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fullscratch_pairing_weil_small.py
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fullscratch_pairing_weil_small.py
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# fullscratch weil pairing
from collections import namedtuple
import hashlib
Point = namedtuple("Point", "x y")
O = 'Origin'
def ext_euc(a, b):
e1 = 0
f1 = 0
c = a // b
d = a - c*b
e2 = 1
f2 = -1*c
if d == 0:
return (a, b, c, d, e1, f1, e2, f2, 0, 0)
a = b
b = d
c = a // b
d = a - c*b
e3 = -1*c*e2
f3 = 1-c*f2
if d == 0:
return (a, b, c, d, e1, f1, e2, f2, e3, f3)
while True:
a = b
b = d
c = a // b
d = a - c*b
e1 = e2
f1 = f2
e2 = e3
f2 = f3
e3 = e1 - c*e2
f3 = f1 - c*f2
if d == 0:
break
return (a, b, c, d, e1, f1, e2, f2, e3, f3)
def tocomp(obj):
if type(obj) == int or type(obj) == Mod:
return Complex(obj, 0)
if type(obj) == Complex:
return obj
class Complex(object):
def __init__(self, r, i):
self.r = r
self.i = i
def __mul__(self, other):
k = tocomp(other)
return Complex(self.r*k.r - self.i*k.i, self.r*k.i + self.i*k.r)
def __rmul__(self, other):
k = tocomp(other)
return k * self
def __add__(self, other):
k = tocomp(other)
return Complex(self.r + k.r, self.i + k.i)
def __truediv__(self, other):
k = tocomp(other)
s = k.r**2 + k.i**2
return Complex((self.r*k.r + self.i*k.i) / s, (self.i*k.r - self.r*k.i) / s)
def __rtruediv__(self, other):
k = tocomp(other)
return k / self
def __sub__(self, other):
k = tocomp(other)
return Complex(self.r - k.r, self.i - k.i)
def __neg__(self):
return Complex(-self.r, -self.i)
def __pow__(self, n):
bs = format(n, 'b')[::-1]
tmp = self
result = Complex(1, 0)
for i in bs:
if i == "1":
result = result * tmp
tmp = tmp*tmp
return result
def __eq__(self, other):
k = tocomp(other)
return self.r == k.r and self.i == k.i
def __repr__(self):
return str(self.r) + " + " + str(self.i) + "i"
def tomod(obj, p):
if type(obj) == Mod:
return Mod(obj.n, p)
if type(obj) == int:
return Mod(obj, p)
class Mod(object):
def __init__(self, n, p):
self.n = n % p
self.p = p
def __mul__(self, other):
k = tomod(other, self.p)
return Mod(self.n * k.n, self.p)
def __rmul__(self, other):
k = tomod(other, self.p)
return k * self
def __add__(self, other):
k = tomod(other, self.p)
return Mod(self.n + k.n, self.p)
def __radd__(self, other):
k = tomod(other, self.p)
return k + self
def __truediv__(self, other):
k = tomod(other, self.p)
if k.n == 1:
return self
return Mod((self.n * ext_euc(self.p, k.n) [7]) % self.p, self.p)
def __rtruediv__(self, other):
k = tomod(other, self.p)
return k / self
def __sub__(self, other):
k = tomod(other, self.p)
return Mod(self.n - k.n, self.p)
def __rsub__(self, other):
k = tomod(other, self.p)
return k - self
def __neg__(self):
return Mod(-self.n, self.p)
def __pow__(self, k):
return Mod(pow(self.n, k, self.p), self.p)
def __eq__(self, other):
k = tomod(other, self.p)
return self.n % self.p == k.n % self.p
def __repr__(self):
return str(self.n % self.p) + " % " + str(self.p)
class Curve:
def __init__(self, a, b):
self.a = a
self.b = b
def valid(self, P):
if P == O:
return True
else:
return (P.y**2 - (P.x**3 + self.a*P.x + self.b)) == 0
def inv(self, P):
if P == O:
return P
return Point(P.x, -P.y)
def add(self, P, Q):
if not (self.valid(P) and self.valid(Q)):
raise ValueError("Invalid inputs")
# Deal with the special cases where either P, Q, or P + Q is
# the origin.
if P == O:
result = Q
elif Q == O:
result = P
elif Q == self.inv(P):
result = O
else:
# Cases not involving the origin.
if P == Q:
dydx = (3 * P.x ** 2 + self.a) / (2 * P.y)
else:
dydx = (Q.y - P.y) / (Q.x - P.x)
x = dydx**2 - P.x - Q.x
y = dydx * (P.x - x) - P.y
result = Point(x, y)
assert self.valid(result)
return result
def mul(self, P, n):
bs = format(n, 'b')[::-1]
tmp = P
result = O
for i in bs:
if i == "1":
result = self.add(result, tmp)
tmp = self.add(tmp, tmp)
return result
def miller(self, P, Q, l):
def g(P1, P2):
if self.inv(P1) == P2:
if P1 == "Origin":
return 1 # div f = 3(O) - 3(O) = 0 → f = non-zero const .
return Q.x - P1.x
if P1 == P2:
if P1 == "Origin":
return 1 # div f = 3(O) - 3(O) = 0 → f = non-zero const .
lam = (3 * P1.x ** 2 + self.a)/(2 * P1.y)
else:
if P1 == "Origin":
raise NotImplementedError("wakaran") # div f = (P2) - (O) → ???
if P2 == "Origin":
raise NotImplementedError("wakaran") # div f = (P1) - (O) → ???
lam = (P2.y - P1.y)/(P2.x - P1.x)
P3 = self.add(P1, P2)
return (Q.y - lam*(Q.x - P1.x) - P1.y)/(Q.x - P3.x)
f = 1
V = P
bs = format(l, 'b')[1:]
for i in bs:
f = f*f*g(V, V)
V = self.add(V, V)
if i == '1':
f = f*g(V, P)
V = self.add(V, P)
assert V == 'Origin'
return f
if __name__ == "__main__":
p = 3191 # a prime s.t. p % 4 == 3, p % 3 == 2, and sqrt(2) exists in F_p, and p + 1 has a big prime factor
E1 = Curve(Complex(Mod(0, p), Mod(0, p)), Complex(Mod(3, p), Mod(0, p)))
P1 = Point(Complex(Mod(1, p), Mod(0, p)), Complex(Mod(2, p), Mod(0, p)))
sqex = (p + 1) // 4
sq2 = Complex(Mod(2, p), Mod(0, p)) ** sqex
assert sq2 ** 2 == Complex(Mod(2, p), Mod(0, p))
sq3 = Complex(Mod(3, p), Mod(0, p)) ** sqex
assert sq3 ** 2 == Complex(Mod(3, p), Mod(0, p))
def mkp(k):
A = Mod(k, p)
B = ( (A**3 + 3)/(3 * A) ) ** sqex
assert B**2 == (A**3 + 3)/(3 * A)
X = Complex(A, B)
sqi = sq2 / 2 + sq2 / 2 * Complex(Mod(0, p), Mod(1, p))
assert (((X**3 + 3).i)**sqex)**2 == (X**3 + 3).i
Y = sqi * ((X**3 + 3).i)**sqex
assert (Y**2 == X**3 + 3)
return Point(X, Y)
P2 = mkp(8)
P3 = mkp(4)
def test(P):
k = 1
TMP = P
while True:
TMP = E1.add(TMP, P)
k += 1
if TMP == "Origin":
print(k)
break
print("P1")
test(P1)
print("P2")
test(P2)
print("P3")
test(P3)
# common order for all points
l = p + 1
assert E1.mul(P1, l) == E1.mul(P2, l) == E1.mul(P3, l) == "Origin"
# weil pairing
w = lambda X, Y: E1.miller(X, Y, l)/E1.miller(Y, X, l)
print(w(P1, P2))
print(w(E1.mul(P1, 123), E1.mul(P2, 947)))
print(w(E1.mul(P1, 116481), E1.mul(P2, 1)))
print(w(E1.mul(P1, 10001), E1.mul(P2, 1)))
print(w(E1.mul(P1, 10003), E1.mul(P2, 1)))
print(w(E1.mul(P1, 10005), E1.mul(P2, 1)))
print(w(E1.mul(P1, 8873), E1.mul(P2, 9291)))
print(w(E1.mul(P1, 1), E1.mul(P2, 8873*9291)))
sanbai = -Complex(Mod(1, p), Mod(0, p)) / 2 + sq3*Complex(Mod(0, p), Mod(1, p)) / 2
s = lambda X: Point(sanbai * X.x, X.y)
ww = lambda X, Y: w(X, s(Y))
print(ww(E1.mul(P1, 943), E1.mul(P2, 917)))
print(ww(E1.mul(P1, 1), E1.mul(P2, 943*917)))
print(ww(E1.mul(P1, 1), E1.mul(P2, 3)))
print(ww(E1.mul(P1, 1), E1.mul(P2, 5)))
print(ww(E1.mul(P1, 1), E1.mul(P2, 7)))
print(ww(E1.mul(P2, 857), E1.mul(P3, 117)))
print(ww(E1.mul(P2, 857*117), E1.mul(P3, 1)))
print(ww(E1.mul(P2, 3), E1.mul(P3, 5)))
print(ww(E1.mul(P2, 15), E1.mul(P3, 1)))