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ccxt/static_dependencies/starkware/crypto/math_utils.py

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+###############################################################################
+# Copyright 2019 StarkWare Industries Ltd.                                    #
+#                                                                             #
+# Licensed under the Apache License, Version 2.0 (the "License").             #
+# You may not use this file except in compliance with the License.            #
+# You may obtain a copy of the License at                                     #
+#                                                                             #
+# https://www.starkware.co/open-source-license/                               #
+#                                                                             #
+# Unless required by applicable law or agreed to in writing,                  #
+# software distributed under the License is distributed on an "AS IS" BASIS,  #
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.    #
+# See the License for the specific language governing permissions             #
+# and limitations under the License.                                          #
+###############################################################################
+
+
+from typing import Tuple
+
+from ...sympy.core.intfunc import igcdex
+
+# A type that represents a point (x,y) on an elliptic curve.
+ECPoint = Tuple[int, int]
+
+def div_mod(n: int, m: int, p: int) -> int:
+    """
+    Finds a nonnegative integer 0 <= x < p such that (m * x) % p == n
+    """
+    a, b, c = igcdex(m, p)
+    assert c == 1
+    return (n * a) % p
+
+def div_ceil(x, y):
+    assert isinstance(x, int) and isinstance(y, int)
+    return -((-x) // y)
+    
+def ec_add(point1: ECPoint, point2: ECPoint, p: int) -> ECPoint:
+    """
+    Gets two points on an elliptic curve mod p and returns their sum.
+    Assumes the points are given in affine form (x, y) and have different x coordinates.
+    """
+    assert (point1[0] - point2[0]) % p != 0
+    m = div_mod(point1[1] - point2[1], point1[0] - point2[0], p)
+    x = (m * m - point1[0] - point2[0]) % p
+    y = (m * (point1[0] - x) - point1[1]) % p
+    return x, y
+
+
+def ec_neg(point: ECPoint, p: int) -> ECPoint:
+    """
+    Given a point (x,y) return (x, -y)
+    """
+    x, y = point
+    return (x, (-y) % p)
+
+
+def ec_double(point: ECPoint, alpha: int, p: int) -> ECPoint:
+    """
+    Doubles a point on an elliptic curve with the equation y^2 = x^3 + alpha*x + beta mod p.
+    Assumes the point is given in affine form (x, y) and has y != 0.
+    """
+    assert point[1] % p != 0
+    m = div_mod(3 * point[0] * point[0] + alpha, 2 * point[1], p)
+    x = (m * m - 2 * point[0]) % p
+    y = (m * (point[0] - x) - point[1]) % p
+    return x, y
+
+
+def ec_mult(m: int, point: ECPoint, alpha: int, p: int) -> ECPoint:
+    """
+    Multiplies by m a point on the elliptic curve with equation y^2 = x^3 + alpha*x + beta mod p.
+    Assumes the point is given in affine form (x, y) and that 0 < m < order(point).
+    """
+    if m == 1:
+        return point
+    if m % 2 == 0:
+        return ec_mult(m // 2, ec_double(point, alpha, p), alpha, p)
+    return ec_add(ec_mult(m - 1, point, alpha, p), point, p)