Raining Primes

hackthebox

Task: RSA service whose primes have form p=a*r+b with a hidden secret prime r, plus a multiplicative key-derivation oracle and AES-ECB flag encryption. Solution: recover r via Approximate Common Divisor LLL lattice, zero the AES key with the k1=2r multiplicative trick, then factor n with an integer-only quadratic identity using known r.