hashtocurve.sage

#!/usr/bin/env sage

# Simplified SWU for a = 0 as described in [WB2019] <https://eprint.iacr.org/2019/403> and
# <https://www.ietf.org/archive/id/draft-irtf-cfrg-hash-to-curve-10.html#name-simplified-swu-for-ab-0-2>.

import sys
from math import ceil, log
from struct import pack
from pyblake2 import blake2b
from hashlib import sha256

if sys.version_info[0] == 2:
    range = xrange
    as_byte = ord
else:
    as_byte = lambda x: x

def as_bytes(x):
    # <https://stackoverflow.com/a/29182294/393146>
    return bytes(bytearray(x))

load('squareroottab.sage')

DEBUG = True
VERBOSE = False
OP_COUNT = False

# E: a short Weierstrass elliptic curve
def find_z_sswu(E):
    (0, 0, 0, A, B) = E.a_invariants()
    F = E.base_field()

    R.<x> = F[]                       # Polynomial ring over F
    g = x^3 + F(A) * x + F(B)         # y^2 = g(x) = x^3 + A * x + B

    # <https://www.ietf.org/archive/id/draft-irtf-cfrg-hash-to-curve-10.html#name-finding-z-for-the-shallue-va> is
    # ambiguous about whether to start with ctr = F.gen() or ctr = 1. In fact they are specified to be the same,
    # since F is a prime field.
    # <https://doc.sagemath.org/html/en/reference/finite_rings/sage/rings/finite_rings/finite_field_prime_modn.html#sage.rings.finite_rings.finite_field_prime_modn.FiniteField_prime_modn.gen>,
    # The note in the I-D ("NOTE: if init_ctr=1 fails to find Z, try setting it to F.gen()") could only make a
    # difference for extension fields that are constructed with an explicit modulus.
    ctr = 1
    while True:
        for Z_cand in (F(ctr), F(-ctr)):
            if is_good_Z(F, g, A, B, Z_cand):
                return Z_cand
        ctr += 1

def is_good_Z(F, g, A, B, Z):
    # Criterion 1: Z is non-square in F.
    if Z.is_square():
        return False

    # Criterion 2: Z != -1 in F.
    if Z == F(-1):
        return False

    # Criterion 3: g(x) - Z is irreducible over F.
    if not (g - Z).is_irreducible():
        return False

    # Criterion 4: g(B / (Z * A)) is square in F.
    if not g(F(B) / (Z * F(A))).is_square():
        return False

    return True


# Point in Chudnovsky coordinates (Jacobian with Z^2 and Z^3 cached).
class ChudnovskyPoint:
    def __init__(self, E, x, y, z, z2=None, z3=None):
        if z2 is None:
            z2 = z^2
        if z3 is None:
            z3 = z^3

        if DEBUG:
            (0, 0, 0, A, B) = E.a_invariants()
            assert z2 == z^2
            assert z3 == z^3
            assert y^2 == x^3 + A*x*z^4 + B*z^6

        (self.x, self.y, self.z, self.z2, self.z3) = (x, y, z, z2, z3)

    def add(self, other, E, c):
        (0, 0, 0, A, B) = E.a_invariants()

        # Unified addition on y^2 = x^3 + Ax + B with Chudnovsky input and output.
        (X1, Y1, Z1, Z1_2, Z1_3) = ( self.x,  self.y,  self.z,  self.z2,  self.z3)
        (X2, Y2, Z2, Z2_2, Z2_3) = (other.x, other.y, other.z, other.z2, other.z3)

        assert Z1 != 0 and Z2 != 0

        # <https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-add-2007-bl>
        U1 = c.mul(X1, Z2_2)
        U2 = c.mul(X2, Z1_2)
        S1 = c.mul(Y1, Z2_3)
        S2 = c.mul(Y2, Z1_3)
        H = U2 - U1

        # now unify doubling <https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl>:
        #   XX = c.sqr(X)
        #   YY = c.sqr(Y)
        # YYYY = c.sqr(YY)
        #    S = 2*(c.sqr(X1 + YY) - XX - YYYY)
        #      = 4*X1*YY
        #    M = 3*XX + c.mul(A, Z1_2)
        #   X3 = c.sqr(M) - 2*S
        #   Y3 = M*(S - X3) - 8*YYYY
        #   Z3 = c.sqr(Y1 + Z1) - YY - ZZ
        #
        # with the rest of addition:
        #    I = c.sqr(2*H)
        #    J = c.mul(H, I)
        #    r = 2*(S2-S1)
        #    V = c.mul(U1, I)
        #   X3 = c.sqr(r) - J - 2*V
        #   Y3 = r*(V - X3) - 2*c.mul(S1, J)
        #   Z3 = c.mul((c.sqr(Z1 + Z2) - Z1_2 - Z2_2), H)

        r = 2*(S2-S1)
        X_or_r = select_z_nz(H, X1, r)
        Y_or_H = select_z_nz(H, Y1, H)
        XX_or_rr = c.sqr(X_or_r)
        if DEBUG: assert XX_or_rr == X1^2 if H == 0 else r^2

        YY_or_HH = c.sqr(Y_or_H)
        if DEBUG: assert YY_or_HH == Y1^2 if H == 0 else H^2

        YYY_or_HHH = c.mul(Y_or_H, YY_or_HH)
        if DEBUG: assert YYY_or_HHH == Y1^3 if H == 0 else H^3

        YYYY_or_S1HHH = c.mul(select_z_nz(H, Y_or_H, S1), YYY_or_HHH)
        if DEBUG: assert YYYY_or_S1HHH == Y1^4 if H == 0 else S1*H^3

        S_or_V = 4*c.mul(select_z_nz(H, X1, U1), YY_or_HH)
        if DEBUG: assert S_or_V == 4*X1*Y1^2 if H == 0 else U1 * (4*H^3)

        J = select_z_nz(H, 0, 4*YYY_or_HHH)
        # W = { (Z + Y)^2 - Z^2    = 2*Y*Z + Y^2    for doubling
        #     { (Z1 + Z2)^2 - Z1^2 = 2*Z1*Z2 + Z2^2 for addition
        W = c.sqr(Z1 + select_z_nz(H, Y1, Z2)) - Z1_2
        if DEBUG: assert W == 2*Y1*Z1 + Y1^2 if H == 0 else 2*Z1*Z2 + Z2^2

        # Another option would be to multiply Z1_2 by A if that's faster than squaring.
        Z1_4 = c.sqr(Z1_2)
        AZ1_4_or_Z3 = c.mul(select_z_nz(H, A, H), select_z_nz(H, Z1_4, W - Z2_2))
        if DEBUG: assert AZ1_4_or_Z3 == A*Z1^4 if H == 0 else 2*Z1*Z2*H

        M_or_r = select_z_nz(H, 3*XX_or_rr + AZ1_4_or_Z3, r)
        if DEBUG: assert M_or_r == 3*X1^2 + A*Z1^4 if H == 0 else r

        X3 = c.sqr(M_or_r) - J - 2*S_or_V
        Y3 = c.mul(M_or_r, S_or_V - X3) - 8*YYYY_or_S1HHH
        # If U1 + U2 = 0 then the result is the point at infinity.
        Z3 = select_z_nz(U1 + U2, 0, select_z_nz(H, W - YY_or_HH, AZ1_4_or_Z3))
        Z3_2 = c.sqr(Z3)
        Z3_3 = c.mul(Z3_2, Z3)
        R = ChudnovskyPoint(E, X3, Y3, Z3, Z3_2, Z3_3)

        if DEBUG: assert R.to_sage(E) == self.to_sage(E) + other.to_sage(E)
        return R

    def to_sage(self, E):
        return E((self.x / self.z2, self.y / self.z3))

    def to_affine(self, E):
        return self.to_sage(E).xy()

    def to_jacobian(self):
        return (self.x, self.y, self.z)

    def __repr__(self):
        return "ChudnovskyPoint {\n  0x%064x\n: 0x%064x\n: 0x%064x\n: 0x%064x\n: 0x%064x\n}" % (int(self.x), int(self.y), int(self.z), int(self.z2), int(self.z3))


assert p == 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001
assert q == 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001
Fp = GF(p)
Fq = GF(q)

IsoEp_A = 10949663248450308183708987909873589833737836120165333298109615750520499732811
IsoEq_A = 17413348858408915339762682399132325137863850198379221683097628341577494210225
IsoEp_B = 1265
IsoEq_B = 1265
IsoEp   = EllipticCurve(Fp, [IsoEp_A, IsoEp_B])
IsoEq   = EllipticCurve(Fq, [IsoEq_A, IsoEq_B])
Ep      = EllipticCurve(Fp, [0, 5])
Eq      = EllipticCurve(Fq, [0, 5])

k = 256
Lp = (len(format(p, 'b')) + k + 7) // 8
Lq = (len(format(q, 'b')) + k + 7) // 8
assert Lp == 64 and Lq == 64
CHUNKLEN = Lp

IsoEpZ = find_z_sswu(IsoEp)
IsoEqZ = find_z_sswu(IsoEq)
assert IsoEpZ == Mod(-13, p)
assert IsoEqZ == Mod(-13, q)


def select_z_nz(s, ifz, ifnz):
    # This should be constant-time in a real implementation.
    return ifz if (s == 0) else ifnz

def map_to_curve_simple_swu(F, E, Z, u, c):
    if VERBOSE: print("map_to_curve(0x%064x)" % (u,))

    # would be precomputed
    h = F.g
    (0, 0, 0, A, B) = E.a_invariants()
    mBdivA = -B / A
    BdivZA = B / (Z * A)
    Z2 = Z^2
    assert (Z/h).is_square()
    theta = sqrt(Z/h)

    # 1. tv1 = inv0(Z^2 * u^4 + Z * u^2)
    # 2. x1 = (-B / A) * (1 + tv1)
    # 3. If tv1 == 0, set x1 = B / (Z * A)
    # 4. gx1 = x1^3 + A * x1 + B
    #
    # We use the "Avoiding inversions" optimization in [WB2019, section 4.2]
    # (not to be confused with section 4.3):
    #
    #   here       [WB2019]
    #   -------    ---------------------------------
    #   Z          \xi
    #   u          t
    #   Z * u^2    \xi * t^2 (called u, confusingly)
    #   x1         X_0(t)
    #   x2         X_1(t)
    #   gx1        g(X_0(t))
    #   gx2        g(X_1(t))
    #
    # Using the "here" names:
    #    x1 = N_x1/D = [B*(Z^2 * u^4 + Z * u^2 + 1)] / [-A*(Z^2 * u^4 + Z * u^2]
    #   gx1 = U/V    = [N_x1^3 + A * N_x1 * D^2 + B * D^3] / D^3

    # Z and B are small so we don't count multiplication by them as a mul; A is large.
    Zu2 = Z * c.sqr(u)
    ta = c.sqr(Zu2) + Zu2
    N_x1 = B * (ta + 1)
    D = c.mul(A, select_z_nz(ta, Z, -ta))
    N2_x1 = c.sqr(N_x1)
    D2 = c.sqr(D)
    D3 = c.mul(D2, D)
    U = c.mul(N2_x1 + c.mul(A, D2), N_x1) + B*D3

    if DEBUG:
        x1 = N_x1/D
        gx1 = U/D3
        tv1 = (0 if ta == 0 else 1/ta)
        assert x1 == (BdivZA if tv1 == 0 else mBdivA * (1 + tv1))
        assert gx1 == x1^3 + A * x1 + B

    # 5. x2 = Z * u^2 * x1
    N_x2 = c.mul(Zu2, N_x1)  # same D

    # 6. gx2 = x2^3 + A * x2 + B  [optimized out; see below]
    # 7. If is_square(gx1), set x = x1 and y = sqrt(gx1)
    # 8. Else set x = x2 and y = sqrt(gx2)
    (y1, zero_if_gx1_square) = F.sarkar_divsqrt(U, D3, c)
    if VERBOSE: print("zero_if_gx1_square = %064x" % (zero_if_gx1_square,))

    # This magic also comes from a generalization of [WB2019, section 4.2].
    #
    # The Sarkar square root algorithm with input s gives us a square root of
    # h * s for free when s is not square, where h is a fixed nonsquare.
    # We know that Z/h is a square since both Z and h are nonsquares.
    # Precompute \theta as a square root of Z/h, or choose Z = h so that \theta = 1.
    #
    # We have gx2 = g(Z * u^2 * x1) = Z^3 * u^6 * gx1
    #                               = (Z * u^3)^2 * (Z/h * h * gx1)
    #                               = (Z * \theta * u^3)^2 * (h * gx1)
    #
    # When gx1 is not square, y1 is a square root of h * gx1, and so Z * \theta * u^3 * y1
    # is a square root of gx2. Note that we don't actually need to compute gx2.

    y2 = c.mul(theta, c.mul(Zu2, c.mul(u, y1)))
    if DEBUG and zero_if_gx1_square != 0:
        x2 = N_x2/D
        assert y1^2 == h * gx1, (y1_2, Z, gx1)
        assert y2^2 == x2^3 + A * x2 + B, (y2, x2, A, B)

    N_x = select_z_nz(zero_if_gx1_square, N_x1, N_x2)
    y = select_z_nz(zero_if_gx1_square, y1, y2)

    # 9. If sgn0(u) != sgn0(y), set y = -y
    y = select_z_nz((int(u) % 2) - (int(y) % 2), y, -y)

    if VERBOSE:
        print("num_x = 0x%064x\ndiv   = 0x%064x\ny     = 0x%064x\ndiv3  = 0x%064x" % (int(N_x), int(D), int(y), int(D3)))

    return ChudnovskyPoint(E, c.mul(N_x, D), c.mul(y, D3), D, D2, D3)


# iso_Ep = Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 10949663248450308183708987909873589833737836120165333298109615750520499732811*x + 1265 over Fp
def isop_map_affine(x, y, c):
    c.muls += 2+1+1 + 2+1+1+2
    # batch inversion
    c.muls += 3
    c.invs += 1
    Nx = ((( 6432893846517566412420610278260439325191790329320346825767705947633326140075 *x +
            23989696149150192365340222745168215001509815558210986772351135915822265203574)*x +
            10492611921771203378452795982353351666191589197598957448093274638589204800759)*x +
            12865787693035132824841220556520878650383580658640693651535411895266652280192)
    Dx =  ((                                                                               x +
            13271109177048389296812780941310096270046944650307955939477485891950613419807)*x +
            22768321103861051515190775253992702316905399997697804654926324362758820947460)

    Ny = (((11793638718615538422771118843477472096184948937087302513907460903994431256804 *x +
            11994848074575096182670111372584107500754907779105493386175567957911132601787)*x +
            28823569610051396102362669851238297121581474897215657071023781420043761726004)*x +
             1072148974419594402070101713043406554198631721553391137627950991272221023311) * y
    Dy = (((                                                                               x +
             5432652610908059517272798285879155923388888734491153551238890455750936314542)*x +
            10408918692925056833786833257634153023990087029210292532869619559576527581706)*x +
            28948022309329048855892746252171976963363056481941560715954676764349967629797)

    return (Nx / Dx, Ny / Dy)

# The same isogeny iso_Ep but with input in Chudnovsky coordinates (Jacobian with z^2 and z^3)
# and output in Jacobian <https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html>,
# according to "Avoiding inversions" in [WB2019, section 4.3].
def isop_map_jacobian(P, c):
    (x, y, z, z2, z3) = (P.x, P.y, P.z, P.z2, P.z3)
    z4 = c.sqr(z2)
    z6 = c.sqr(z3)

    if VERBOSE: print("IsoEp { x: 0x%064x, y: 0x%064x, z: 0x%064x }" % (x, y, z))
    Nx = ((( 6432893846517566412420610278260439325191790329320346825767705947633326140075    *x +
            23989696149150192365340222745168215001509815558210986772351135915822265203574*z2)*x +
            10492611921771203378452795982353351666191589197598957448093274638589204800759*z4)*x +
            12865787693035132824841220556520878650383580658640693651535411895266652280192*z6)
    c.muls += 6
    Dx =  ((                                                                              z2 *x +
            13271109177048389296812780941310096270046944650307955939477485891950613419807*z4)*x +
            22768321103861051515190775253992702316905399997697804654926324362758820947460*z6)
    c.muls += 4

    Ny = (((11793638718615538422771118843477472096184948937087302513907460903994431256804    *x +
            11994848074575096182670111372584107500754907779105493386175567957911132601787*z2)*x +
            28823569610051396102362669851238297121581474897215657071023781420043761726004*z4)*x +
             1072148974419594402070101713043406554198631721553391137627950991272221023311*z6) * y
    c.muls += 7
    Dy = (((                                                                                  x +
             5432652610908059517272798285879155923388888734491153551238890455750936314542*z2)*x +
            10408918692925056833786833257634153023990087029210292532869619559576527581706*z4)*x +
            28948022309329048855892746252171976963363056481941560715954676764349967629797*z6) * z3
    c.muls += 6

    if VERBOSE: print("num_x = 0x%064x\ndiv_x = 0x%064x\nnum_y = 0x%064x\ndiv_y = 0x%064x" % (Nx, Dx, Ny, Dy))

    zo = c.mul(Dx, Dy)
    xo = c.mul(c.mul(Nx, Dy), zo)
    yo = c.mul(c.mul(Ny, Dx), c.sqr(zo))

    assert isop_map_affine(x / z2, y / z3, Cost()) == (xo / zo^2, yo / zo^3)
    return (xo, yo, zo)


# iso_Eq = Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 17413348858408915339762682399132325137863850198379221683097628341577494210225*x + 1265 over Fp
def isoq_map_affine(x, y, c):
    c.muls += 2+1+1 + 2+1+1+2
    # batch inversion
    c.muls += 3
    c.invs += 1

    Nx = (((25731575386070265649682441113041757300767161317281464337493104665238544842753 *x +
            13377367003779316331268047403600734872799183885837485433911493934102207511749)*x +
            11064082577423419940183149293632076317553812518550871517841037420579891210813)*x +
            22515128462811482443472135973911537638171266152621281295306466582083726737451)
    Dx =  ((                                                                               x +
             4604213796697651557841441623718706001740429044770779386484474413346415813353)*x +
             9250006497141849826017568406346290940322373181457057184910582871723433210981)

    Ny = ((( 8577191795356755216560813704347252433589053772427154779164368221746181614251 *x +
            21162694656554182593580396827886355918081120183889566406795618341247785229923)*x +
            11620280474556824258112134491145636201000922752744881519070727793732904824884)*x +
            13937936667454727226911322269564285204582212380194126516142098360337545123123) * y
    Dy = (((                                                                               x +
            21380331849711001764708535561664047484292171808126992769566582994216305194078)*x +
            27750019491425549478052705219038872820967119544371171554731748615170299632943)*x +
            28948022309329048855892746252171976963363056481941647379679742748393362947557)

    return (Nx / Dx, Ny / Dy)

# The same isogeny iso_Eq but with input in Chudnovsky coordinates (Jacobian with z^2 and z^3)
# and output in Jacobian <https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html>,
# according to "Avoiding inversions" in [WB2019, section 4.3].
def isoq_map_jacobian(P, c):
    (x, y, z, z2, z3) = (P.x, P.y, P.z, P.z2, P.z3)
    z4 = c.sqr(z2)
    z6 = c.sqr(z3)

    Nx = (((25731575386070265649682441113041757300767161317281464337493104665238544842753    *x +
            13377367003779316331268047403600734872799183885837485433911493934102207511749*z2)*x +
            11064082577423419940183149293632076317553812518550871517841037420579891210813*z4)*x +
            22515128462811482443472135973911537638171266152621281295306466582083726737451*z6)
    c.muls += 6
    Dx =  ((                                                                              z2 *x +
             4604213796697651557841441623718706001740429044770779386484474413346415813353*z4)*x +
             9250006497141849826017568406346290940322373181457057184910582871723433210981*z6)
    c.muls += 4

    Ny = ((( 8577191795356755216560813704347252433589053772427154779164368221746181614251    *x +
            21162694656554182593580396827886355918081120183889566406795618341247785229923*z2)*x +
            11620280474556824258112134491145636201000922752744881519070727793732904824884*z4)*x +
            13937936667454727226911322269564285204582212380194126516142098360337545123123*z6) * y
    c.muls += 7
    Dy = (((                                                                                  x +
            21380331849711001764708535561664047484292171808126992769566582994216305194078*z2)*x +
            27750019491425549478052705219038872820967119544371171554731748615170299632943*z4)*x +
            28948022309329048855892746252171976963363056481941647379679742748393362947557*z6) * z3
    c.muls += 6

    zo = c.mul(Dx, Dy)
    xo = c.mul(c.mul(Nx, Dy), zo)
    yo = c.mul(c.mul(Ny, Dx), c.sqr(zo))

    assert isoq_map_affine(x / z2, y / z3, Cost()) == (xo / zo^2, yo / zo^3)
    return (xo, yo, zo)

def hex_bytes(bs):
    return "[%s]" % (", ".join(["%02x" % (as_byte(b),) for b in bs]),)

def hash(hasher, msg):
    if VERBOSE: print(hex_bytes(msg))
    h = hasher()
    h.update(msg)
    return h.digest()

SHA256 = (sha256, 32, 64)
BLAKE2b = (blake2b, 64, 128)

def hash_to_field(modulus, message, DST, count):
    outlen = int(count * CHUNKLEN)
    uniform_bytes = expand_message_xmd(BLAKE2b, message, DST, outlen)
    if VERBOSE:
        print("uniform_bytes:")
        print(hex_bytes(uniform_bytes[: 64]))
        print(hex_bytes(uniform_bytes[64 :]))

    return [Mod(OS2IP(uniform_bytes[CHUNKLEN*i : CHUNKLEN*(i+1)]), modulus) for i in range(count)]

def OS2IP(bs):
    acc = 0
    for b in bs:
        acc = (acc<<8) + as_byte(b)
    return acc

def expand_message_xmd(H, msg, DST, len_in_bytes):
    assert isinstance(DST, bytes)
    assert isinstance(msg, bytes)

    (hasher, b_in_bytes, r_in_bytes) = H
    assert len(DST) <= 255
    ell = (len_in_bytes + b_in_bytes - 1)//b_in_bytes
    assert ell <= 255

    DST_prime = DST + as_bytes([len(DST)])
    msg_prime = b"\x00"*r_in_bytes + msg + as_bytes([len_in_bytes >> 8, len_in_bytes & 0xFF, 0]) + DST_prime

    if VERBOSE: print("b_0:")
    b_0 = hash(hasher, msg_prime)
    if VERBOSE: print("b_1:")
    b = hash(hasher, b_0 + b"\x01" + DST_prime)
    for i in range(2, ell+1):
        if VERBOSE: print("b_%d:" % (i,))
        b += hash(hasher, as_bytes(as_byte(x) ^^ as_byte(y) for x, y in zip(b_0, b[-64 :])) + as_bytes([i]) + DST_prime)

    return b[: len_in_bytes]


def hash_to_pallas_jacobian(msg, DST):
    c = Cost()
    us = hash_to_field(p, msg, DST, 2)
    if VERBOSE: print("us = [0x%064x, 0x064%x]" % (us[0], us[1]))
    Q0 = map_to_curve_simple_swu(F_p, IsoEp, IsoEpZ, us[0], c)
    Q1 = map_to_curve_simple_swu(F_p, IsoEp, IsoEpZ, us[1], c)

    R = Q0.add(Q1, IsoEp, c)
    # Q0.add(Q0, IsoEp, Cost())  # check that unified addition works

    # no cofactor clearing needed since Pallas is prime-order
    (Px, Py, Pz) = isop_map_jacobian(R, c)
    P = Ep((Px / Pz^2, Py / Pz^3))
    return (P, (Px, Py, Pz), c)

def hash_to_vesta_jacobian(msg, DST):
    c = Cost()
    us = hash_to_field(q, msg, DST, 2)
    if VERBOSE: print("us = [0x%064x, 0x064%x]" % (us[0], us[1]))
    Q0 = map_to_curve_simple_swu(F_q, IsoEq, IsoEqZ, us[0], c)
    Q1 = map_to_curve_simple_swu(F_q, IsoEq, IsoEqZ, us[1], c)

    R = Q0.add(Q1, IsoEq, c)
    # Q0.add(Q0, IsoEq, Cost())  # check that unified addition works

    # no cofactor clearing needed since Vesta is prime-order
    (Px, Py, Pz) = isoq_map_jacobian(R, c)
    P = Eq((Px / Pz^2, Py / Pz^3))
    return (P, (Px, Py, Pz), c)


print("")
P0 = map_to_curve_simple_swu(F_p, IsoEp, IsoEpZ, Mod(0, p), Cost())
print("Fp: map_to_curve_simple_swu(0) = %r" % (P0.to_affine(IsoEp),))
P1 = map_to_curve_simple_swu(F_p, IsoEp, IsoEpZ, Mod(1, p), Cost())
print("Fp: map_to_curve_simple_swu(1) = %r" % (P1.to_affine(IsoEp),))
Pa = map_to_curve_simple_swu(F_p, IsoEp, IsoEpZ, Mod(0x123456789abcdef123456789abcdef123456789abcdef123456789abcdef0123, p), Cost())
print("Fp: map_to_curve_simple_swu(0x123456789abcdef123456789abcdef123456789abcdef123456789abcdef0123) = %r" % (Pa.to_affine(IsoEp),))
print("")

Q0 = map_to_curve_simple_swu(F_q, IsoEq, IsoEqZ, Mod(0, q), Cost())
print("Fq: map_to_curve_simple_swu(0) = %r" % (Q0.to_affine(IsoEq),))
Q1 = map_to_curve_simple_swu(F_q, IsoEq, IsoEqZ, Mod(1, q), Cost())
print("Fq: map_to_curve_simple_swu(1) = %r" % (Q1.to_affine(IsoEq),))
Qa = map_to_curve_simple_swu(F_q, IsoEq, IsoEqZ, Mod(0x123456789abcdef123456789abcdef123456789abcdef123456789abcdef0123, q), Cost())
print("Fq: map_to_curve_simple_swu(0x123456789abcdef123456789abcdef123456789abcdef123456789abcdef0123) = %r" % (Qa.to_affine(IsoEq),))
print("")

xyz = isop_map_jacobian(
    ChudnovskyPoint(IsoEp,
                    Mod(0x0a881e4d556945aa9c6cfc47bce1aba6593c053e5e2337adc37f111df5c4419e, p),
                    Mod(0x035e5c8a06d5cfb4a62eec46f662cb4e6979f7f2b0acf188f234e04434502b47, p),
                    Mod(0x3af37975b09331256ac4e343558dcbf3575baa717958ef1f11ab791d4fb6f6b4, p)),
    Cost())
print("Ep { x: 0x%064x, y: 0x%064x, z: 0x%064x }" % xyz)
print("")

# This test vector is chosen so that the first map_to_curve_simple_swu takes the gx1 square
# "branch" and the second takes the gx1 non-square "branch" (opposite to the Vesta test vector).
(P, xyz, c) = hash_to_pallas_jacobian(b"Trans rights now!", b"z.cash:test-pallas_XMD:BLAKE2b_SSWU_RO_")
print("Ep { x: 0x%064x, y: 0x%064x, z: 0x%064x }" % xyz)
print("")

# This test vector is chosen so that the first map_to_curve_simple_swu takes the gx1 non-square
# "branch" and the second takes the gx1 square "branch" (opposite to the Pallas test vector).
(P, xyz, c) = hash_to_vesta_jacobian(b"hello", b"z.cash:test-vesta_XMD:BLAKE2b_SSWU_RO_")
print("Eq { x: 0x%064x, y: 0x%064x, z: 0x%064x }" % xyz)
print("")

if OP_COUNT:
    iters = 100
    for i in range(iters):
        (P, xyz, cost) = hash_to_pallas_jacobian(pack(">I", i), b"z.cash:test-pallas_XMD:BLAKE2b_SSWU_RO_")
        print(xyz, cost)