Tags: misc math
Rating:
# INS'hAck 2018: Math killer - hard -
**Category:** MISC |
**Points:** 60 |
**Description:**
> There's another way to solve the 'Math killer - easy' challenge, if you find it you'll get another flag!
>
> If you solve it the easy way first, its flag is a hint.
>
> Reminder: https://math-killer.ctf.insecurity-insa.fr/solve?a=...&b=...&c=...
>
> [CHALLENGE'S FILE](https://static.ctf.insecurity-insa.fr/chall.png)
>
> 
## Write-up
Solving the easy one first give us a hint, the easy one flag was `INSA{try_positive_solutions_now}`
So, we need to solve the eqation by using only positive numbers. After doing some research I found that [paper](http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf) by Bremner and MacLeod called **An unusual cubic representation problem**.
Also, I found a `CoCalc` code implementation written by Alon Amit that solving a similar equation.
I modified the code to be able to solve any a/(b+c)+b/(a+c)+c/(a+b)=N eqation with even N number.
```python
N=6
e=(4*(N^2)) + ((12*N)-3)
f=32*(N+3)
eq=EllipticCurve([0,e,0,f,0]) # Define the elliptic curve corresponding to the equation a/(b+c)+b/(a+c)+c/(a+b)=N
eq.rank()
eq.gens()
P=eq(-200,680) # This is a generator for the group of rational points from ee.gens() result
def orig(P,N):
x=P[0]
y=P[1]
a=(8*(N+3)-x+y)/(2*(N+3)*(4-x))
b=(8*(N+3)-x-y)/(2*(N+3)*(4-x))
c=(-4*(N+3)-(N+2)*x)/((N+3)*(4-x))
da=denominator(a)
db=denominator(b)
dc=denominator(c)
l=lcm(da,lcm(db,dc))
return [a*l,b*l,c*l]
orig(P,N)
m=11 # The smallest integer m such that one of the points mP + T, T ∈ Tor(EN(Q))
u=orig(m*P,N) # Bremner and MacLeod noticed that 11P yields a positive solution for N=6
(a,b,c)=(u[0],u[1],u[2])
```
That give us a 134 digits a,b and c numbers which satisfy a given functional equation.
```python
...
(a,b,c)
(20260869859883222379931520298326390700152988332214525711323500132179943287700005601210288797153868533207131302477269470450828233936557,
2250324022012683866886426461942494811141200084921223218461967377588564477616220767789632257358521952443049813799712386367623925971447,
1218343242702905855792264237868803223073090298310121297526752830558323845503910071851999217959704024280699759290559009162035102974023)
a/(b+c)+b/(a+c)+c/(a+b)
6
```
By adding the numbers to the given [URL](https://math-killer.ctf.insecurity-insa.fr/solve?a=20260869859883222379931520298326390700152988332214525711323500132179943287700005601210288797153868533207131302477269470450828233936557&b=2250324022012683866886426461942494811141200084921223218461967377588564477616220767789632257358521952443049813799712386367623925971447&c=1218343242702905855792264237868803223073090298310121297526752830558323845503910071851999217959704024280699759290559009162035102974023) and BINGO!
```css
{
"flags": {
"easy": "INSA{try_positive_solutions_now}",
"hard": "INSA{OMG_you_actually_killed_math}"
},
"message": "Congrats for solving hard mode ! you get both flags for free :)",
"status": "success"
}
```
The flag is `INSA{OMG_you_actually_killed_math}`
