Add Elliptic Curve and docs

examples/elliptic/curve.py

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+#!/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()

examples/elliptic/curve_add.py

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+#!/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()

examples/elliptic/curve_scalar.py

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+#!/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()

examples/elliptic/point_at_infinity.py

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+#!/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()