh = Sha256::new();
        for i in 0..20 {
            // Build pseudorandom A, u and v
            sh.update(((3 * i + 0) as u64).to_le_bytes());
            let v1 = sh.finalize_reset();
            sh.update(((3 * i + 1) as u64).to_le_bytes());
            let v2 = sh.finalize_reset();
            sh.update(((3 * i + 2) as u64).to_le_bytes());
            let v3 = sh.finalize_reset();
            let A = Point::mulgen(&Scalar::decode_reduce(&v1));
            let u = Scalar::decode_reduce(&v2);
            let v = Scalar::decode_reduce(&v3);

            // Compute u*A + v*B in two different ways; check that they
            // match.
            let R1 = u * A + Point::mulgen(&v);
            let R2 = A.mul_add_mulgen_vartime(&u, &v);
            assert!(R1.equals(R2) == 0xFFFFFFFF);
        }
    }

    #[test]
    fn verify_helper() {
        // Low-order points (encoded).
        const LOW_ENC: [[u8; 32]; 8] = [
            [ 0x01, 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 ],
            [ 0x26, 0xE8, 0x95, 0x8F, 0xC2, 0xB2, 0x27, 0xB0,
              0x45, 0xC3, 0xF4, 0x89, 0xF2, 0xEF, 0x98, 0xF0,
              0xD5, 0xDF, 0xAC, 0x05, 0xD3, 0xC6, 0x33, 0x39,
              0xB1, 0x38, 0x02, 0x88, 0x6D, 0x53, 0xFC, 0x85 ],
            [ 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, 0x80 ],
            [ 0xC7, 0x17, 0x6A, 0x70, 0x3D, 0x4D, 0xD8, 0x4F,
              0xBA, 0x3C, 0x0B, 0x76, 0x0D, 0x10, 0x67, 0x0F,
              0x2A, 0x20, 0x53, 0xFA, 0x2C, 0x39, 0xCC, 0xC6,
              0x4E, 0xC7, 0xFD, 0x77, 0x92, 0xAC, 0x03, 0xFA ],
            [ 0xEC, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
              0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
              0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF,
              0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0x7F ],
            [ 0xC7, 0x17, 0x6A, 0x70, 0x3D, 0x4D, 0xD8, 0x4F,
              0xBA, 0x3C, 0x0B, 0x76, 0x0D, 0x10, 0x67, 0x0F,
              0x2A, 0x20, 0x53, 0xFA, 0x2C, 0x39, 0xCC, 0xC6,
              0x4E, 0xC7, 0xFD, 0x77, 0x92, 0xAC, 0x03, 0x7A ],
            [ 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, 0x00 ],
            [ 0x26, 0xE8, 0x95, 0x8F, 0xC2, 0xB2, 0x27, 0xB0,
              0x45, 0xC3, 0xF4, 0x89, 0xF2, 0xEF, 0x98, 0xF0,
              0xD5, 0xDF, 0xAC, 0x05, 0xD3, 0xC6, 0x33, 0x39,
              0xB1, 0x38, 0x02, 0x88, 0x6D, 0x53, 0xFC, 0x05 ],
        ];

        let mut low = [Point::NEUTRAL; 8];
        for i in 0..8 {
            low[i] = Point::decode(&LOW_ENC[i]).unwrap();
            assert!(low[i].has_low_order() == 0xFFFFFFFF);
        }

        let mut sh = Sha256::new();
        for i in 0..20 {
            // Build pseudorandom A, s and k
            // Compute R = s*B - k*A
            sh.update(((3 * i + 0) as u64).to_le_bytes());
            let v1 = sh.finalize_reset();
            sh.update(((3 * i + 1) as u64).to_le_bytes());
            let v2 = sh.finalize_reset();
            sh.update(((3 * i + 2) as u64).to_le_bytes());
            let v3 = sh.finalize_reset();
            let A = Point::mulgen(&Scalar::decode_reduce(&v1));
            let s = Scalar::decode_reduce(&v2);
            let k = Scalar::decode_reduce(&v3);
            let R = Point::mulgen(&s) - k * A;

            for j in 0..8 {
                // The equation must be verified even if we add
                // low-order points to either A or R.
                assert!(A.verify_helper_vartime(&(R + low[j]), &s, &k));
                assert!((A + low[j]).verify_helper_vartime(&R, &s, &k));
                let j2 = (j + i) & 7;
                assert!((A + low[j]).verify_helper_vartime(&(R + low[j2]), &s, &k));
            }

            // The equation must NOT match if we change k or s.
            assert!(!A.verify_helper_vartime(&R, &(s + Scalar::ONE), &k));
            assert!(!A.verify_helper_vartime(&R, &s, &(k + Scalar::ONE)));
        }
    }

    struct Ed25519TestVector<'a> {
        s: &'a str,
        Q: &'a str,
        m: &'a str,
        dom: bool,
        ph: bool,
        ctx: &'a str,
        sig: &'a str,
    }

    // Test vectors from RFC 8032.
    const TEST_VECTORS: [Ed25519TestVector; 6] = [
        Ed25519TestVector {
            s:   "9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60",
            Q:   "d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a",
            m:   "",
            dom: false,
            ph:  false,
            ctx: "",
            sig: "e5564300c360ac729086e2cc806e828a84877f1eb8e5d974d873e065224901555fb8821590a33bacc61e39701cf9b46bd25bf5f0595bbe24655141438e7a100b",
        },
        Ed25519TestVector {
            s:   "4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb",
            Q:   "3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c",
            m:   "72",
            dom: false,
            ph:  false,
            ctx: "",
            sig: "92a009a9f0d4cab8720e820b5f642540a2b27b5416503f8fb3762223ebdb69da085ac1e43e15996e458f3613d0f11d8c387b2eaeb4302aeeb00d291612bb0c00",
        },
        Ed25519TestVector {
            s:   "c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7",
            Q:   "fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025",
            m:   "af82",
            dom: false,
            ph:  false,
            ctx: "",
            sig: "6291d657deec24024827e69c3abe01a30ce548a284743a445e3680d7db5ac3ac18ff9b538d16f290ae67f760984dc6594a7c15e9716ed28dc027beceea1ec40a",
        },
        Ed25519TestVector {
            s:   "833fe62409237b9d62ec77587520911e9a759cec1d19755b7da901b96dca3d42",
            Q:   "ec172b93ad5e563bf4932c70e1245034c35467ef2efd4d64ebf819683467e2bf",
            m:   "ddaf35a193617abacc417349ae20413112e6fa4e89a97ea20a9eeee64b55d39a2192992a274fc1a836ba3c23a3feebbd454d4423643ce80e2a9ac94fa54ca49f",
            dom: false,
            ph:  false,
            ctx: "",
            sig: "dc2a4459e7369633a52b1bf277839a00201009a3efbf3ecb69bea2186c26b58909351fc9ac90b3ecfdfbc7c66431e0303dca179c138ac17ad9bef1177331a704",
        },
        Ed25519TestVector {
            s:   "ab9c2853ce297ddab85c993b3ae14bcad39b2c682beabc27d6d4eb20711d6560",
            Q:   "0f1d1274943b91415889152e893d80e93275a1fc0b65fd71b4b0dda10ad7d772",
            m:   "f726936d19c800494e3fdaff20b276a8",
            dom: true,
            ph:  false,
            ctx: "666f6f",
            sig: "21655b5f1aa965996b3f97b3c849eafba922a0a62992f73b3d1b73106a84ad85e9b86a7b6005ea868337ff2d20a7f5fbd4cd10b0be49a68da2b2e0dc0ad8960f",
        },
        Ed25519TestVector {
            s:   "833fe62409237b9d62ec77587520911e9a759cec1d19755b7da901b96dca3d42",
            Q:   "ec172b93ad5e563bf4932c70e1245034c35467ef2efd4d64ebf819683467e2bf",
            m:   "616263",
            dom: true,
            ph:  true,
            ctx: "",
            sig: "98a70222f0b8121aa9d30f813d683f809e462b469c7ff87639499bb94e6dae4131f85042463c2a355a2003d062adf5aaa10b8c61e636062aaad11c2a26083406",
        },
    ];

    #[test]
    fn signatures() {
        for tv in TEST_VECTORS.iter() {
            let seed = hex::decode(tv.s).unwrap();
            let Q_enc = hex::decode(tv.Q).unwrap();
            let msg = hex::decode(tv.m).unwrap();
            let ctx = hex::decode(tv.ctx).unwrap();
            let mut sig = [0u8; 64];
            hex::decode_to_slice(tv.sig, &mut sig[..]).unwrap();

            let skey = PrivateKey::from_seed(&seed[..]);
            assert!(&Q_enc[..] == skey.public_key.encode());
            if tv.dom {
                if tv.ph {
                    let mut sh = Sha512::new();
                    sh.update(&msg[..]);
                    let hm = sh.finalize();
                    assert!(skey.sign_ph(&ctx[..], &hm) == sig);
                } else {
                    assert!(skey.sign_ctx(&ctx[..], &msg[..]) == sig);
                }
            } else {
                assert!(skey.sign_raw(&msg[..]) == sig);
            }

            let pkey = PublicKey::decode(&Q_enc[..]).unwrap();
            if tv.dom {
                if tv.ph {
                    let mut sh = Sha512::new();
                    sh.update(&msg[..]);
                    let mut hm = sh.finalize();
                    assert!(pkey.verify_ph(&sig, &ctx[..], &hm));
                    assert!(!pkey.verify_ph(&sig, &[1u8], &hm));
                    hm[42] ^= 0x08;
                    assert!(!pkey.verify_ph(&sig, &ctx[..], &hm));
                } else {
                    assert!(pkey.verify_ctx(&sig, &ctx[..], &msg[..]));
                    assert!(!pkey.verify_ctx(&sig, &[1u8], &msg[..]));
                    assert!(!pkey.verify_ctx(&sig, &ctx[..], &[0u8]));
                }
            } else {
                assert!(pkey.verify_raw(&sig, &msg[..]));
                assert!(!pkey.verify_raw(&sig, &[0u8]));
            }
        }
    }

    #[test]
    fn signatures_frost() {
        // Test vector from draft-irtf-cfrg-frost-05, section C.1
        let d_enc = hex::decode("7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a790c6e13a98304").unwrap();
        let (d, _) = Scalar::decode32(&d_enc[..]);
        let Q_enc = hex::decode("15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040d380fb9738673").unwrap();
        let Q = Point::decode(&Q_enc[..]).unwrap();
        assert!(Point::mulgen(&d).equals(Q) == 0xFFFFFFFF);
        let pkey = PublicKey::from_point(&Q);
        let msg = hex::decode("74657374").unwrap();
        let sig = hex::decode("2b8d9c6995333c5990e3a3dd6568785539d3322f7f0376452487ea35cfda587b75650edb12b1a8619c88ed1f8463d6baeefb18d3fed3c279102fdfecb255fa0e").unwrap();
        assert!(pkey.verify_raw(&sig[..], &msg[..]));
    }

    #[test]
    fn signatures_trunc() {
        let skey = PrivateKey::from_seed(&[0u8; 32]);
        let pkey = skey.public_key;
        for i in 0..10 {
            let mut msg = [0u8; 8];
            msg[..].copy_from_slice(&(i as u64).to_le_bytes());
            let mut sig = skey.sign_raw(&msg);
            sig[63] = 0;
            for rm in 8..33 {
                let n = 512 - rm;
                sig[n >> 3] &= !(0x01u8 << (n & 7));
                let vv = pkey.verify_trunc_raw(&sig, rm, &msg);
                assert!(vv.is_some());
                let sig2 = vv.unwrap();
                assert!(pkey.verify_raw(&sig2, &msg));
                msg[0] ^= 1;
                assert!(pkey.verify_trunc_raw(&sig, rm, &msg).is_none());
                msg[0] ^= 1;
            }
        }
    }

    #[test]
    fn in_subgroup() {
        let T8_enc: [u8; 32] = [
            0x26, 0xE8, 0x95, 0x8F, 0xC2, 0xB2, 0x27, 0xB0,
            0x45, 0xC3, 0xF4, 0x89, 0xF2, 0xEF, 0x98, 0xF0,
            0xD5, 0xDF, 0xAC, 0x05, 0xD3, 0xC6, 0x33, 0x39,
            0xB1, 0x38, 0x02, 0x88, 0x6D, 0x53, 0xFC, 0x85,
        ];
        let T8 = Point::decode(&T8_enc).unwrap();
        let mut sh = Sha256::new();
        for i in 0..30 {
            let mut P = Point::NEUTRAL;
            if i > 0 {
                sh.update(&(i as u64).to_le_bytes());
                let v = sh.finalize_reset();
                P.set_mulgen(&Scalar::decode_reduce(&v[..]));
            }
            assert!(P.is_in_subgroup() == 0xFFFFFFFF);
            for _ in 0..7 {
                P += T8;
                assert!(P.is_in_subgroup() == 0);
            }
        }
    }
}