324 lines
6.9 KiB
Python
324 lines
6.9 KiB
Python
# Ed25519 digital signatures
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# Based on https://ed25519.cr.yp.to/python/ed25519.py
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# See also https://ed25519.cr.yp.to/software.html
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# Adapted by Ron Garret
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# Sped up considerably using coordinate transforms found on:
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# https://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
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# Specifically add-2008-hwcd-4 and dbl-2008-hwcd
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import hashlib
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import random
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try: # pragma nocover
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unicode
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PY3 = False
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def asbytes(b):
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"""Convert array of integers to byte string"""
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return ''.join(chr(x) for x in b)
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def joinbytes(b):
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"""Convert array of bytes to byte string"""
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return ''.join(b)
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def bit(h, i):
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"""Return i'th bit of bytestring h"""
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return (ord(h[i // 8]) >> (i % 8)) & 1
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except NameError: # pragma nocover
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PY3 = True
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asbytes = bytes
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joinbytes = bytes
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def bit(h, i):
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return (h[i // 8] >> (i % 8)) & 1
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b = 256
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q = 2 ** 255 - 19
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l = 2 ** 252 + 27742317777372353535851937790883648493 # noqa: E741
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def H(m):
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return hashlib.sha512(m).digest()
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def expmod(b, e, m):
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if e == 0:
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return 1
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t = expmod(b, e // 2, m) ** 2 % m
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if e & 1:
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t = (t * b) % m
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return t
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# Can probably get some extra speedup here by replacing this with
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# an extended-euclidean, but performance seems OK without that
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def inv(x):
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return expmod(x, q - 2, q)
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d = -121665 * inv(121666)
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I = expmod(2, (q - 1) // 4, q) # noqa: E741
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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 = expmod(xx, (q + 3) // 8, q)
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if (x * x - xx) % q != 0:
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x = (x * I) % q
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if x % 2 != 0:
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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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# def edwards(P,Q):
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# x1 = P[0]
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# y1 = P[1]
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# x2 = Q[0]
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# y2 = Q[1]
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# x3 = (x1*y2+x2*y1) * inv(1+d*x1*x2*y1*y2)
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# y3 = (y1*y2+x1*x2) * inv(1-d*x1*x2*y1*y2)
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# return (x3 % q,y3 % q)
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# def scalarmult(P,e):
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# if e == 0: return [0,1]
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# Q = scalarmult(P,e/2)
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# Q = edwards(Q,Q)
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# if e & 1: Q = edwards(Q,P)
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# return Q
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# Faster (!) version based on:
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# https://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
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def xpt_add(pt1, pt2):
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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 xpt_double(pt):
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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 pt_xform(pt):
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(x, y) = pt
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return x, y, 1, (x * y) % q
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def pt_unxform(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 xpt_mult(pt, n):
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if n == 0:
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return pt_xform((0, 1))
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_ = xpt_double(xpt_mult(pt, n >> 1))
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return xpt_add(_, pt) if n & 1 else _
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def scalarmult(pt, e):
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return pt_unxform(xpt_mult(pt_xform(pt), e))
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def encodeint(y):
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bits = [(y >> i) & 1 for i in range(b)]
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e = [(sum([bits[i * 8 + j] << j for j in range(8)]))
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for i in range(b // 8)]
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return asbytes(e)
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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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bits = [(y >> i) & 1 for i in range(b - 1)] + [x & 1]
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e = [(sum([bits[i * 8 + j] << j for j in range(8)]))
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for i in range(b // 8)]
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return asbytes(e)
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def publickey(sk):
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h = H(sk)
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a = 2 ** (b - 2) + sum(2 ** i * bit(h, i) for i in range(3, b - 2))
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A = scalarmult(B, a)
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return encodepoint(A)
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def Hint(m):
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h = H(m)
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return sum(2 ** i * bit(h, i) for i in range(2 * b))
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def signature(m, sk, pk):
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h = H(sk)
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a = 2 ** (b - 2) + sum(2 ** i * bit(h, i) for i in range(3, b - 2))
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inter = joinbytes([h[i] for i in range(b // 8, b // 4)])
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r = Hint(inter + m)
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R = scalarmult(B, r)
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S = (r + Hint(encodepoint(R) + pk + m) * a) % l
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return encodepoint(R) + encodeint(S)
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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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def decodeint(s):
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return sum(2 ** i * bit(s, i) for i in range(0, b))
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def decodepoint(s):
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y = sum(2 ** i * bit(s, i) for i in range(0, b - 1))
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x = xrecover(y)
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if x & 1 != bit(s, b - 1):
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x = q - x
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P = [x, y]
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if not isoncurve(P):
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raise Exception("decoding point that is not on curve")
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return P
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def checkvalid(s, m, pk):
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if len(s) != b // 4:
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raise Exception("signature length is wrong")
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if len(pk) != b // 8:
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raise Exception("public-key length is wrong")
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R = decodepoint(s[0:b // 8])
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A = decodepoint(pk)
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S = decodeint(s[b // 8:b // 4])
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h = Hint(encodepoint(R) + pk + m)
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v1 = scalarmult(B, S)
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# v2 = edwards(R,scalarmult(A,h))
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v2 = pt_unxform(xpt_add(pt_xform(R), pt_xform(scalarmult(A, h))))
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return v1 == v2
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##########################################################
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#
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# Curve25519 reference implementation by Matthew Dempsky, from:
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# https://cr.yp.to/highspeed/naclcrypto-20090310.pdf
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# P = 2 ** 255 - 19
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P = q
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A = 486662
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# def expmod(b, e, m):
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# if e == 0: return 1
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# t = expmod(b, e / 2, m) ** 2 % m
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# if e & 1: t = (t * b) % m
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# return t
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# def inv(x): return expmod(x, P - 2, P)
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def add(n, m, d):
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(xn, zn) = n
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(xm, zm) = m
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(xd, zd) = d
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x = 4 * (xm * xn - zm * zn) ** 2 * zd
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z = 4 * (xm * zn - zm * xn) ** 2 * xd
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return (x % P, z % P)
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def double(n):
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(xn, zn) = n
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x = (xn ** 2 - zn ** 2) ** 2
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z = 4 * xn * zn * (xn ** 2 + A * xn * zn + zn ** 2)
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return (x % P, z % P)
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def curve25519(n, base=9):
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one = (base, 1)
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two = double(one)
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# f(m) evaluates to a tuple
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# containing the mth multiple and the
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# (m+1)th multiple of base.
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def f(m):
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if m == 1:
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return (one, two)
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(pm, pm1) = f(m // 2)
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if m & 1:
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return (add(pm, pm1, one), double(pm1))
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return (double(pm), add(pm, pm1, one))
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((x, z), _) = f(n)
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return (x * inv(z)) % P
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def genkey(n=0):
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n = n or random.randint(0, P)
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n &= ~7
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n &= ~(128 << 8 * 31)
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n |= 64 << 8 * 31
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return n
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# def str2int(s):
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# return int(hexlify(s), 16)
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# # return sum(ord(s[i]) << (8 * i) for i in range(32))
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#
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# def int2str(n):
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# return unhexlify("%x" % n)
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# # return ''.join([chr((n >> (8 * i)) & 255) for i in range(32)])
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#################################################
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def dsa_test():
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import os
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msg = str(random.randint(q, q + q)).encode('utf-8')
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sk = os.urandom(32)
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pk = publickey(sk)
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sig = signature(msg, sk, pk)
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return checkvalid(sig, msg, pk)
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def dh_test():
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sk1 = genkey()
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sk2 = genkey()
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return curve25519(sk1, curve25519(sk2)) == curve25519(sk2, curve25519(sk1))
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