text: # The reference Ed25519 software is in the public domain.
  #     Source: https://ed25519.cr.yp.to/python/ed25519.py
  #
  
  from binascii import hexlify, unhexlify
  import hashlib
  import operator as _oper
  import sys as _sys
  
  # Set up byte handling for Python 2 or 3
  if _sys.version_info.major == 2:    # pragma: no cover
      int2byte = chr
      range = xrange
  
      def indexbytes(buf, i):
          return ord(buf[i])
  
      def intlist2bytes(l):
          return b"".join(chr(c) for c in l)
  else:                               # pragma: no cover
      indexbytes = _oper.getitem
      intlist2bytes = bytes
      int2byte = _oper.methodcaller("to_bytes", 1, "big")
  
  b = 256
  q = 2**255 - 19
  l = 2**252 + 27742317777372353535851937790883648493
  
  def expmod(b, e, m):
      if e == 0: return 1
      t = expmod(b, e//2, m)**2 % m
      if e & 1: t = (t*b) % m
      return t
  
  def inv(x):
    return expmod(x, q-2, q)
  
  d = -121665 * inv(121666)
  I = expmod(2, (q-1)//4, q)
  
  def xrecover(y):
      xx = (y*y-1) * inv(d*y*y+1)
      x = expmod(xx, (q+3)//8, q)
      if (x*x - xx) % q != 0: x = (x*I) % q
      if x % 2 != 0: x = q-x
      return x
  
  def compress(P):
      zinv = inv(P[2])
      return (P[0] * zinv % q, P[1] * zinv % q)
  
  def decompress(P):
      return (P[0], P[1], 1, P[0]*P[1] % q)
  
  By = 4 * inv(5)
  Bx = xrecover(By)
  B = [Bx%q, By%q]
  
  def edwards(P, Q):
      x1 = P[0]
      y1 = P[1]
      x2 = Q[0]
      y2 = Q[1]
      x3 = (x1*y2+x2*y1) * inv(1+d*x1*x2*y1*y2)
      y3 = (y1*y2+x1*x2) * inv(1-d*x1*x2*y1*y2)
      return [x3%q, y3%q]
  
  def add(P, Q):
      A = (P[1]-P[0])*(Q[1]-Q[0]) % q
      B = (P[1]+P[0])*(Q[1]+Q[0]) % q
      C = 2 * P[3] * Q[3] * d % q
      D = 2 * P[2] * Q[2] % q
      E = B-A
      F = D-C
      G = D+C
      H = B+A
      return (E*F, G*H, F*G, E*H)
  
  def add_compressed(P, Q):
      return compress(add(decompress(P), decompress(Q)))
  
  def scalarmult(P, e):
      if e == 0: return [0, 1]
      Q = scalarmult(P, e//2)
      Q = edwards(Q, Q)
      if e & 1: Q = edwards(Q, P)
      return Q
  
  def encodeint(y):
      bits = [(y >> i) & 1 for i in range(b)]
      return b''.join([int2byte(sum([bits[i*8 + j] << j for j in range(8)])) for i in range(b//8)])
  
  def encodepoint(P):
      x = P[0]
      y = P[1]
      bits = [(y >> i) & 1 for i in range(b-1)] + [x & 1]
      return b''.join([int2byte(sum([bits[i * 8 + j] << j for j in range(8)])) for i in range(b//8)])
  
  def bit(h, i):
      return (indexbytes(h, i//8) >> (i%8)) & 1
  
  def isoncurve(P):
      x = P[0]
      y = P[1]
      return (-x*x + y*y - 1 - d*x*x*y*y) % 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]
      if not isoncurve(P): raise Exception("decoding point that is not on curve")
      return P
  
  # These are unused but let's keep them
  #def H(m):
  #    return hashlib.sha512(m).digest()
  #
  #def Hint(m):
  #    h = H(m)
  #    return sum(2**i * bit(h, i) for i in range(2*b))
  #
  #def publickey(sk):
  #    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 signature(m, sk, pk):
  #    h = H(sk)
  #    a = 2**(b-2) + sum(2**i * bit(h, i) for i in range(3, b-2))
  #    r = Hint(intlist2bytes([indexbytes(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 checkvalid(s, m, pk):
  #    if len(s) != b//4: raise Exception("signature length is wrong")
  #    if len(pk) != b//8: raise Exception("public-key length is wrong")
  #    R = decodepoint(s[0:b//8])
  #    A = decodepoint(pk)
  #    S = decodeint(s[b//8:b//4])
  #    h = Hint(encodepoint(R) + pk + m)
  #    if scalarmult(B, S) != edwards(R, scalarmult(A, h)):
  #        raise Exception("signature does not pass verification")
  
  def public_from_secret(k):
      keyInt = decodeint(k)
      aB = scalarmult(B, keyInt)
      return encodepoint(aB)
  
  def public_from_secret_hex(hk):
      return hexlify(public_from_secret(unhexlify(hk))).decode()
  