Add Elliptic Curve and docs
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159
Cryptotools/Groups/curve.py
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159
Cryptotools/Groups/curve.py
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#!/usr/bin/env python3
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from Cryptotools.Groups.elliptic import Point
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import numpy as np
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from math import sqrt
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class Curve:
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# Curve
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WEIERSTRASS = 0
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MONTGOMERY = 1
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def __init__(self, a, b, t):
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self._a = a
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self._b = b
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if t not in (Curve.WEIERSTRASS, Curve.MONTGOMERY):
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raise Exception(f"The type of the curve is not recognized")
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self._type = t
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self._xtmp = np.linspace(-5, 5, 500).tolist()
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self._x = list()
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self._y = list()
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self._yn = list()
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self._points = list()
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self._pointsSym = list()
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def f(self, x):
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if self._type == Curve.WEIERSTRASS:
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y = pow(x, 3) + (self._a * x) + self._b
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if self._type == Curve.MONTGOMERY:
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y = pow(x, 3) + (3 * pow(x, 2)) + x
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if y > 0:
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return sqrt(y)
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return None
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def generatePoints(self):
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for x in self._xtmp:
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y = self.f(x)
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if y is None:
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continue
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self._x.append(x)
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self._y.append(y)
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self._yn.append(-y)
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self._points.append(Point(
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x,
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y
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))
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self._pointsSym.append(Point(
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x,
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-y
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))
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@property
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def x(self):
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return self._x
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@property
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def y(self):
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return self._y
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@property
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def yn(self):
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return self._yn
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def getPoints(self):
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return self._points
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def getPointsSym(self):
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return self._pointsSym
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def add(self, P, Q) -> Point:
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"""
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This function operathe addition operation on two points P and Q
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Args:
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P (Object): The first Point on the curve
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Q (Object): The second Point on the curve
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Returns:
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Return the Point object R
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"""
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## Check if P or Q are infinity
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if (P.x, P.y) == (0, 0) and (Q.x, Q.y) == (0, 0):
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return Point(0, 0)
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elif (P.x, P.y) == (0, 0):
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return Point(Q.x, Q.y)
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elif (Q.x, Q.y) == (0, 0):
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return Point(P.x, P.y)
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# point doubling
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if P.x == Q.x:
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# Infinity
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if P.y != Q.y or Q.y == 0:
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return Point(0, 0)
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# Point doubling
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try:
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inv = pow(2 * P.y, -1); # It's working with the inverse modular, WHY ???
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m = ((3 * pow(P.x, 2)) + self._a) * inv
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except ValueError:
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return Point(0, 0)
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else:
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try:
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inv = pow(Q.x - P.x, -1)
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m = (Q.y - P.y) * inv
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except ValueError:
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# May call this Exception: base is not invertible for the given modulus
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# I return an Infinity point until I fixed that
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return Point(0, 0)
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xr = (pow(m, 2) - P.x - Q.x)
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yr = (m * (P.x - xr)) - P.y
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return Point(xr, yr)
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def scalar(self, P, n) -> Point:
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"""
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This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add.
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Args:
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P (point): the Point to multiplication
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n (Integer): multiplicate n time P
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Returns:
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Return the result of the Scalar multiplication
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"""
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binary = bin(n)[2:]
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binary = binary[::-1] # We need to reverse the binary
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nP = Point(0, 0)
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Rtmp = P
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for b in binary:
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if b == '1':
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nP = self.add(nP, Rtmp)
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Rtmp = self.add(Rtmp, Rtmp) # Double P
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return nP
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def find_reverse(self, P):
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"""
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This function return the reverse of the Point P
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Args:
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P (Point): Point object to find
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Returns:
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Return the object Pr, which is the reverse point of P
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"""
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Pr = None
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for p in self._pointsSym:
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if P.x == p.x and -P.y == p.y:
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Pr = Point(p.x, p.y)
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break
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return Pr
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34
Cryptotools/Groups/point.py
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34
Cryptotools/Groups/point.py
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#!/usr/bin/env python3
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class Point:
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"""
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This simple class represent the Point at the coordinate x and y in a plan
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Attributes:
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x (Integer): Position at the x
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y (Integer): Position at the y
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"""
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def __init__(self, x, y):
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self._x = x
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self._y = y
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@property
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def x(self):
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return self._x
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@property
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def y(self):
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return self._y
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@x.setter
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def x(self, x):
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self._x = x
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@y.setter
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def y(self, y):
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self._y = y
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def __eq__(self, other):
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print(self._x, other.x)
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return self._x == other.x and self._y == other.y
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6
docs/curves.md
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docs/curves.md
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# Elliptic Curves
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## Points
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::: Cryptotools.Groups.point
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## Curves
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::: Cryptotools.Groups.curve
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docs/example-curves.md
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docs/example-curves.md
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# Generating Elliptic Curves
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In the following section, we are going to see how to generate Elliptic Curves and draw the curve with Python.
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## A Weierstrass Curve
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The python code below draw a Weierstrass Curve:
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```
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#!/usr/bin/env python3
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import matplotlib.pyplot as plt
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from Cryptotools.Groups.curve import Curve
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a = 3
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b = 8
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curve = Curve(a, b, Curve.WEIERSTRASS)
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x = curve.x
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curve.generatePoints()
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y = curve.y
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yn = curve.yn
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points = curve.getPoints()
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plt.figure(figsize=(10, 6))
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plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$')
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plt.plot(x, yn, color='b', )
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plt.legend()
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plt.show()
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```
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* Low-Level Cryptographic
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* [Number Theory](/number-theory)
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* [Group Theory](/group-theory)
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* [Curves](/curves)
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* Public Keys:
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* [RSA](/rsa)
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* Utils
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* [Utils](/utils)
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* Examples:
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* [Generating RSA keys](/examples-rsa-keys)
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* [Generating RSA keys](/example-rsa-keys)
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* [Generate curves](/example-curves)
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24
examples/elliptic/curve.py
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24
examples/elliptic/curve.py
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#!/usr/bin/env python3
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import matplotlib.pyplot as plt
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from Cryptotools.Groups.curve import Curve
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a = 3
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b = 8
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curve = Curve(a, b, Curve.WEIERSTRASS)
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x = curve.x
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curve.generatePoints()
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y = curve.y
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yn = curve.yn
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points = curve.getPoints()
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#print(x)
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#print(y)
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plt.figure(figsize=(10, 6))
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plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$')
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plt.plot(x, yn, color='b', )
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plt.legend()
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plt.show()
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48
examples/elliptic/curve_add.py
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examples/elliptic/curve_add.py
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#!/usr/bin/env python3
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import matplotlib.pyplot as plt
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from Cryptotools.Groups.curve import Curve
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a = 3
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b = 8
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curve = Curve(a, b, Curve.WEIERSTRASS)
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x = curve.x
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curve.generatePoints()
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y = curve.y
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yn = curve.yn
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points = curve.getPoints()
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P = points[10]
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Q = points[55]
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R = points[247]
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#print(f"{P.x} {P.y}")
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#print(f"{Q.x} {Q.y}")
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# Make an addition
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Rp = curve.add(P, Q)
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#print(f"{Rp.x} {Rp.y}")
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#print(yn)
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#print(x)
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#print(y)
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plt.figure(figsize=(10, 6))
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plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$')
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plt.plot(x, yn, color='b', )
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plt.plot(P.x, P.y, marker='o', color="red")
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plt.annotate('P', (P.x, P.y + 0.5))
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plt.plot(Q.x, Q.y, marker='o', color="red")
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plt.annotate('Q', (Q.x, Q.y + 0.5))
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plt.plot(R.x, R.y, marker='o', color="red")
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plt.annotate('R', (R.x - 0.2, R.y + 0.5))
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plt.plot(Rp.x, Rp.y, marker='o', color="red")
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plt.annotate('R\'', (Rp.x - 0.2, Rp.y + 0.5))
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plt.axline([P.x, P.y], [Q.x, Q.y], color="red")
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plt.axline([R.x, R.y], [Rp.x, Rp.y], linestyle="--", color="red")
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plt.legend()
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plt.show()
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44
examples/elliptic/curve_scalar.py
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examples/elliptic/curve_scalar.py
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#!/usr/bin/env python3
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import matplotlib.pyplot as plt
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from Cryptotools.Groups.curve import Curve
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from Cryptotools.Groups.point import Point
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a = 3
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b = 8
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curve = Curve(a, b, Curve.WEIERSTRASS)
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x = curve.x
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curve.generatePoints()
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y = curve.y
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yn = curve.yn
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points = curve.getPoints()
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P = points[10]
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nP = curve.scalar(P, 5)
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print(f"{nP.x} {nP.y}")
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# For testing, we may add n times with the addition operation for the Scalar Multiplication
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tmp = Point(0, 0)
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for i in range(5):
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tmp = curve.add(P, tmp)
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# Unfortunately, the result is approximatively the same
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# the multiplication need to be more accurate
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# if tmp == nP:
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# print(True)
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plt.figure(figsize=(10, 6))
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plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$')
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plt.plot(x, yn, color='b', )
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plt.plot(P.x, P.y, marker='o', color="red")
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plt.annotate('P', (P.x, P.y + 0.5))
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plt.plot(nP.x, nP.y, marker='o', color="red")
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plt.annotate('nP', (nP.x, nP.y + 0.5))
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plt.legend()
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plt.show()
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43
examples/elliptic/point_at_infinity.py
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examples/elliptic/point_at_infinity.py
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#!/usr/bin/env python3
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from Cryptotools.Groups.point import Point
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from Cryptotools.Groups.curve import Curve
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import numpy as np
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import matplotlib.pyplot as plt
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from math import sqrt
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a = 3
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b = 8
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curve = Curve(a, b, Curve.WEIERSTRASS)
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x = curve.x
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curve.generatePoints()
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y = curve.y
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yn = curve.yn
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points = curve.getPoints()
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pointsReverse = curve.getPointsSym()
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P = points[10]
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Q = curve.find_reverse(P)
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# We made the addition
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# The result is the point at infinity
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R = curve.add(P, Q)
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print(f"{R.x} {R.y}")
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plt.figure(figsize=(10, 6))
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plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$')
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plt.plot(x, yn, color='b', )
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plt.plot(P.x, P.y, marker='o', color="red")
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plt.annotate('P', (P.x, P.y + 0.5))
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plt.plot(Q.x, Q.y, marker='o', color="red")
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plt.text(-1 , 11, f"R = infinity")
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plt.annotate('Q', (Q.x, Q.y + 0.5))
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plt.axline([P.x, P.y], [Q.x, Q.y], color="red")
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plt.legend()
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plt.show()
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@ -11,9 +11,11 @@ nav:
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- Low-level cryptographic:
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- Number theory: number-theory.md
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- Group theory: group-theory.md
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- Curves: curves.md
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- Public Keys:
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- RSA: rsa.md
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- Utils:
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- Utils: utils.md
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- Examples:
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- Generating RSA Keys: examples-rsa-keys.md
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- Generating RSA Keys: example-rsa-keys.md
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- Generating Curves: example-curves.md
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@ -43,15 +43,24 @@
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</li>
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<li class="toctree-l1"><a class="reference internal" href="/group-theory/">Group theory</a>
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</li>
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<li class="toctree-l1"><a class="reference internal" href="/curves/">Curves</a>
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</li>
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</ul>
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<p class="caption"><span class="caption-text">Public Keys</span></p>
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<ul>
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<li class="toctree-l1"><a class="reference internal" href="/rsa/">RSA</a>
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</li>
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</ul>
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<p class="caption"><span class="caption-text">Utils</span></p>
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<ul>
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<li class="toctree-l1"><a class="reference internal" href="/utils/">Utils</a>
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</li>
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</ul>
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<p class="caption"><span class="caption-text">Examples</span></p>
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<ul>
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<li class="toctree-l1"><a class="reference internal" href="/examples-rsa-keys/">Generating RSA Keys</a>
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<li class="toctree-l1"><a class="reference internal" href="/example-rsa-keys/">Generating RSA Keys</a>
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</li>
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<li class="toctree-l1"><a class="" href="/example-curves.md">Generating Curves</a>
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</li>
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</ul>
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</div>
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1096
site/curves/index.html
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1096
site/curves/index.html
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File diff suppressed because it is too large
Load Diff
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<script>
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// Current page data
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var mkdocs_page_name = "Generating RSA Keys";
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var mkdocs_page_input_path = "examples-rsa-keys.md";
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var mkdocs_page_input_path = "example-rsa-keys.md";
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var mkdocs_page_url = null;
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</script>
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@ -50,18 +50,27 @@
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</li>
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<li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
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</li>
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<li class="toctree-l1"><a class="reference internal" href="../curves/">Curves</a>
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</li>
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</ul>
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<p class="caption"><span class="caption-text">Public Keys</span></p>
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<ul>
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<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
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</li>
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</ul>
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<p class="caption"><span class="caption-text">Utils</span></p>
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<ul>
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<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
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</li>
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</ul>
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<p class="caption"><span class="caption-text">Examples</span></p>
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<ul class="current">
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<li class="toctree-l1 current"><a class="reference internal current" href="#">Generating RSA Keys</a>
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<ul class="current">
|
||||
</ul>
|
||||
</li>
|
||||
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
|
||||
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|
||||
</ul>
|
||||
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|
||||
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|
||||
@ -116,7 +125,7 @@ print(f"Plaintext: {plaintext}")
|
||||
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|
||||
</div><footer>
|
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|
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|
||||
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|
||||
|
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|
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@ -139,7 +148,7 @@ print(f"Plaintext: {plaintext}")
|
||||
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|
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|
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|
||||
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|
||||
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|
||||
<li class="toctree-l1"><a class="reference internal" href="../curves/">Curves</a>
|
||||
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|
||||
</ul>
|
||||
<p class="caption"><span class="caption-text">Public Keys</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
|
||||
</li>
|
||||
</ul>
|
||||
<p class="caption"><span class="caption-text">Utils</span></p>
|
||||
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|
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|
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|
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<p class="caption"><span class="caption-text">Examples</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
|
||||
</li>
|
||||
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
@ -3015,7 +3024,7 @@ In Group Theory, an identity element is an element in the group which do not cha
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|
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|
||||
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<hr/>
|
||||
@ -3041,7 +3050,7 @@ In Group Theory, an identity element is an element in the group which do not cha
|
||||
<span><a href="../number-theory/" style="color: #fcfcfc">« Previous</a></span>
|
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@ -50,15 +50,24 @@
|
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</li>
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<li class="toctree-l1"><a class="reference internal" href="group-theory/">Group theory</a>
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<li class="toctree-l1"><a class="reference internal" href="curves/">Curves</a>
|
||||
</li>
|
||||
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||||
<p class="caption"><span class="caption-text">Public Keys</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="rsa/">RSA</a>
|
||||
</li>
|
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<p class="caption"><span class="caption-text">Utils</span></p>
|
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<li class="toctree-l1"><a class="reference internal" href="utils/">Utils</a>
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|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="examples-rsa-keys/">Generating RSA Keys</a>
|
||||
<li class="toctree-l1"><a class="reference internal" href="example-rsa-keys/">Generating RSA Keys</a>
|
||||
</li>
|
||||
<li class="toctree-l1"><a class="" href="example-curves.md">Generating Curves</a>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
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|
||||
<li>Low-Level Cryptographic<ul>
|
||||
<li><a href="/number-theory">Number Theory</a></li>
|
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<li><a href="/group-theory">Group Theory</a></li>
|
||||
<li><a href="/curves">Curves</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li>Public Keys:<ul>
|
||||
<li><a href="/rsa">RSA</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li>Utils<ul>
|
||||
<li><a href="/utils">Utils</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li>Examples:<ul>
|
||||
<li><a href="/examples-rsa-keys">Generating RSA keys</a></li>
|
||||
<li><a href="/example-rsa-keys">Generating RSA keys</a></li>
|
||||
<li><a href="/example-curves">Generate curves</a></li>
|
||||
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|
||||
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|
||||
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|
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|
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<p class="caption"><span class="caption-text">Public Keys</span></p>
|
||||
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|
||||
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
|
||||
</li>
|
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<p class="caption"><span class="caption-text">Utils</span></p>
|
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|
||||
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|
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</li>
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<p class="caption"><span class="caption-text">Examples</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
|
||||
</li>
|
||||
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
|
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||||
@ -52,15 +52,24 @@
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|
||||
</li>
|
||||
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|
||||
<p class="caption"><span class="caption-text">Public Keys</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
|
||||
</li>
|
||||
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|
||||
<p class="caption"><span class="caption-text">Utils</span></p>
|
||||
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|
||||
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|
||||
</li>
|
||||
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||||
<p class="caption"><span class="caption-text">Examples</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
|
||||
</li>
|
||||
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
|
||||
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||||
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||||
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|
||||
@ -80,15 +80,24 @@
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|
||||
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|
||||
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|
||||
<p class="caption"><span class="caption-text">Public Keys</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
|
||||
</li>
|
||||
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|
||||
<p class="caption"><span class="caption-text">Utils</span></p>
|
||||
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|
||||
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|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
|
||||
</li>
|
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<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
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||||
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||||
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||||
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<p class="caption"><span class="caption-text">Utils</span></p>
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||||
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<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
|
||||
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||||
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<p class="caption"><span class="caption-text">Examples</span></p>
|
||||
<ul>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
|
||||
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
|
||||
</li>
|
||||
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
@ -1041,8 +1050,8 @@ The key is a tuple of the key and the modulus n</p>
|
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<hr/>
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@ -1065,10 +1074,10 @@ The key is a tuple of the key and the modulus n</p>
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||||
<p>CryptoTools provides several utils function wich can be used for cryptography purposes</p>
|
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|
||||
<div class="doc doc-object doc-module">
|
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||||
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||||
<div class="doc doc-object doc-function">
|
||||
|
||||
|
||||
<h2 id="Cryptotools.Utils.utils.bin_expo" class="doc doc-heading">
|
||||
<code class="highlight language-python"><span class="n">bin_expo</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span></code>
|
||||
|
||||
</h2>
|
||||
|
||||
|
||||
<div class="doc doc-contents ">
|
||||
|
||||
<p>This function perform an binary exponentiation, also known as Exponentiation squaring.
|
||||
A binary exponentiation is the process for computing an integer power of a number, such as a ** n.
|
||||
For doing that, the first step is to convert the exponent into a binary representation
|
||||
And for each 1 bit, we compute the exponent.</p>
|
||||
|
||||
|
||||
<table class="field-list">
|
||||
<colgroup>
|
||||
<col class="field-name" />
|
||||
<col class="field-body" />
|
||||
</colgroup>
|
||||
<tbody valign="top">
|
||||
<tr class="field">
|
||||
<th class="field-name">Parameters:</th>
|
||||
<td class="field-body">
|
||||
<ul class="first simple">
|
||||
<li>
|
||||
<b><code>n</code></b>
|
||||
(<code><span title="Integer">Integer</span></code>)
|
||||
–
|
||||
<div class="doc-md-description">
|
||||
<p>it's the base</p>
|
||||
</div>
|
||||
</li>
|
||||
<li>
|
||||
<b><code>e</code></b>
|
||||
(<code><span title="Integer">Integer</span></code>)
|
||||
–
|
||||
<div class="doc-md-description">
|
||||
<p>it's the exponent</p>
|
||||
</div>
|
||||
</li>
|
||||
</ul>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<table class="field-list">
|
||||
<colgroup>
|
||||
<col class="field-name" />
|
||||
<col class="field-body" />
|
||||
</colgroup>
|
||||
<tbody valign="top">
|
||||
<tr class="field">
|
||||
<th class="field-name">Returns:</th>
|
||||
<td class="field-body">
|
||||
<ul class="first simple">
|
||||
<li>
|
||||
<code><span title="int">int</span></code>
|
||||
–
|
||||
<div class="doc-md-description">
|
||||
<p>Return the result of the exponentation of n ** e</p>
|
||||
</div>
|
||||
</li>
|
||||
</ul>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<details class="quote">
|
||||
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
|
||||
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">35</span>
|
||||
<span class="normal">36</span>
|
||||
<span class="normal">37</span>
|
||||
<span class="normal">38</span>
|
||||
<span class="normal">39</span>
|
||||
<span class="normal">40</span>
|
||||
<span class="normal">41</span>
|
||||
<span class="normal">42</span>
|
||||
<span class="normal">43</span>
|
||||
<span class="normal">44</span>
|
||||
<span class="normal">45</span>
|
||||
<span class="normal">46</span>
|
||||
<span class="normal">47</span>
|
||||
<span class="normal">48</span>
|
||||
<span class="normal">49</span>
|
||||
<span class="normal">50</span>
|
||||
<span class="normal">51</span>
|
||||
<span class="normal">52</span>
|
||||
<span class="normal">53</span>
|
||||
<span class="normal">54</span>
|
||||
<span class="normal">55</span>
|
||||
<span class="normal">56</span>
|
||||
<span class="normal">57</span>
|
||||
<span class="normal">58</span>
|
||||
<span class="normal">59</span>
|
||||
<span class="normal">60</span>
|
||||
<span class="normal">61</span>
|
||||
<span class="normal">62</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span> <span class="nf">bin_expo</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="o">-></span> <span class="nb">int</span><span class="p">:</span>
|
||||
<span class="w"> </span><span class="sd">"""</span>
|
||||
<span class="sd"> This function perform an binary exponentiation, also known as Exponentiation squaring.</span>
|
||||
<span class="sd"> A binary exponentiation is the process for computing an integer power of a number, such as a ** n.</span>
|
||||
<span class="sd"> For doing that, the first step is to convert the exponent into a binary representation</span>
|
||||
<span class="sd"> And for each 1 bit, we compute the exponent.</span>
|
||||
|
||||
<span class="sd"> Args:</span>
|
||||
<span class="sd"> n (Integer): it's the base</span>
|
||||
<span class="sd"> e (Integer): it's the exponent</span>
|
||||
|
||||
<span class="sd"> Returns:</span>
|
||||
<span class="sd"> Return the result of the exponentation of n ** e</span>
|
||||
|
||||
<span class="sd"> """</span>
|
||||
<span class="n">binary</span> <span class="o">=</span> <span class="nb">bin</span><span class="p">(</span><span class="n">e</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span> <span class="c1"># Remove the prefix 0b</span>
|
||||
|
||||
<span class="n">r</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="n">exp</span> <span class="o">=</span> <span class="mi">1</span>
|
||||
<span class="c1"># We need to reverse, and to start from the right to left</span>
|
||||
<span class="c1"># Otherwise, we do on left to the right</span>
|
||||
<span class="c1"># It's dirty, maybe we can find another way to do that</span>
|
||||
<span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">binary</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
|
||||
<span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="s1">'1'</span><span class="p">:</span>
|
||||
<span class="n">r</span> <span class="o">*=</span> <span class="n">n</span> <span class="o">**</span> <span class="n">exp</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">n</span> <span class="o">**</span> <span class="n">exp</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
|
||||
<span class="n">exp</span> <span class="o">*=</span> <span class="mi">2</span>
|
||||
<span class="k">return</span> <span class="n">r</span>
|
||||
</code></pre></div></td></tr></table></div>
|
||||
</details>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
<div class="doc doc-object doc-function">
|
||||
|
||||
|
||||
<h2 id="Cryptotools.Utils.utils.egcd" class="doc doc-heading">
|
||||
<code class="highlight language-python"><span class="n">egcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span></code>
|
||||
|
||||
</h2>
|
||||
|
||||
|
||||
<div class="doc doc-contents ">
|
||||
|
||||
<p>This function compute the Extended Euclidean algorithm
|
||||
https://user.eng.umd.edu/~danadach/Cryptography_20/ExtEuclAlg.pdf
|
||||
https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm</p>
|
||||
|
||||
|
||||
<table class="field-list">
|
||||
<colgroup>
|
||||
<col class="field-name" />
|
||||
<col class="field-body" />
|
||||
</colgroup>
|
||||
<tbody valign="top">
|
||||
<tr class="field">
|
||||
<th class="field-name">Parameters:</th>
|
||||
<td class="field-body">
|
||||
<ul class="first simple">
|
||||
<li>
|
||||
<b><code>a</code></b>
|
||||
(<code><span title="Integer">Integer</span></code>)
|
||||
–
|
||||
<div class="doc-md-description">
|
||||
<p>the number a</p>
|
||||
</div>
|
||||
</li>
|
||||
<li>
|
||||
<b><code>b</code></b>
|
||||
(<code><span title="Integer">Integer</span></code>)
|
||||
–
|
||||
<div class="doc-md-description">
|
||||
<p>the number b</p>
|
||||
</div>
|
||||
</li>
|
||||
</ul>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<details class="quote">
|
||||
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
|
||||
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">18</span>
|
||||
<span class="normal">19</span>
|
||||
<span class="normal">20</span>
|
||||
<span class="normal">21</span>
|
||||
<span class="normal">22</span>
|
||||
<span class="normal">23</span>
|
||||
<span class="normal">24</span>
|
||||
<span class="normal">25</span>
|
||||
<span class="normal">26</span>
|
||||
<span class="normal">27</span>
|
||||
<span class="normal">28</span>
|
||||
<span class="normal">29</span>
|
||||
<span class="normal">30</span>
|
||||
<span class="normal">31</span>
|
||||
<span class="normal">32</span>
|
||||
<span class="normal">33</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span> <span class="nf">egcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
|
||||
<span class="w"> </span><span class="sd">"""</span>
|
||||
<span class="sd"> This function compute the Extended Euclidean algorithm</span>
|
||||
<span class="sd"> https://user.eng.umd.edu/~danadach/Cryptography_20/ExtEuclAlg.pdf</span>
|
||||
<span class="sd"> https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm</span>
|
||||
|
||||
<span class="sd"> Args:</span>
|
||||
<span class="sd"> a (Integer): the number a</span>
|
||||
<span class="sd"> b (Integer): the number b</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
|
||||
<span class="n">gcd</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">y1</span> <span class="o">=</span> <span class="n">egcd</span><span class="p">(</span><span class="n">b</span> <span class="o">%</span> <span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">y1</span> <span class="o">-</span> <span class="p">(</span><span class="n">b</span> <span class="o">//</span> <span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="n">x1</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">x1</span>
|
||||
<span class="k">return</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
|
||||
</code></pre></div></td></tr></table></div>
|
||||
</details>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
<div class="doc doc-object doc-function">
|
||||
|
||||
|
||||
<h2 id="Cryptotools.Utils.utils.exponent_squaring" class="doc doc-heading">
|
||||
<code class="highlight language-python"><span class="n">exponent_squaring</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span></code>
|
||||
|
||||
</h2>
|
||||
|
||||
|
||||
<div class="doc doc-contents ">
|
||||
|
||||
<p>This function perform an exponentiation squaring, which compute an integer power of a number based on the following algorithm:
|
||||
n ** e = {
|
||||
1 # if n is 0
|
||||
(n ** (e / 2)) ** 2 # if n is even
|
||||
((n ** (e - 1 / 2)) ** 2) * n # if n is odd
|
||||
}</p>
|
||||
|
||||
|
||||
<table class="field-list">
|
||||
<colgroup>
|
||||
<col class="field-name" />
|
||||
<col class="field-body" />
|
||||
</colgroup>
|
||||
<tbody valign="top">
|
||||
<tr class="field">
|
||||
<th class="field-name">Parameters:</th>
|
||||
<td class="field-body">
|
||||
<ul class="first simple">
|
||||
<li>
|
||||
<b><code>n</code></b>
|
||||
(<code><span title="Integer">Integer</span></code>)
|
||||
–
|
||||
<div class="doc-md-description">
|
||||
<p>n is the base of n ** e</p>
|
||||
</div>
|
||||
</li>
|
||||
<li>
|
||||
<b><code>e</code></b>
|
||||
(<code><span title="Integer">Integer</span></code>)
|
||||
–
|
||||
<div class="doc-md-description">
|
||||
<p>e is the exponent</p>
|
||||
</div>
|
||||
</li>
|
||||
</ul>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
|
||||
<details class="quote">
|
||||
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
|
||||
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">64</span>
|
||||
<span class="normal">65</span>
|
||||
<span class="normal">66</span>
|
||||
<span class="normal">67</span>
|
||||
<span class="normal">68</span>
|
||||
<span class="normal">69</span>
|
||||
<span class="normal">70</span>
|
||||
<span class="normal">71</span>
|
||||
<span class="normal">72</span>
|
||||
<span class="normal">73</span>
|
||||
<span class="normal">74</span>
|
||||
<span class="normal">75</span>
|
||||
<span class="normal">76</span>
|
||||
<span class="normal">77</span>
|
||||
<span class="normal">78</span>
|
||||
<span class="normal">79</span>
|
||||
<span class="normal">80</span>
|
||||
<span class="normal">81</span>
|
||||
<span class="normal">82</span>
|
||||
<span class="normal">83</span>
|
||||
<span class="normal">84</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span> <span class="nf">exponent_squaring</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
|
||||
<span class="w"> </span><span class="sd">"""</span>
|
||||
<span class="sd"> This function perform an exponentiation squaring, which compute an integer power of a number based on the following algorithm:</span>
|
||||
<span class="sd"> n ** e = {</span>
|
||||
<span class="sd"> 1 # if n is 0</span>
|
||||
<span class="sd"> (n ** (e / 2)) ** 2 # if n is even</span>
|
||||
<span class="sd"> ((n ** (e - 1 / 2)) ** 2) * n # if n is odd</span>
|
||||
<span class="sd"> }</span>
|
||||
|
||||
<span class="sd"> Args:</span>
|
||||
<span class="sd"> n (Integer): n is the base of n ** e</span>
|
||||
<span class="sd"> e (Integer): e is the exponent</span>
|
||||
<span class="sd"> """</span>
|
||||
<span class="k">if</span> <span class="n">e</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">exponent_squaring</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">e</span><span class="p">)</span>
|
||||
<span class="k">elif</span> <span class="n">e</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="mi">1</span>
|
||||
<span class="k">elif</span> <span class="n">e</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span> <span class="c1"># n is odd</span>
|
||||
<span class="k">return</span> <span class="n">exponent_squaring</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">e</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">n</span>
|
||||
<span class="k">elif</span> <span class="n">e</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># n is even</span>
|
||||
<span class="k">return</span> <span class="n">exponent_squaring</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">n</span><span class="p">,</span> <span class="n">e</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span>
|
||||
</code></pre></div></td></tr></table></div>
|
||||
</details>
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
<div class="doc doc-object doc-function">
|
||||
|
||||
|
||||
<h2 id="Cryptotools.Utils.utils.gcd" class="doc doc-heading">
|
||||
<code class="highlight language-python"><span class="n">gcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span></code>
|
||||
|
||||
</h2>
|
||||
|
||||
|
||||
<div class="doc doc-contents ">
|
||||
|
||||
<p>This function calculate the GCD (Greatest Common Divisor of the number a of b
|
||||
Args:
|
||||
a (Integer): the number a
|
||||
b (integer): the number b</p>
|
||||
|
||||
|
||||
<details class="return" open>
|
||||
<summary>Return</summary>
|
||||
<p>the GCD</p>
|
||||
</details>
|
||||
|
||||
<details class="quote">
|
||||
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
|
||||
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"> 3</span>
|
||||
<span class="normal"> 4</span>
|
||||
<span class="normal"> 5</span>
|
||||
<span class="normal"> 6</span>
|
||||
<span class="normal"> 7</span>
|
||||
<span class="normal"> 8</span>
|
||||
<span class="normal"> 9</span>
|
||||
<span class="normal">10</span>
|
||||
<span class="normal">11</span>
|
||||
<span class="normal">12</span>
|
||||
<span class="normal">13</span>
|
||||
<span class="normal">14</span>
|
||||
<span class="normal">15</span>
|
||||
<span class="normal">16</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span> <span class="nf">gcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
|
||||
<span class="w"> </span><span class="sd">"""</span>
|
||||
<span class="sd"> This function calculate the GCD (Greatest Common Divisor of the number a of b</span>
|
||||
<span class="sd"> Args:</span>
|
||||
<span class="sd"> a (Integer): the number a</span>
|
||||
<span class="sd"> b (integer): the number b</span>
|
||||
|
||||
<span class="sd"> Return:</span>
|
||||
<span class="sd"> the GCD </span>
|
||||
<span class="sd"> """</span>
|
||||
|
||||
<span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||||
<span class="k">return</span> <span class="n">a</span>
|
||||
<span class="k">return</span> <span class="n">gcd</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="o">%</span><span class="n">b</span><span class="p">)</span>
|
||||
</code></pre></div></td></tr></table></div>
|
||||
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