Paper 2025/073

Conditional Constant Function Problem and Its Quantum Solutions: Attacking Feistel Ciphers

Abstract

In this paper, we define the conditional constant function problem (CCFP) and, for a special case of CCFP, we propose a quantum algorithm for solving it efficiently. Such an algorithm enables us to make new evaluations to the quantum security of Feistel block cipher in the case where a quantum adversary only has the ability to make online queries in a classical manner, which is relatively realistic. Specifically, we significantly improved the chosen-plaintext key recovery attacks on two Feistel block cipher variants known as Feistel-KF and Feistel-FK. For Feistel-KF, we construct a 3-round distinguisher based on the special case of CCFP and propose key recovery attacks mounting to $r>3$ rounds. For Feistel-FK, our CCFP based distinguisher covers 4 rounds and the key recovery attacks are applicable for $r>4$ rounds. Utilizing our CCFP solving algorithm, we are able to reduce the classical memory complexity of our key recovery attacks from the previous exponential $O(2^{cn})$ to $O(1)$. The query complexity of our key recovery attacks on Feistel-KF is also significantly reduced from $O(2^{cn})$ to $O(1)$ where $c$'s are constants. Our key recovery results enjoy the current optimal complexities. They also indicate that quantum algorithms solving CCFP could be more promising than those solving the period finding problem.