TL;DR: recover prime factors of `λ(n)` from `r_j = prod(s_i) + 1` where `s_i` are factors of the period. Then use `d = e^{-1} mod(prod(r_j))` as decryption exponent, since (hopefully) `prod(r_j) = k * λ(n)`

Full writeup and analysis at <https://ur4ndom.dev/posts/2022-07-04-gctf-cycling/>