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368 lines
13 KiB
368 lines
13 KiB
# MIT License
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#
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# Copyright (c) 2015 Brian Warner and other contributors
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# Permission is hereby granted, free of charge, to any person obtaining a copy
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# of this software and associated documentation files (the "Software"), to deal
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# in the Software without restriction, including without limitation the rights
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# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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# copies of the Software, and to permit persons to whom the Software is
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# furnished to do so, subject to the following conditions:
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#
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# The above copyright notice and this permission notice shall be included in all
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# copies or substantial portions of the Software.
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#
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# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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# SOFTWARE.
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import binascii, hashlib, itertools
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Q = 2**255 - 19
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L = 2**252 + 27742317777372353535851937790883648493
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def inv(x):
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return pow(x, Q-2, Q)
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d = -121665 * inv(121666)
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I = pow(2,(Q-1)//4,Q)
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def xrecover(y):
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xx = (y*y-1) * inv(d*y*y+1)
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x = pow(xx,(Q+3)//8,Q)
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if (x*x - xx) % Q != 0: x = (x*I) % Q
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if x % 2 != 0: x = Q-x
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return x
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By = 4 * inv(5)
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Bx = xrecover(By)
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B = [Bx % Q,By % Q]
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# Extended Coordinates: x=X/Z, y=Y/Z, x*y=T/Z
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# http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
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def xform_affine_to_extended(pt):
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(x, y) = pt
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return (x%Q, y%Q, 1, (x*y)%Q) # (X,Y,Z,T)
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def xform_extended_to_affine(pt):
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(x, y, z, _) = pt
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return ((x*inv(z))%Q, (y*inv(z))%Q)
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def double_element(pt): # extended->extended
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# dbl-2008-hwcd
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(X1, Y1, Z1, _) = pt
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A = (X1*X1)
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B = (Y1*Y1)
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C = (2*Z1*Z1)
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D = (-A) % Q
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J = (X1+Y1) % Q
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E = (J*J-A-B) % Q
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G = (D+B) % Q
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F = (G-C) % Q
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H = (D-B) % Q
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X3 = (E*F) % Q
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Y3 = (G*H) % Q
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Z3 = (F*G) % Q
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T3 = (E*H) % Q
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return (X3, Y3, Z3, T3)
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def add_elements(pt1, pt2): # extended->extended
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# add-2008-hwcd-3 . Slightly slower than add-2008-hwcd-4, but -3 is
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# unified, so it's safe for general-purpose addition
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(X1, Y1, Z1, T1) = pt1
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(X2, Y2, Z2, T2) = pt2
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A = ((Y1-X1)*(Y2-X2)) % Q
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B = ((Y1+X1)*(Y2+X2)) % Q
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C = T1*(2*d)*T2 % Q
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D = Z1*2*Z2 % Q
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E = (B-A) % Q
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F = (D-C) % Q
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G = (D+C) % Q
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H = (B+A) % Q
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X3 = (E*F) % Q
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Y3 = (G*H) % Q
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T3 = (E*H) % Q
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Z3 = (F*G) % Q
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return (X3, Y3, Z3, T3)
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def scalarmult_element_safe_slow(pt, n):
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# this form is slightly slower, but tolerates arbitrary points, including
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# those which are not in the main 1*L subgroup. This includes points of
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# order 1 (the neutral element Zero), 2, 4, and 8.
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assert n >= 0
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if n==0:
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return xform_affine_to_extended((0,1))
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_ = double_element(scalarmult_element_safe_slow(pt, n>>1))
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return add_elements(_, pt) if n&1 else _
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def _add_elements_nonunfied(pt1, pt2): # extended->extended
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# add-2008-hwcd-4 : NOT unified, only for pt1!=pt2. About 10% faster than
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# the (unified) add-2008-hwcd-3, and safe to use inside scalarmult if you
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# aren't using points of order 1/2/4/8
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(X1, Y1, Z1, T1) = pt1
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(X2, Y2, Z2, T2) = pt2
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A = ((Y1-X1)*(Y2+X2)) % Q
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B = ((Y1+X1)*(Y2-X2)) % Q
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C = (Z1*2*T2) % Q
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D = (T1*2*Z2) % Q
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E = (D+C) % Q
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F = (B-A) % Q
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G = (B+A) % Q
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H = (D-C) % Q
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X3 = (E*F) % Q
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Y3 = (G*H) % Q
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Z3 = (F*G) % Q
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T3 = (E*H) % Q
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return (X3, Y3, Z3, T3)
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def scalarmult_element(pt, n): # extended->extended
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# This form only works properly when given points that are a member of
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# the main 1*L subgroup. It will give incorrect answers when called with
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# the points of order 1/2/4/8, including point Zero. (it will also work
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# properly when given points of order 2*L/4*L/8*L)
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assert n >= 0
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if n==0:
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return xform_affine_to_extended((0,1))
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_ = double_element(scalarmult_element(pt, n>>1))
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return _add_elements_nonunfied(_, pt) if n&1 else _
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# points are encoded as 32-bytes little-endian, b255 is sign, b2b1b0 are 0
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def encodepoint(P):
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x = P[0]
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y = P[1]
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# MSB of output equals x.b0 (=x&1)
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# rest of output is little-endian y
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assert 0 <= y < (1<<255) # always < 0x7fff..ff
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if x & 1:
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y += 1<<255
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return binascii.unhexlify("%064x" % y)[::-1]
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def isoncurve(P):
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x = P[0]
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y = P[1]
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return (-x*x + y*y - 1 - d*x*x*y*y) % Q == 0
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class NotOnCurve(Exception):
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pass
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def decodepoint(s):
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unclamped = int(binascii.hexlify(s[:32][::-1]), 16)
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clamp = (1 << 255) - 1
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y = unclamped & clamp # clear MSB
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x = xrecover(y)
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if bool(x & 1) != bool(unclamped & (1<<255)): x = Q-x
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P = [x,y]
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if not isoncurve(P): raise NotOnCurve("decoding point that is not on curve")
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return P
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# scalars are encoded as 32-bytes little-endian
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def bytes_to_scalar(s):
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assert len(s) == 32, len(s)
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return int(binascii.hexlify(s[::-1]), 16)
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def bytes_to_clamped_scalar(s):
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# Ed25519 private keys clamp the scalar to ensure two things:
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# 1: integer value is in L/2 .. L, to avoid small-logarithm
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# non-wraparaound
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# 2: low-order 3 bits are zero, so a small-subgroup attack won't learn
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# any information
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# set the top two bits to 01, and the bottom three to 000
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a_unclamped = bytes_to_scalar(s)
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AND_CLAMP = (1<<254) - 1 - 7
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OR_CLAMP = (1<<254)
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a_clamped = (a_unclamped & AND_CLAMP) | OR_CLAMP
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return a_clamped
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def random_scalar(entropy_f): # 0..L-1 inclusive
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# reduce the bias to a safe level by generating 256 extra bits
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oversized = int(binascii.hexlify(entropy_f(32+32)), 16)
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return oversized % L
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def password_to_scalar(pw):
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oversized = hashlib.sha512(pw).digest()
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return int(binascii.hexlify(oversized), 16) % L
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def scalar_to_bytes(y):
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y = y % L
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assert 0 <= y < 2**256
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return binascii.unhexlify("%064x" % y)[::-1]
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# Elements, of various orders
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def is_extended_zero(XYTZ):
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# catch Zero
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(X, Y, Z, T) = XYTZ
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Y = Y % Q
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Z = Z % Q
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if X==0 and Y==Z and Y!=0:
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return True
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return False
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class ElementOfUnknownGroup:
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# This is used for points of order 2,4,8,2*L,4*L,8*L
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def __init__(self, XYTZ):
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assert isinstance(XYTZ, tuple)
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assert len(XYTZ) == 4
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self.XYTZ = XYTZ
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def add(self, other):
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if not isinstance(other, ElementOfUnknownGroup):
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raise TypeError("elements can only be added to other elements")
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sum_XYTZ = add_elements(self.XYTZ, other.XYTZ)
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if is_extended_zero(sum_XYTZ):
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return Zero
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return ElementOfUnknownGroup(sum_XYTZ)
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def scalarmult(self, s):
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if isinstance(s, ElementOfUnknownGroup):
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raise TypeError("elements cannot be multiplied together")
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assert s >= 0
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product = scalarmult_element_safe_slow(self.XYTZ, s)
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return ElementOfUnknownGroup(product)
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def to_bytes(self):
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return encodepoint(xform_extended_to_affine(self.XYTZ))
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def __eq__(self, other):
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return self.to_bytes() == other.to_bytes()
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def __ne__(self, other):
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return not self == other
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class Element(ElementOfUnknownGroup):
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# this only holds elements in the main 1*L subgroup. It never holds Zero,
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# or elements of order 1/2/4/8, or 2*L/4*L/8*L.
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def add(self, other):
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if not isinstance(other, ElementOfUnknownGroup):
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raise TypeError("elements can only be added to other elements")
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sum_element = ElementOfUnknownGroup.add(self, other)
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if sum_element is Zero:
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return sum_element
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if isinstance(other, Element):
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# adding two subgroup elements results in another subgroup
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# element, or Zero, and we've already excluded Zero
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return Element(sum_element.XYTZ)
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# not necessarily a subgroup member, so assume not
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return sum_element
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def scalarmult(self, s):
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if isinstance(s, ElementOfUnknownGroup):
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raise TypeError("elements cannot be multiplied together")
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# scalarmult of subgroup members can be done modulo the subgroup
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# order, and using the faster non-unified function.
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s = s % L
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# scalarmult(s=0) gets you Zero
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if s == 0:
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return Zero
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# scalarmult(s=1) gets you self, which is a subgroup member
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# scalarmult(s<grouporder) gets you a different subgroup member
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return Element(scalarmult_element(self.XYTZ, s))
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# negation and subtraction only make sense for the main subgroup
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def negate(self):
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# slow. Prefer e.scalarmult(-pw) to e.scalarmult(pw).negate()
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return Element(scalarmult_element(self.XYTZ, L-2))
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def subtract(self, other):
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return self.add(other.negate())
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class _ZeroElement(ElementOfUnknownGroup):
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def add(self, other):
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return other # zero+anything = anything
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def scalarmult(self, s):
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return self # zero*anything = zero
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def negate(self):
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return self # -zero = zero
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def subtract(self, other):
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return self.add(other.negate())
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Base = Element(xform_affine_to_extended(B))
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Zero = _ZeroElement(xform_affine_to_extended((0,1))) # the neutral (identity) element
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_zero_bytes = Zero.to_bytes()
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def arbitrary_element(seed): # unknown DL
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# TODO: if we don't need uniformity, maybe use just sha256 here?
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hseed = hashlib.sha512(seed).digest()
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y = int(binascii.hexlify(hseed), 16) % Q
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# we try successive Y values until we find a valid point
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for plus in itertools.count(0):
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y_plus = (y + plus) % Q
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x = xrecover(y_plus)
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Pa = [x,y_plus] # no attempt to use both "positive" and "negative" X
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# only about 50% of Y coordinates map to valid curve points (I think
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# the other half give you points on the "twist").
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if not isoncurve(Pa):
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continue
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P = ElementOfUnknownGroup(xform_affine_to_extended(Pa))
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# even if the point is on our curve, it may not be in our particular
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# (order=L) subgroup. The curve has order 8*L, so an arbitrary point
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# could have order 1,2,4,8,1*L,2*L,4*L,8*L (everything which divides
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# the group order).
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# [I MAY BE COMPLETELY WRONG ABOUT THIS, but my brief statistical
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# tests suggest it's not too far off] There are phi(x) points with
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# order x, so:
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# 1 element of order 1: [(x=0,y=1)=Zero]
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# 1 element of order 2 [(x=0,y=-1)]
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# 2 elements of order 4
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# 4 elements of order 8
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# L-1 elements of order L (including Base)
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# L-1 elements of order 2*L
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# 2*(L-1) elements of order 4*L
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# 4*(L-1) elements of order 8*L
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# So 50% of random points will have order 8*L, 25% will have order
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# 4*L, 13% order 2*L, and 13% will have our desired order 1*L (and a
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# vanishingly small fraction will have 1/2/4/8). If we multiply any
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# of the 8*L points by 2, we're sure to get an 4*L point (and
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# multiplying a 4*L point by 2 gives us a 2*L point, and so on).
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# Multiplying a 1*L point by 2 gives us a different 1*L point. So
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# multiplying by 8 gets us from almost any point into a uniform point
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# on the correct 1*L subgroup.
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P8 = P.scalarmult(8)
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# if we got really unlucky and picked one of the 8 low-order points,
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# multiplying by 8 will get us to the identity (Zero), which we check
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# for explicitly.
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if is_extended_zero(P8.XYTZ):
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continue
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# Test that we're finally in the right group. We want to scalarmult
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# by L, and we want to *not* use the trick in Group.scalarmult()
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# which does x%L, because that would bypass the check we care about.
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# P is still an _ElementOfUnknownGroup, which doesn't use x%L because
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# that's not correct for points outside the main group.
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assert is_extended_zero(P8.scalarmult(L).XYTZ)
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return Element(P8.XYTZ)
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# never reached
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def bytes_to_unknown_group_element(bytes):
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# this accepts all elements, including Zero and wrong-subgroup ones
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if bytes == _zero_bytes:
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return Zero
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XYTZ = xform_affine_to_extended(decodepoint(bytes))
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return ElementOfUnknownGroup(XYTZ)
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def bytes_to_element(bytes):
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# this strictly only accepts elements in the right subgroup
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P = bytes_to_unknown_group_element(bytes)
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if P is Zero:
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raise ValueError("element was Zero")
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if not is_extended_zero(P.scalarmult(L).XYTZ):
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raise ValueError("element is not in the right group")
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# the point is in the expected 1*L subgroup, not in the 2/4/8 groups,
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# or in the 2*L/4*L/8*L groups. Promote it to a correct-group Element.
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return Element(P.XYTZ)
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