# Examples of Elliptic Curves, with Graphical Representations
#
# plot([],figsize=(4,4),title='Here is a Graph',frame=True,axes_labels=['$x$-axis, units','$y$-axis, units'])
#
# The examples are from Stein's book ...
# Example 6.1.2 from Ch.6
E=EllipticCurve([-5,4]); E # y2=x^3-5x+4 (implicitely) over the rational field
E.plot(thickness=4, rgbcolor=(0.1,0.7,0.1),title="Elliptic Curve",frame=True)

# Elliptic curve over finite fields, a.k.a. Galois Fields: GF(p)=Z/pZ=Fp
E=EllipticCurve(GF(37),[1,0]); E
E.plot(pointsize=45)

# ... what a mess :) ...

EC2=EllipticCurve(GF(7),[1,0]); EC2
EC2.plot(pointsize=45)

# Example with prescribed roots: y^2=x^3-x => x=-1,0,1 roots
#  intersects the axes at (0,-1),(0,0),(0,1)
EC3=EllipticCurve(GF(7),[-1,0]); EC3
EC3.plot(pointsize=25,color="red",figsize=(6,2))

# Addition on Elliptic Curves
#
# Ex. 6.2.3 - The Group Law
E=EllipticCurve([-5,4])
P=E([1,0]); Q=E([0,2]); 
print P, Q, P+Q

# Plot with colors the points and the sum ...

P+P

P+Q+Q+Q+Q

P+4*Q

4*Q

Q+Q

# Example of addition for E(GF(7)): roots -1,0,1 => x^3+x => a=-1, b=0
E=EllipticCurve(GF(7),[-1,0]); E
E.plot(pointsize=45)

 # Selecting points on the curve and adding them ...
P=E([1,0]); Q=E([4,2]); R=E([5,1])
P+P
P+Q

# Changing the Finite Field from F7 to F5 ...
#
# Example K=GF(5), EC: roots -1,0,1
E=EllipticCurve(GF(5),[-1,0]); E
E.plot(pointsize=45)

P=E([1,0]); Q=E([0,0]); R=E([2,1]); S=E([2,-1])
O=Q+Q
O
2*P

# P and Q have order 2
2*E([-1,0])
2*R

# 2R=Q, so ord(R)=4; since |E|=7+1 (point at infinity: O), either it's Z4xZ2 or Z8 ...
2*S

# of course S=-R has the same order and 3S=R and they form one orbit / 4-cycle
3*S
2*E([-2,2])

# [-2,2] has order 4 too, so ... pick two generators: R and [-1,0]?
E([-2,2])-R

# yep! now sketch nicely the 4x2-1 "curved pointed torus" inside the 5x5 torus ... if you can :) where is "infinity" O?

# the point is: "continuity", i.e. smooth connecting dots, should be replaced by the (cyclic) order defined by orbits of generators; so join points as a graph ... (Cayley graph of a group with respect to a set of generators; but this is another story :)

# Another example
E=EllipticCurve(GF(5),[1,1]); E
E.plot(pointsize=45)

A=E([0,1])
3*A
6*A

B=E([2,1])
B*3

3*E([2,4])
E([3,1])+E([4,2])

# So A=(0,1) is a generator

# 
# Another example: y^2=x^3+1 over GF(5)
E=EllipticCurve(GF(5),[0,1])
E.plot(pointsize=45)
E([0,1])+E([2,2])

# Note: the geometric construction fails: (2,2) is a double point of int. with the line => 
# 3rd point: -(2,2)=(2,-2)=(2,3)

# *****************************************************************
#             6.4 Example of Discrete Log Problem
# *****************************************************************
E=EllipticCurve(GF(7),[1,1]); E
E.plot(pointsize=45)

3*E([2,2])
2*E([2,2])