RedPwn 2021 - Blecc [Crypto]

CTF: RedPwn 2021

Challenge information:

Name: Blecc
Category: Crypto
Description: Blecc! What are all these numbers? This doesn't look like RSA...
Author: AdnanSlef
Points: 119
Solves: 146 out of 1418 teams
Flag: flag{m1n1_3cc}

The challenge contains a file named blecc.txt with the following content:
p = 17459102747413984477
a = 2
b = 3
G = (15579091807671783999, 4313814846862507155)
Q = (8859996588597792495, 2628834476186361781)
d = ???
Can you help me find d?
Decode it as a string and wrap in flag format.


The description, the file contents and the challenge's name itself Blecc gives us a big hint "ECC" regarding Elliptic-Curve Cryptography. We are given:
p the prime modulus.
a, b, G and Q

Usually a capital letter contains the point on the elliptic curve and lowercase letters mean integer scalar.

Once we understand the information given, we are ready to compute d. I am using SageMath to do the maths. The finite field can be defined as F = GF(P), and we can define an Elliptic Curve E in Sage using the EllipticCurve command: E = EllipticCurve(GF(p),[a, b]).

In this case Q = k*G so Q = d*G or Q = GF(p)*G, and solve d from the equation d = G.discrete_log(Q).

We can use long_to_bytes from pycryptodome or use x.to_bytes((x.bit_length() + 7)//8, 'big') to convert 7868191182322623331 to a string format:

And the flag for this crypto challenge is flag{m1n1_3cc}.