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+# ed25519.py - Optimized version of the reference implementation of Ed25519
+#
+# Written in 2011? by Daniel J. Bernstein <djb@cr.yp.to>
+#            2013 by Donald Stufft <donald@stufft.io>
+#            2013 by Alex Gaynor <alex.gaynor@gmail.com>
+#            2013 by Greg Price <price@mit.edu>
+#
+# To the extent possible under law, the author(s) have dedicated all copyright
+# and related and neighboring rights to this software to the public domain
+# worldwide. This software is distributed without any warranty.
+#
+# You should have received a copy of the CC0 Public Domain Dedication along
+# with this software. If not, see
+# <http://creativecommons.org/publicdomain/zero/1.0/>.
+#
+# Downloaded from https://raw.githubusercontent.com/pyca/ed25519/main/ed25519.py
+# on April 1, 2024.
+
+"""
+NB: This code is not safe for use with secret keys or secret data.
+The only safe use of this code is for verifying signatures on public messages.
+
+Functions for computing the public key of a secret key and for signing
+a message are included, namely publickey_unsafe and signature_unsafe,
+for testing purposes only.
+
+The root of the problem is that Python's long-integer arithmetic is
+not designed for use in cryptography.  Specifically, it may take more
+or less time to execute an operation depending on the values of the
+inputs, and its memory access patterns may also depend on the inputs.
+This opens it to timing and cache side-channel attacks which can
+disclose data to an attacker.  We rely on Python's long-integer
+arithmetic, so we cannot handle secrets without risking their disclosure.
+"""
+
+import hashlib
+
+
+__version__ = "1.0.dev0"
+
+
+b = 256
+q = 2**255 - 19
+l = 2**252 + 27742317777372353535851937790883648493
+
+
+def H(m):
+    return hashlib.sha512(m).digest()
+
+
+def pow2(x, p):
+    """== pow(x, 2**p, q)"""
+    while p > 0:
+        x = x * x % q
+        p -= 1
+    return x
+
+
+def inv(z):
+    r"""$= z^{-1} \mod q$, for z != 0"""
+    # Adapted from curve25519_athlon.c in djb's Curve25519.
+    z2 = z * z % q  # 2
+    z9 = pow2(z2, 2) * z % q  # 9
+    z11 = z9 * z2 % q  # 11
+    z2_5_0 = (z11 * z11) % q * z9 % q  # 31 == 2^5 - 2^0
+    z2_10_0 = pow2(z2_5_0, 5) * z2_5_0 % q  # 2^10 - 2^0
+    z2_20_0 = pow2(z2_10_0, 10) * z2_10_0 % q  # ...
+    z2_40_0 = pow2(z2_20_0, 20) * z2_20_0 % q
+    z2_50_0 = pow2(z2_40_0, 10) * z2_10_0 % q
+    z2_100_0 = pow2(z2_50_0, 50) * z2_50_0 % q
+    z2_200_0 = pow2(z2_100_0, 100) * z2_100_0 % q
+    z2_250_0 = pow2(z2_200_0, 50) * z2_50_0 % q  # 2^250 - 2^0
+    return pow2(z2_250_0, 5) * z11 % q  # 2^255 - 2^5 + 11 = q - 2
+
+
+d = -121665 * inv(121666) % q
+I = pow(2, (q - 1) // 4, q)
+
+
+def xrecover(y):
+    xx = (y * y - 1) * inv(d * y * y + 1)
+    x = pow(xx, (q + 3) // 8, q)
+
+    if (x * x - xx) % q != 0:
+        x = (x * I) % q
+
+    if x % 2 != 0:
+        x = q - x
+
+    return x
+
+
+By = 4 * inv(5)
+Bx = xrecover(By)
+B = (Bx % q, By % q, 1, (Bx * By) % q)
+ident = (0, 1, 1, 0)
+
+
+def edwards_add(P, Q):
+    # This is formula sequence 'addition-add-2008-hwcd-3' from
+    # http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
+    (x1, y1, z1, t1) = P
+    (x2, y2, z2, t2) = Q
+
+    a = (y1 - x1) * (y2 - x2) % q
+    b = (y1 + x1) * (y2 + x2) % q
+    c = t1 * 2 * d * t2 % q
+    dd = z1 * 2 * z2 % q
+    e = b - a
+    f = dd - c
+    g = dd + c
+    h = b + a
+    x3 = e * f
+    y3 = g * h
+    t3 = e * h
+    z3 = f * g
+
+    return (x3 % q, y3 % q, z3 % q, t3 % q)
+
+
+def edwards_double(P):
+    # This is formula sequence 'dbl-2008-hwcd' from
+    # http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
+    (x1, y1, z1, t1) = P
+
+    a = x1 * x1 % q
+    b = y1 * y1 % q
+    c = 2 * z1 * z1 % q
+    # dd = -a
+    e = ((x1 + y1) * (x1 + y1) - a - b) % q
+    g = -a + b  # dd + b
+    f = g - c
+    h = -a - b  # dd - b
+    x3 = e * f
+    y3 = g * h
+    t3 = e * h
+    z3 = f * g
+
+    return (x3 % q, y3 % q, z3 % q, t3 % q)
+
+
+def scalarmult(P, e):
+    if e == 0:
+        return ident
+    Q = scalarmult(P, e // 2)
+    Q = edwards_double(Q)
+    if e & 1:
+        Q = edwards_add(Q, P)
+    return Q
+
+
+# Bpow[i] == scalarmult(B, 2**i)
+Bpow = []
+
+
+def make_Bpow():
+    P = B
+    for i in range(253):
+        Bpow.append(P)
+        P = edwards_double(P)
+
+
+make_Bpow()
+
+
+def scalarmult_B(e):
+    """
+    Implements scalarmult(B, e) more efficiently.
+    """
+    # scalarmult(B, l) is the identity
+    e = e % l
+    P = ident
+    for i in range(253):
+        if e & 1:
+            P = edwards_add(P, Bpow[i])
+        e = e // 2
+    assert e == 0, e
+    return P
+
+
+def encodeint(y):
+    bits = [(y >> i) & 1 for i in range(b)]
+    return bytes(
+        [sum([bits[i * 8 + j] << j for j in range(8)]) for i in range(b // 8)]
+    )
+
+
+def encodepoint(P):
+    (x, y, z, t) = P
+    zi = inv(z)
+    x = (x * zi) % q
+    y = (y * zi) % q
+    bits = [(y >> i) & 1 for i in range(b - 1)] + [x & 1]
+    return bytes(
+        [sum([bits[i * 8 + j] << j for j in range(8)]) for i in range(b // 8)]
+    )
+
+
+def bit(h, i):
+    return (h[i // 8] >> (i % 8)) & 1
+
+
+def publickey_unsafe(sk):
+    """
+    Not safe to use with secret keys or secret data.
+
+    See module docstring.  This function should be used for testing only.
+    """
+    h = H(sk)
+    a = 2 ** (b - 2) + sum(2**i * bit(h, i) for i in range(3, b - 2))
+    A = scalarmult_B(a)
+    return encodepoint(A)
+
+
+def Hint(m):
+    h = H(m)
+    return sum(2**i * bit(h, i) for i in range(2 * b))
+
+
+def signature_unsafe(m, sk, pk):
+    """
+    Not safe to use with secret keys or secret data.
+
+    See module docstring.  This function should be used for testing only.
+    """
+    h = H(sk)
+    a = 2 ** (b - 2) + sum(2**i * bit(h, i) for i in range(3, b - 2))
+    r = Hint(bytes([h[j] for j in range(b // 8, b // 4)]) + m)
+    R = scalarmult_B(r)
+    S = (r + Hint(encodepoint(R) + pk + m) * a) % l
+    return encodepoint(R) + encodeint(S)
+
+
+def isoncurve(P):
+    (x, y, z, t) = P
+    return (
+        z % q != 0
+        and x * y % q == z * t % q
+        and (y * y - x * x - z * z - d * t * t) % q == 0
+    )
+
+
+def decodeint(s):
+    return sum(2**i * bit(s, i) for i in range(0, b))
+
+
+def decodepoint(s):
+    y = sum(2**i * bit(s, i) for i in range(0, b - 1))
+    x = xrecover(y)
+    if x & 1 != bit(s, b - 1):
+        x = q - x
+    P = (x, y, 1, (x * y) % q)
+    if not isoncurve(P):
+        raise ValueError("decoding point that is not on curve")
+    return P
+
+
+class SignatureMismatch(Exception):
+    pass
+
+
+def checkvalid(s, m, pk):
+    """
+    Not safe to use when any argument is secret.
+
+    See module docstring.  This function should be used only for
+    verifying public signatures of public messages.
+    """
+    if len(s) != b // 4:
+        raise ValueError("signature length is wrong")
+
+    if len(pk) != b // 8:
+        raise ValueError("public-key length is wrong")
+
+    R = decodepoint(s[: b // 8])
+    A = decodepoint(pk)
+    S = decodeint(s[b // 8 : b // 4])
+    h = Hint(encodepoint(R) + pk + m)
+
+    (x1, y1, z1, t1) = P = scalarmult_B(S)
+    (x2, y2, z2, t2) = Q = edwards_add(R, scalarmult(A, h))
+
+    if (
+        not isoncurve(P)
+        or not isoncurve(Q)
+        or (x1 * z2 - x2 * z1) % q != 0
+        or (y1 * z2 - y2 * z1) % q != 0
+    ):
+        raise SignatureMismatch("signature does not pass verification")