diff --git a/Cryptotools/Groups/curve.py b/Cryptotools/Groups/curve.py new file mode 100644 index 0000000..0cb697e --- /dev/null +++ b/Cryptotools/Groups/curve.py @@ -0,0 +1,159 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.elliptic import Point +import numpy as np +from math import sqrt + + +class Curve: + # Curve + WEIERSTRASS = 0 + MONTGOMERY = 1 + + + def __init__(self, a, b, t): + self._a = a + self._b = b + + if t not in (Curve.WEIERSTRASS, Curve.MONTGOMERY): + raise Exception(f"The type of the curve is not recognized") + + self._type = t + self._xtmp = np.linspace(-5, 5, 500).tolist() + self._x = list() + self._y = list() + self._yn = list() + self._points = list() + self._pointsSym = list() + + def f(self, x): + if self._type == Curve.WEIERSTRASS: + y = pow(x, 3) + (self._a * x) + self._b + if self._type == Curve.MONTGOMERY: + y = pow(x, 3) + (3 * pow(x, 2)) + x + if y > 0: + return sqrt(y) + return None + + def generatePoints(self): + for x in self._xtmp: + y = self.f(x) + if y is None: + continue + + self._x.append(x) + self._y.append(y) + self._yn.append(-y) + self._points.append(Point( + x, + y + )) + self._pointsSym.append(Point( + x, + -y + )) + + @property + def x(self): + return self._x + + @property + def y(self): + return self._y + + @property + def yn(self): + return self._yn + + def getPoints(self): + return self._points + + def getPointsSym(self): + return self._pointsSym + + def add(self, P, Q) -> Point: + """ + This function operathe addition operation on two points P and Q + + Args: + P (Object): The first Point on the curve + Q (Object): The second Point on the curve + + Returns: + Return the Point object R + """ + + ## Check if P or Q are infinity + if (P.x, P.y) == (0, 0) and (Q.x, Q.y) == (0, 0): + return Point(0, 0) + elif (P.x, P.y) == (0, 0): + return Point(Q.x, Q.y) + elif (Q.x, Q.y) == (0, 0): + return Point(P.x, P.y) + + # point doubling + if P.x == Q.x: + # Infinity + if P.y != Q.y or Q.y == 0: + return Point(0, 0) + + # Point doubling + try: + inv = pow(2 * P.y, -1); # It's working with the inverse modular, WHY ??? + m = ((3 * pow(P.x, 2)) + self._a) * inv + except ValueError: + return Point(0, 0) + + else: + try: + inv = pow(Q.x - P.x, -1) + m = (Q.y - P.y) * inv + except ValueError: + # May call this Exception: base is not invertible for the given modulus + # I return an Infinity point until I fixed that + return Point(0, 0) + + xr = (pow(m, 2) - P.x - Q.x) + + yr = (m * (P.x - xr)) - P.y + return Point(xr, yr) + + def scalar(self, P, n) -> Point: + """ + This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add. + + Args: + P (point): the Point to multiplication + n (Integer): multiplicate n time P + + Returns: + Return the result of the Scalar multiplication + """ + binary = bin(n)[2:] + binary = binary[::-1] # We need to reverse the binary + + nP = Point(0, 0) + Rtmp = P + + for b in binary: + if b == '1': + nP = self.add(nP, Rtmp) + Rtmp = self.add(Rtmp, Rtmp) # Double P + + return nP + + def find_reverse(self, P): + """ + This function return the reverse of the Point P + Args: + P (Point): Point object to find + + Returns: + Return the object Pr, which is the reverse point of P + """ + Pr = None + for p in self._pointsSym: + if P.x == p.x and -P.y == p.y: + Pr = Point(p.x, p.y) + break + return Pr diff --git a/Cryptotools/Groups/point.py b/Cryptotools/Groups/point.py new file mode 100644 index 0000000..0ed5c7c --- /dev/null +++ b/Cryptotools/Groups/point.py @@ -0,0 +1,34 @@ +#!/usr/bin/env python3 + + +class Point: + """ + This simple class represent the Point at the coordinate x and y in a plan + + Attributes: + x (Integer): Position at the x + y (Integer): Position at the y + """ + def __init__(self, x, y): + self._x = x + self._y = y + + @property + def x(self): + return self._x + + @property + def y(self): + return self._y + + @x.setter + def x(self, x): + self._x = x + + @y.setter + def y(self, y): + self._y = y + + def __eq__(self, other): + print(self._x, other.x) + return self._x == other.x and self._y == other.y diff --git a/docs/curves.md b/docs/curves.md new file mode 100644 index 0000000..608143e --- /dev/null +++ b/docs/curves.md @@ -0,0 +1,6 @@ +# Elliptic Curves +## Points +::: Cryptotools.Groups.point + +## Curves +::: Cryptotools.Groups.curve diff --git a/docs/example-curves.md b/docs/example-curves.md new file mode 100644 index 0000000..42382ef --- /dev/null +++ b/docs/example-curves.md @@ -0,0 +1,30 @@ +# Generating Elliptic Curves + +In the following section, we are going to see how to generate Elliptic Curves and draw the curve with Python. + +## A Weierstrass Curve +The python code below draw a Weierstrass Curve: + +``` +#!/usr/bin/env python3 + +import matplotlib.pyplot as plt +from Cryptotools.Groups.curve import Curve + + +a = 3 +b = 8 + +curve = Curve(a, b, Curve.WEIERSTRASS) +x = curve.x +curve.generatePoints() +y = curve.y +yn = curve.yn +points = curve.getPoints() + +plt.figure(figsize=(10, 6)) +plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$') +plt.plot(x, yn, color='b', ) +plt.legend() +plt.show() +``` diff --git a/docs/examples-rsa-keys.md b/docs/example-rsa-keys.md similarity index 100% rename from docs/examples-rsa-keys.md rename to docs/example-rsa-keys.md diff --git a/docs/index.md b/docs/index.md index f434fae..5e9e02b 100644 --- a/docs/index.md +++ b/docs/index.md @@ -5,11 +5,12 @@ * Low-Level Cryptographic * [Number Theory](/number-theory) * [Group Theory](/group-theory) + * [Curves](/curves) * Public Keys: * [RSA](/rsa) * Utils * [Utils](/utils) * Examples: - * [Generating RSA keys](/examples-rsa-keys) - + * [Generating RSA keys](/example-rsa-keys) + * [Generate curves](/example-curves) diff --git a/examples/elliptic/curve.py b/examples/elliptic/curve.py new file mode 100644 index 0000000..05fbfcc --- /dev/null +++ b/examples/elliptic/curve.py @@ -0,0 +1,24 @@ +#!/usr/bin/env python3 + +import matplotlib.pyplot as plt +from Cryptotools.Groups.curve import Curve + + +a = 3 +b = 8 + +curve = Curve(a, b, Curve.WEIERSTRASS) +x = curve.x +curve.generatePoints() +y = curve.y +yn = curve.yn +points = curve.getPoints() + +#print(x) +#print(y) + +plt.figure(figsize=(10, 6)) +plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$') +plt.plot(x, yn, color='b', ) +plt.legend() +plt.show() diff --git a/examples/elliptic/curve_add.py b/examples/elliptic/curve_add.py new file mode 100644 index 0000000..de70dd9 --- /dev/null +++ b/examples/elliptic/curve_add.py @@ -0,0 +1,48 @@ +#!/usr/bin/env python3 + +import matplotlib.pyplot as plt +from Cryptotools.Groups.curve import Curve + + +a = 3 +b = 8 + +curve = Curve(a, b, Curve.WEIERSTRASS) +x = curve.x +curve.generatePoints() +y = curve.y +yn = curve.yn +points = curve.getPoints() + +P = points[10] +Q = points[55] +R = points[247] +#print(f"{P.x} {P.y}") +#print(f"{Q.x} {Q.y}") + +# Make an addition +Rp = curve.add(P, Q) +#print(f"{Rp.x} {Rp.y}") + +#print(yn) +#print(x) +#print(y) + +plt.figure(figsize=(10, 6)) +plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$') +plt.plot(x, yn, color='b', ) +plt.plot(P.x, P.y, marker='o', color="red") +plt.annotate('P', (P.x, P.y + 0.5)) + +plt.plot(Q.x, Q.y, marker='o', color="red") +plt.annotate('Q', (Q.x, Q.y + 0.5)) + +plt.plot(R.x, R.y, marker='o', color="red") +plt.annotate('R', (R.x - 0.2, R.y + 0.5)) + +plt.plot(Rp.x, Rp.y, marker='o', color="red") +plt.annotate('R\'', (Rp.x - 0.2, Rp.y + 0.5)) +plt.axline([P.x, P.y], [Q.x, Q.y], color="red") +plt.axline([R.x, R.y], [Rp.x, Rp.y], linestyle="--", color="red") +plt.legend() +plt.show() diff --git a/examples/elliptic/curve_scalar.py b/examples/elliptic/curve_scalar.py new file mode 100644 index 0000000..6f1d893 --- /dev/null +++ b/examples/elliptic/curve_scalar.py @@ -0,0 +1,44 @@ +#!/usr/bin/env python3 + +import matplotlib.pyplot as plt +from Cryptotools.Groups.curve import Curve +from Cryptotools.Groups.point import Point + + +a = 3 +b = 8 + +curve = Curve(a, b, Curve.WEIERSTRASS) +x = curve.x +curve.generatePoints() +y = curve.y +yn = curve.yn +points = curve.getPoints() + +P = points[10] +nP = curve.scalar(P, 5) +print(f"{nP.x} {nP.y}") + +# For testing, we may add n times with the addition operation for the Scalar Multiplication +tmp = Point(0, 0) +for i in range(5): + tmp = curve.add(P, tmp) + +# Unfortunately, the result is approximatively the same +# the multiplication need to be more accurate +# if tmp == nP: +# print(True) + +plt.figure(figsize=(10, 6)) +plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$') +plt.plot(x, yn, color='b', ) +plt.plot(P.x, P.y, marker='o', color="red") +plt.annotate('P', (P.x, P.y + 0.5)) + +plt.plot(nP.x, nP.y, marker='o', color="red") +plt.annotate('nP', (nP.x, nP.y + 0.5)) + + + +plt.legend() +plt.show() diff --git a/examples/elliptic/point_at_infinity.py b/examples/elliptic/point_at_infinity.py new file mode 100644 index 0000000..40b9e09 --- /dev/null +++ b/examples/elliptic/point_at_infinity.py @@ -0,0 +1,43 @@ +#!/usr/bin/env python3 + +from Cryptotools.Groups.point import Point +from Cryptotools.Groups.curve import Curve +import numpy as np +import matplotlib.pyplot as plt +from math import sqrt + + +a = 3 +b = 8 + +curve = Curve(a, b, Curve.WEIERSTRASS) +x = curve.x +curve.generatePoints() +y = curve.y +yn = curve.yn +points = curve.getPoints() +pointsReverse = curve.getPointsSym() + +P = points[10] +Q = curve.find_reverse(P) + +# We made the addition +# The result is the point at infinity +R = curve.add(P, Q) +print(f"{R.x} {R.y}") + + +plt.figure(figsize=(10, 6)) +plt.plot(x, y, color='b', label=f'$y^2 = x^3 + {a}x + {b}$') +plt.plot(x, yn, color='b', ) +plt.plot(P.x, P.y, marker='o', color="red") +plt.annotate('P', (P.x, P.y + 0.5)) + +plt.plot(Q.x, Q.y, marker='o', color="red") +plt.text(-1 , 11, f"R = infinity") +plt.annotate('Q', (Q.x, Q.y + 0.5)) + + +plt.axline([P.x, P.y], [Q.x, Q.y], color="red") +plt.legend() +plt.show() diff --git a/mkdocs.yml b/mkdocs.yml index f04ea4d..ca3986a 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -11,9 +11,11 @@ nav: - Low-level cryptographic: - Number theory: number-theory.md - Group theory: group-theory.md + - Curves: curves.md - Public Keys: - RSA: rsa.md - Utils: - Utils: utils.md - Examples: - - Generating RSA Keys: examples-rsa-keys.md + - Generating RSA Keys: example-rsa-keys.md + - Generating Curves: example-curves.md diff --git a/site/404.html b/site/404.html index 7a70374..45594df 100644 --- a/site/404.html +++ b/site/404.html @@ -43,15 +43,24 @@
  • Group theory
    • +
  • Curves +

    • Public Keys

    +

    Utils

    +

    Examples

    diff --git a/site/curves/index.html b/site/curves/index.html new file mode 100644 index 0000000..bef9f0d --- /dev/null +++ b/site/curves/index.html @@ -0,0 +1,1096 @@ + + + + + + + + Curves - CryptoTools documentation + + + + + + + + + + + + + + +
    + + +
    + +
    +
    +
      +
      • +
    • Low-level cryptographic
      • +
    • Curves
      • +
    • +
      • +
    +
    +
    +
    +
    + +

    Elliptic Curves

    +

    Points

    + + +
    + + + + +
    + + + + + + + + + + +
    + + + + + + + + + +
    + + + +

    + Point + + +

    + + +
    + + +

    This simple class represent the Point at the coordinate x and y in a plan

    + + + + + + + + + + + + + +
    Attributes: +
      +
    • + x + (Integer) + – +
      +

      Position at the x

      +
      +
      • +
    • + y + (Integer) + – +
      +

      Position at the y

      +
      +
      • +
    +
    + + + + + + + +
    + Source code in Cryptotools/Groups/point.py + 
     4
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    class Point:
+    """
+        This simple class represent the Point at the coordinate x and y in a plan
+
+        Attributes:
+            x (Integer): Position at the x
+            y (Integer): Position at the y
+    """
+    def __init__(self, x, y):
+        self._x = x
+        self._y = y
+
+    @property
+    def x(self):
+        return self._x
+
+    @property
+    def y(self):
+        return self._y
+
+    @x.setter
+    def x(self, x):
+        self._x = x
+
+    @y.setter
+    def y(self, y):
+        self._y = y
+
+    def __eq__(self, other):
+        print(self._x, other.x)
+        return self._x == other.x and self._y == other.y
+
    +
    + + + +
    + + + + + + + + + + + + +
    + +
    + +
    + + + + +
    + +
    + +

    Curves

    + + +
    + + + + +
    + + + + + + + + + + +
    + + + + + + + + + +
    + + + +

    + Curve + + +

    + + +
    + + + + + + + + + +
    + Source code in Cryptotools/Groups/curve.py + 
      8
+  9
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+159
    class Curve:
+    # Curve
+    WEIERSTRASS = 0
+    MONTGOMERY = 1
+
+
+    def __init__(self, a, b, t):
+        self._a = a
+        self._b = b
+
+        if t not in (Curve.WEIERSTRASS, Curve.MONTGOMERY):
+            raise Exception(f"The type of the curve is not recognized")
+
+        self._type = t
+        self._xtmp = np.linspace(-5, 5, 500).tolist()
+        self._x = list()
+        self._y = list()
+        self._yn = list()
+        self._points = list()
+        self._pointsSym = list()
+
+    def f(self, x):
+        if self._type == Curve.WEIERSTRASS:
+            y = pow(x, 3) + (self._a * x) + self._b
+        if self._type == Curve.MONTGOMERY:
+            y = pow(x, 3) + (3 * pow(x, 2)) + x
+        if y > 0:
+            return sqrt(y)
+        return None
+
+    def generatePoints(self):
+        for x in self._xtmp:
+            y = self.f(x)
+            if y is None:
+                continue
+
+            self._x.append(x)
+            self._y.append(y)
+            self._yn.append(-y)
+            self._points.append(Point(
+                x,
+                y
+            ))
+            self._pointsSym.append(Point(
+                x,
+                -y 
+            ))
+
+    @property
+    def x(self):
+        return self._x
+
+    @property
+    def y(self):
+        return self._y
+
+    @property
+    def yn(self):
+        return self._yn
+
+    def getPoints(self):
+        return self._points
+
+    def getPointsSym(self):
+        return self._pointsSym
+
+    def add(self, P, Q) -> Point:
+        """
+            This function operathe addition operation on two points P and Q
+
+            Args:
+                P (Object): The first Point on the curve
+                Q (Object): The second Point on the curve
+
+            Returns:
+                Return the Point object R
+        """
+
+        ## Check if P or Q are infinity
+        if (P.x, P.y) == (0, 0) and (Q.x, Q.y) == (0, 0):
+            return Point(0, 0)
+        elif (P.x, P.y) == (0, 0):
+            return Point(Q.x, Q.y)
+        elif (Q.x, Q.y) == (0, 0):
+            return Point(P.x, P.y)
+
+        # point doubling
+        if P.x == Q.x:
+            # Infinity
+            if P.y != Q.y or Q.y == 0:
+                return Point(0, 0)
+
+            # Point doubling
+            try:
+                inv = pow(2 * P.y, -1); # It's working with the inverse modular, WHY ???
+                m = ((3 * pow(P.x, 2)) + self._a) * inv 
+            except ValueError:
+                return Point(0, 0)
+
+        else:
+            try:
+                inv = pow(Q.x - P.x, -1)
+                m = (Q.y - P.y) * inv
+            except ValueError:
+                # May call this Exception: base is not invertible for the given modulus
+                # I return an Infinity point until I fixed that
+                return Point(0, 0)
+
+        xr = (pow(m, 2) - P.x - Q.x)
+
+        yr = (m * (P.x - xr)) - P.y
+        return Point(xr, yr)
+
+    def scalar(self, P, n) -> Point:
+        """
+            This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add.
+
+            Args:
+                P (point): the Point to multiplication
+                n (Integer): multiplicate n time P
+
+            Returns:
+                Return the result of the Scalar multiplication
+        """
+        binary = bin(n)[2:]
+        binary = binary[::-1] # We need to reverse the binary
+
+        nP = Point(0, 0)
+        Rtmp = P
+
+        for b in binary:
+            if b == '1':
+                nP = self.add(nP, Rtmp)
+            Rtmp = self.add(Rtmp, Rtmp)  # Double P
+
+        return nP
+
+    def find_reverse(self, P):
+        """
+            This function return the reverse of the Point P
+            Args:
+                P (Point): Point object to find 
+
+            Returns:
+                Return the object Pr, which is the reverse point of P
+        """
+        Pr = None
+        for p in self._pointsSym:
+             if P.x == p.x and -P.y == p.y:
+                 Pr = Point(p.x, p.y)
+                 break
+        return Pr
+
    +
    + + + +
    + + + + + + + + + + +
    + + +

    + add(P, Q) + +

    + + +
    + +

    This function operathe addition operation on two points P and Q

    + + + + + + + + + + + + + +
    Parameters: +
      +
    • + P + (Object) + – +
      +

      The first Point on the curve

      +
      +
      • +
    • + Q + (Object) + – +
      +

      The second Point on the curve

      +
      +
      • +
    +
    + + + + + + + + + + + + +
    Returns: +
      +
    • + Point + – +
      +

      Return the Point object R

      +
      +
      • +
    +
    + +
    + Source code in Cryptotools/Groups/curve.py + 
     74
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    def add(self, P, Q) -> Point:
+    """
+        This function operathe addition operation on two points P and Q
+
+        Args:
+            P (Object): The first Point on the curve
+            Q (Object): The second Point on the curve
+
+        Returns:
+            Return the Point object R
+    """
+
+    ## Check if P or Q are infinity
+    if (P.x, P.y) == (0, 0) and (Q.x, Q.y) == (0, 0):
+        return Point(0, 0)
+    elif (P.x, P.y) == (0, 0):
+        return Point(Q.x, Q.y)
+    elif (Q.x, Q.y) == (0, 0):
+        return Point(P.x, P.y)
+
+    # point doubling
+    if P.x == Q.x:
+        # Infinity
+        if P.y != Q.y or Q.y == 0:
+            return Point(0, 0)
+
+        # Point doubling
+        try:
+            inv = pow(2 * P.y, -1); # It's working with the inverse modular, WHY ???
+            m = ((3 * pow(P.x, 2)) + self._a) * inv 
+        except ValueError:
+            return Point(0, 0)
+
+    else:
+        try:
+            inv = pow(Q.x - P.x, -1)
+            m = (Q.y - P.y) * inv
+        except ValueError:
+            # May call this Exception: base is not invertible for the given modulus
+            # I return an Infinity point until I fixed that
+            return Point(0, 0)
+
+    xr = (pow(m, 2) - P.x - Q.x)
+
+    yr = (m * (P.x - xr)) - P.y
+    return Point(xr, yr)
+
    +
    +
    + +
    + +
    + + +

    + find_reverse(P) + +

    + + +
    + +

    This function return the reverse of the Point P +Args: + P (Point): Point object to find

    + + + + + + + + + + + + + +
    Returns: +
      +
    • + – +
      +

      Return the object Pr, which is the reverse point of P

      +
      +
      • +
    +
    + +
    + Source code in Cryptotools/Groups/curve.py + 
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+159
    def find_reverse(self, P):
+    """
+        This function return the reverse of the Point P
+        Args:
+            P (Point): Point object to find 
+
+        Returns:
+            Return the object Pr, which is the reverse point of P
+    """
+    Pr = None
+    for p in self._pointsSym:
+         if P.x == p.x and -P.y == p.y:
+             Pr = Point(p.x, p.y)
+             break
+    return Pr
+
    +
    +
    + +
    + +
    + + +

    + scalar(P, n) + +

    + + +
    + +

    This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add.

    + + + + + + + + + + + + + +
    Parameters: +
      +
    • + P + (point) + – +
      +

      the Point to multiplication

      +
      +
      • +
    • + n + (Integer) + – +
      +

      multiplicate n time P

      +
      +
      • +
    +
    + + + + + + + + + + + + +
    Returns: +
      +
    • + Point + – +
      +

      Return the result of the Scalar multiplication

      +
      +
      • +
    +
    + +
    + Source code in Cryptotools/Groups/curve.py + 
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    def scalar(self, P, n) -> Point:
+    """
+        This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add.
+
+        Args:
+            P (point): the Point to multiplication
+            n (Integer): multiplicate n time P
+
+        Returns:
+            Return the result of the Scalar multiplication
+    """
+    binary = bin(n)[2:]
+    binary = binary[::-1] # We need to reverse the binary
+
+    nP = Point(0, 0)
+    Rtmp = P
+
+    for b in binary:
+        if b == '1':
+            nP = self.add(nP, Rtmp)
+        Rtmp = self.add(Rtmp, Rtmp)  # Double P
+
+    return nP
+
    +
    +
    + +
    + + + +
    + +
    + +
    + + + + +
    + +
    + +
    + +
    +
    + +
    +
    + +
    + +
    + +
    + + + + « Previous + + + Next » + + +
    + + + + + + + + diff --git a/site/examples-rsa-keys/index.html b/site/example-rsa-keys/index.html similarity index 87% rename from site/examples-rsa-keys/index.html rename to site/example-rsa-keys/index.html index af1636d..3e520fd 100644 --- a/site/examples-rsa-keys/index.html +++ b/site/example-rsa-keys/index.html @@ -14,7 +14,7 @@ @@ -50,18 +50,27 @@
  • Group theory
    • +
  • Curves +

    • Public Keys

    +

    Utils

    +

    Examples

    @@ -116,7 +125,7 @@ print(f"Plaintext: {plaintext}")