Moved py_pairing components here

LICENSE

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+
+The MIT License (MIT)
+
+Copyright (c) 2015 Vitalik Buterin
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in
+all copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
+THE SOFTWARE.

README.md

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+Implements optimal ate pairings over the bn\_128 curve.

py_pairing/__init__.py

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+from .optimized_curve import *
+from .optimized_field_elements import *
+from .optimized_pairing import *

py_pairing/bn128_curve.py

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+from bn128_field_elements import field_modulus, FQ
+from optimized_field_elements import FQ2, FQ12
+# from bn128_field_elements import FQ2, FQ12
+
+curve_order = 21888242871839275222246405745257275088548364400416034343698204186575808495617
+
+# Curve order should be prime
+assert pow(2, curve_order, curve_order) == 2
+# Curve order should be a factor of field_modulus**12 - 1
+assert (field_modulus ** 12 - 1) % curve_order == 0
+
+# Curve is y**2 = x**3 + 3
+b = FQ(3)
+# Twisted curve over FQ**2
+b2 = FQ2([3, 0]) / FQ2([9, 1])
+# Extension curve over FQ**12; same b value as over FQ
+b12 = FQ12([3] + [0] * 11)
+
+# Generator for curve over FQ
+G1 = (FQ(1), FQ(2))
+# Generator for twisted curve over FQ2
+G2 = (FQ2([10857046999023057135944570762232829481370756359578518086990519993285655852781, 11559732032986387107991004021392285783925812861821192530917403151452391805634]),
+      FQ2([8495653923123431417604973247489272438418190587263600148770280649306958101930, 4082367875863433681332203403145435568316851327593401208105741076214120093531]))
+
+# Check that a point is on the curve defined by y**2 == x**3 + b
+def is_on_curve(pt, b):
+    if pt is None:
+        return True
+    x, y = pt
+    return y**2 - x**3 == b
+
+assert is_on_curve(G1, b)
+assert is_on_curve(G2, b2)
+
+# Elliptic curve doubling
+def double(pt):
+    x, y = pt
+    l = 3 * x**2 / (2 * y)
+    newx = l**2 - 2 * x
+    newy = -l * newx + l * x - y
+    return newx, newy
+
+# Elliptic curve addition
+def add(p1, p2):
+    if p1 is None or p2 is None:
+        return p1 if p2 is None else p2
+    x1, y1 = p1
+    x2, y2 = p2
+    if x2 == x1 and y2 == y1:
+        return double(p1)
+    elif x2 == x1:
+        return None
+    else:
+        l = (y2 - y1) / (x2 - x1)
+    newx = l**2 - x1 - x2
+    newy = -l * newx + l * x1 - y1
+    assert newy == (-l * newx + l * x2 - y2)
+    return (newx, newy)
+
+# Elliptic curve point multiplication
+def multiply(pt, n):
+    if n == 0:
+        return None
+    elif n == 1:
+        return pt
+    elif not n % 2:
+        return multiply(double(pt), n // 2)
+    else:
+        return add(multiply(double(pt), int(n // 2)), pt)
+
+# Check that the G1 curve works fine
+assert add(add(double(G1), G1), G1) == double(double(G1))
+assert double(G1) != G1
+assert add(multiply(G1, 9), multiply(G1, 5)) == add(multiply(G1, 12), multiply(G1, 2))
+assert multiply(G1, curve_order) is None
+
+# Check that the G2 curve works fine
+assert add(add(double(G2), G2), G2) == double(double(G2))
+assert double(G2) != G2
+assert add(multiply(G2, 9), multiply(G2, 5)) == add(multiply(G2, 12), multiply(G2, 2))
+assert multiply(G2, curve_order) is None
+assert multiply(G2, 2 * field_modulus - curve_order) is not None
+assert is_on_curve(multiply(G2, 9), b2)
+
+# "Twist" a point in E(FQ2) into a point in E(FQ12)
+w = FQ12([0, 1] + [0] * 10)
+
+# Convert P => -P
+def neg(pt):
+    if pt is None:
+        return None
+    x, y = pt
+    return (x, -y)
+
+def twist(pt):
+    if pt is None:
+        return None
+    _x, _y = pt
+    # Field isomorphism from Z[p] / x**2 to Z[p] / x**2 - 18*x + 82
+    xcoeffs = [_x.coeffs[0] - _x.coeffs[1] * 9, _x.coeffs[1]]
+    ycoeffs = [_y.coeffs[0] - _y.coeffs[1] * 9, _y.coeffs[1]]
+    # Isomorphism into subfield of Z[p] / w**12 - 18 * w**6 + 82,
+    # where w**6 = x
+    nx = FQ12([xcoeffs[0]] + [0] * 5 + [xcoeffs[1]] + [0] * 5)
+    ny = FQ12([ycoeffs[0]] + [0] * 5 + [ycoeffs[1]] + [0] * 5)
+    # Divide x coord by w**2 and y coord by w**3
+    return (nx * w **2, ny * w**3)
+
+# Check that the twist creates a point that is on the curve
+assert is_on_curve(twist(G2), b12)
+
+# Check that the G12 curve works fine
+
+G12 = twist(G2)
+assert add(add(double(G12), G12), G12) == double(double(G12))
+assert double(G12) != G12
+assert add(multiply(G12, 9), multiply(G12, 5)) == add(multiply(G12, 12), multiply(G12, 2))
+assert is_on_curve(multiply(G12, 9), b12)
+assert multiply(G12, curve_order) is None