Quantum cryptanalysis on some generalized Feistel schemes

Abstract

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 5-round 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 ${2^{\left( {\frac{1}{2}{d^2} - \frac{3}{2}d + 2} \right) \cdot \frac{n}{2}}}$, which is better than the quantum brute force search (Grover search) by a factor ${2^{\left( {\frac{1}{4}{d^2} + \frac{1}{4}d} \right)n}}$. For 4d-round Type-2 GFS with 2d branches, the time complexity is ${2^{\frac{{{d^2}n}}{2}}}$, which is better than the quantum brute force search by a factor ${2^{\frac{{3{d^2}n}}{2}}}$.