Quantum Truncated Differential and Boomerang Attack

Abstract: Facing the worldwide steady progress in building quantum computers, it is crucial for cryptographic community to design quantum-safe cryptographic primitives. To achieve this, we need to investigate the capability of cryptographic analysis tools when used by the adversaries with quantum computers. In this article, we concentrate on truncated differential and boomerang cryptanalysis. We first present a quantum algorithm which is designed for finding truncated differentials of symmetric ciphers. We prove that, with a overwhelming probability, the truncated differentials output by our algorithm must have high differential probability for the vast majority of keys in key space. Afterwards, based on this algorithm, we design a quantum algorithm which can be used to find boomerang distinguishers. The quantum circuits of both quantum algorithms contain only polynomial quantum gates. Compared to classical tools for searching truncated differentials or boomerang distinguishers, our algorithms fully utilize the strengths of quantum computing, and can maintain the polynomial complexity while fully considering the impact of S-boxes and key scheduling.