# curve12833227

Apparently curves with numbers in their name are now cool, so I decided to make my own.

2 hours

# Behavior

We have the flag encrypted by key x, and a point xG on the elliptic curve.

# Solution

## TL;DR

1. Recover the equation of the elliptic curve.
2. Map points on the curve to an multiplicative group.
3. Calculate discrete logarithm to find x.

The slope of the curve is

3x^2 + 4x+ 1 / 2y
compare to the derivative of y^2 = x^3 + cx^2 + ax + b
3x^2 + 2cx + a / 2y
E: y^2 = x^3 + 2x^2 + x + b
Evaluate on the point (4, 10), we found that b = 0
E: y^2 = x^3 + 2x^2 + x = x (x + 1)^2


That is a singular curve, which is isomorphic to a multiplicative group.

Here's a great article about how to solve discrete log on these curve.

First, we translate the curve to

y = x^2 (x - 1)


and find the square root of -1:

z^2 = -1 mod p
z = 189460722536962819362366221450020958767
f(x, y) = (y + zx) / (y - zx)


where f is an isomorphism from E to Fp.

And then we can get x by solving the discrete logarithm using sage:

Fp = IntegerModRing(p)
g = Fp(f(Gx, Gy))
t = Fp(f(Tx, Ty))
x = t.log(g)


It takes several minutes to solve, and then we can decrypt our flag with x :)