Breaking The FF3 Format-Preserving Encryption Standard Over Small Domains

The National Institute of Standards and Technology (NIST) recently published a Format-Preserving Encryption standard accepting two Feistel structure based schemes called FF1 and FF3. Particularly, FF3 is a tweakable block cipher based on an 8-round Feistel network. In CCS2016, Bellare et. al. gave an attack to break FF3 (and FF1) with time and data complexity $O(N^5\log(N))$, which is much larger than the code book (but using many tweaks), where $N^2$ is domain size to the Feistel network. In this work, we give a new practical total break attack to the FF3 scheme (also known as BPS scheme). Our FF3 attack requires $O(N^{\frac{11}{6}})$ chosen plaintexts with time complexity $O(N^{5})$. Our attack was successfully tested with $N\leq2^9$. It is a slide attack (using two tweaks) that exploits the bad domain separation of the FF3 design. Due to this weakness, we reduced the FF3 attack to an attack on 4-round Feistel network. Biryukov et. al. already gave a 4-round Feistel structure attack in SAC2015. However, it works with chosen plaintexts and ciphertexts whereas we need a known-plaintext attack. Therefore, we developed a new generic known-plaintext attack to 4-round Feistel network that reconstructs the entire tables for all round functions. It works with $N^{\frac{3}{2}} \left( \frac{N}{2} \right)^{\frac{1}{6}}$ known plaintexts and time complexity $O(N^{3})$. Our 4-round attack is simple to extend to five and more rounds with complexity $N^{(r-5)N+o(N)}$. It shows that FF1 with $N=7$ and FF3 with $7\leq N\leq10$ do not offer a 128-bit security. Finally, we provide an easy and intuitive fix to prevent the FF3 scheme from our $O(N^{5})$ attack.