Quantum Algorithm for the Multicollision Problem
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The current paper presents a new quantum algorithm for finding multicollisions, often denoted by ℓ-collisions, where an ℓ-collision for a function is a set of ℓ distinct inputs that are mapped by the function to the same value. The tight bound of quantum query complexity for finding a 2-collisions of a random function has been revealed to be Θ(N^1/3), where N is the size of the range of the function, but neither the lower nor upper bounds are known for general ℓ-collisions. The paper first integrates the results from existing research to derive several new observations, e.g., ℓ-collisions can be generated only with O(N^1/2) quantum queries for any integer constant ℓ. It then provides a quantum algorithm that finds an ℓ-collision for a random function with the average quantum query complexity of O(N^(2^ℓ-1-1) / (2^ℓ-1)), which matches the tight bound of Θ(N^1/3) for ℓ=2 and improves upon the known bounds, including the above simple bound of O(N^1/2). More generally, the algorithm achieves the average quantum query complexity of O(c_N · N^(2^ℓ-1-1)/( 2^ℓ-1)) and runs over Õ(c_N · N^(2^ℓ-1-1)/( 2^ℓ-1)) qubits in Õ(c_N · N^(2^ℓ-1-1)/( 2^ℓ-1)) expected time for a random function F X→ Y such that |X| ≥ℓ· |Y| / c_N for any 1< c_N ∈ o(N^1/(2^ℓ - 1)). With the same complexities, it is actually able to find a multiclaw for random functions, which is harder to find than a multicollision.
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