Quantum Cryptanalysis on Some Generalized Feistel Schemes

Post-quantum cryptography has attracted much attention from worldwide cryptologists. In ISIT 2010, Kuwakado and Morii gave a quantum distinguisher with polynomial time against 3-round Feistel networks. However, generalized Feistel schemes (GFS) have not been systematically investigated against quantum attacks. In this paper, we study the quantum distinguishers about some generalized Feistel schemes. For d-branch Type-1 GFS (CAST256-like Feistel structure), we introduce (2d−1)-round quantum distinguishers with polynomial time. For 2d-branch Type-2 GFS (RC6/CLEFIA-like Feistel structure), we give (2d+1)-round quantum distinguishers with polynomial time. Classically, Moriai and Vaudenay proved that a 7-round 4-branch Type-1 GFS and 5round 4-branch Type-2 GFS are secure pseudo-random permutations. Obviously, they are no longer secure in quantum setting. Using the above quantum distinguishers, we introduce generic quantum key-recovery attacks by applying the combination of Simon’s and Grover’s algorithms recently proposed by Leander and May. We denote n as the bit length of a branch. For (d2−d+ 2)-round Type-1 GFS with d branches, the time complexity is 2 1 2 d− 3 2 d+2)·n 2 , which is better than the quantum brute force search (Grover search) by a factor 2 1 4 d+ 1 4 . For 4d-round Type-2 GFS with 2d branches, the time complexity is 2 d2n 2 , which is better than the quantum brute force search by a factor 2 3d2n 2 .