tl;dr:
1. Start with `e*d == 1 mod phi(N)` and thus `e*d - k*phi(N) == 1`
2. Notice that `e*d0 - k*phi(N) == 1 mod 2**1050`
3. Using the definition of `phi` for given 4 public keys formulate equation with a single variable and solve it to recover prime factors

Full writeup: https://github.com/p4-team/ctf/tree/master/2019-09-21-dragonctf/rsachained

Original writeup (https://github.com/p4-team/ctf/tree/master/2019-09-21-dragonctf/rsachained).