Quantum Algorithm for the Collision Problem

Years and Authors of Summarized Original Work

• 1998; Brassard, Høyer, Tapp

Problem Definition

A function F is said to be r-to-one if every element in its image has exactly r distinct preimages.

Input ::

an r-to-one function F.

Output ::

x₁ and x₂ such that F(x₁) = F(x₂).

Key Results

The algorithm presented here finds collisions in arbitrary r-to-one functions F after only $O(\root{3}\of{\mbox{ $N/r$}}\,)$ expected evaluations of F. The algorithm uses the function as a black box, that is, the only thing the algorithm requires is the capacity to evaluate the function. Again assuming the function is given by a black box, the algorithm is optimal [1], and it is more efficient than the best possible classical algorithm which has query complexity $\varOmega (\sqrt{N/r})$. The result is stated precisely in the following theorem and corollary.

Theorem 1

Given an r-to-one function F:X→Y with r ≥ 2 and an integer 1 ≤ k ≤ N = |X|, algorithmCollision(F,k) returns a collision after an expected...