On Finding Quantum Multi-collisions
This work addresses a fundamental problem in quantum cryptography for researchers and practitioners, providing tight bounds for quantum multi-collision finding.
The paper tackled the problem of finding k-collisions for compressing hash functions using quantum queries, showing that Θ(N^(1/2 * (1 - 1/(2^k - 1)))) queries are both necessary and sufficient for constant k, which improves prior bounds and resolves the problem.
A $k$-collision for a compressing hash function $H$ is a set of $k$ distinct inputs that all map to the same output. In this work, we show that for any constant $k$, $Θ\left(N^{\frac{1}{2}(1-\frac{1}{2^k-1})}\right)$ quantum queries are both necessary and sufficient to achieve a $k$-collision with constant probability. This improves on both the best prior upper bound (Hosoyamada et al., ASIACRYPT 2017) and provides the first non-trivial lower bound, completely resolving the problem.