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ed25519.py
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ed25519.py
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#################################################################################
# Source https://datatracker.ietf.org/doc/html/rfc8032#section-6
#################################################################################
## First, some preliminaries that will be needed.
import hashlib
from .tmac import tmac
def sha512(s):
return hashlib.sha512(s).digest()
# Base field Z_p
p = 2**255 - 19
def modp_inv(x):
return pow(x, p-2, p)
# Curve constant
d = -121665 * modp_inv(121666) % p
# Group order
q = 2**252 + 27742317777372353535851937790883648493
def sha512_modq(s):
digest = sha512(s)
return int.from_bytes(digest, "little") % q
## Then follows functions to perform point operations.
# Points are represented as tuples (X, Y, Z, T) of extended
# coordinates, with x = X/Z, y = Y/Z, x*y = T/Z
def point_add(P, Q):
A, B = (P[1]-P[0]) * (Q[1]-Q[0]) % p, (P[1]+P[0]) * (Q[1]+Q[0]) % p;
C, D = 2 * P[3] * Q[3] * d % p, 2 * P[2] * Q[2] % p;
E, F, G, H = B-A, D-C, D+C, B+A;
return (E*F, G*H, F*G, E*H);
# Computes Q = s * Q
def point_mul(s, P):
Q = (0, 1, 1, 0) # Neutral element
while s > 0:
if s & 1:
Q = point_add(Q, P)
P = point_add(P, P)
s >>= 1
return Q
def point_equal(P, Q):
# x1 / z1 == x2 / z2 <==> x1 * z2 == x2 * z1
if (P[0] * Q[2] - Q[0] * P[2]) % p != 0:
return False
if (P[1] * Q[2] - Q[1] * P[2]) % p != 0:
return False
return True
## Now follows functions for point compression.
# Square root of -1
modp_sqrt_m1 = pow(2, (p-1) // 4, p)
# Compute corresponding x-coordinate, with low bit corresponding to
# sign, or return None on failure
def recover_x(y, sign):
if y >= p:
return None
x2 = (y*y-1) * modp_inv(d*y*y+1)
if x2 == 0:
if sign:
return None
else:
return 0
# Compute square root of x2
x = pow(x2, (p+3) // 8, p)
if (x*x - x2) % p != 0:
x = x * modp_sqrt_m1 % p
if (x*x - x2) % p != 0:
return None
if (x & 1) != sign:
x = p - x
return x
# Base point
g_y = 4 * modp_inv(5) % p
g_x = recover_x(g_y, 0)
G = (g_x, g_y, 1, g_x * g_y % p)
def to_affine(P):
zinv = modp_inv(P[2])
x = P[0] * zinv % p
y = P[1] * zinv % p
return x, y
def point_compress(P):
x, y = to_affine(P)
return int.to_bytes(y | ((x & 1) << 255), 32, "little")
def point_decompress(s):
if len(s) != 32:
raise Exception("Invalid input length for decompression")
y = int.from_bytes(s, "little")
sign = y >> 255
y &= (1 << 255) - 1
x = recover_x(y, sign)
if x is None:
return None
else:
return (x, y, 1, x*y % p)
## These are functions for manipulating the private key.
def secret_expand(secret):
if len(secret) != 32:
raise Exception("Bad size of private key")
h = sha512(secret)
a = int.from_bytes(h[:32], "little")
a &= (1 << 254) - 8
a |= (1 << 254)
return (a, h[32:])
def secret_to_public(secret):
(a, dummy) = secret_expand(secret)
return point_compress(point_mul(a, G))
## The signature function works as below.
def sign_standard(secret, msg):
a, prefix = secret_expand(secret)
A = point_compress(point_mul(a, G))
r = sha512_modq(prefix + msg)
R = point_mul(r, G)
Rs = point_compress(R)
h = sha512_modq(Rs + A + msg)
s = (r + h * a) % q
signature = Rs + int.to_bytes(s, 32, "little")
return signature, A
def sign(s, prefix, A, sch, scn, m):
r_bytes = tmac(prefix, sch + scn + m, b"\x0C")
r = int.from_bytes(r_bytes, byteorder='big') % q
R = point_mul(r, G)
Rs = point_compress(R)
h = sha512_modq(Rs + A + m)
s = (r + h * s) % q
signature = Rs + int.to_bytes(s, 32, "little")
return signature
## And finally the verification function.
def verify(public, msg, signature):
if len(public) != 32:
raise Exception("Bad public key length")
if len(signature) != 64:
Exception("Bad signature length")
A = point_decompress(public)
if not A:
return False
Rs = signature[:32]
R = point_decompress(Rs)
if not R:
return False
s = int.from_bytes(signature[32:], "little")
if s >= q: return False
h = sha512_modq(Rs + public + msg)
sB = point_mul(s, G)
hA = point_mul(h, A)
return point_equal(sB, point_add(R, hA))