## ECC - 100 (Cryptography)

#### Writeup by ZIceZ

Created: 2014-11-07 21:04:18

Last modified: 2014-11-11 19:29:05

### Problem

We found a weird piece of paper with this written on it. I can't make heads or tails of it, but it seems to be talking about an encoded message. Can you get the message for us?

### Hint

Use a computer algebra system with builtin support for elliptic curves, such as Sage. This elliptic curve is defined over the group of integers Z/nZ (Zmod(n) in Sage) and is in Weierstrass form (a1 = a2 = a3 = 0).

## Answer

### Overview

Elliptical curve encryption.

### Details

First solved for the variable b to complete the elliptic curve cryptosystem.
$y^2 = x^3 + a(x) + b \bmod n$ where a = 0
Since we have a coordinate of the system, b can be solved by plugging in X and Y.
$12418605208975891779391^2 = 236857987845294655469221^3 + b \bmod 928669833265826932708591$
Wolfram alpha can solve this equation for you giving
$b = 268892790095131465246420$
With the all the variables found, the cryptosystem can be built to decrypt your message. As the hint suggested, Sage has a built in library that can handle all the calculations. There is an online version of Sage, so downloading it is not necessary. For information on how to setup an elliptic curve, refer to here.

```
F = FiniteField(928669833265826932708591)
E = EllipticCurve(F,[0,268892790095131465246420])
G = E.point((236857987845294655469221, 12418605208975891779391))
d = 87441340171043308346177
G*d
```

The result is (6976767380847367326785 : 828669833265826932708578 : 1). Decode the X and Y coordinate with the given function.

```
def STR(a):
a = str(a)
for i in range(0, len(a) - 1, 2):
print(chr(int(a[i:i+2])))
print STR(6976767380847367326785)
print STR(828669833265826932708578)
```

### Flag

```
ELLIPTIC CURVES ARE FUN
```