# PicoCTF 2022 — Crypto: Sum-O-Primes

# Description

We have so much faith in RSA we give you not just the product of the primes, but their sum as well!

gen.py output.txt

# Solve

This problem can be solved as we have the product and sum of the 2 primes. This can be achieve by using a quadratic equation.

Here we can calculate the square root value as `theta`

`theta = ((bb**2) - (4*nn)).sqrt()`

Then solve for `p`

and `q`

`p = (bb+theta)//2`

q = (bb-theta)//2

With `p`

and `q`

we can rebuild the private exponent and recover the plain text flag

`phi = (int(p)-1)*(int(q)-1)`

d = pow(e,-1,phi)

pt = pow(int(c,16),d,n)

print( long_to_bytes(pt))

# Flag

`picoCTF{674b189f}`