# Quantum Algorithm for the Collision Problem

@inproceedings{Brassard2016QuantumAF, title={Quantum Algorithm for the Collision Problem}, author={Gilles Brassard and Peter H{\o}yer and Alain Tapp}, booktitle={Encyclopedia of Algorithms}, year={2016} }

In this note, we give a quantum algorithm that finds collisions in arbitrary r-to-one functions after only O( 3 √ N/r ) expected evaluations of the function. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover’s quantum searching algorithm in a…

## 83 Citations

### Quantum algorithms and learning theory

- Computer Science
- 2018

A quantum algorithm to solve a search space of N elements using essentially sqrt{N} queries and other operations, improving over the gate count of Grover's algorithm is described.

### Quantum Collision-Resistance of Non-uniformly Distributed Functions

- Computer Science, MathematicsPQCrypto
- 2016

It is proved that quantum queries are necessary to find a collision for function f whose outputs are chosen according to a distribution with min-entropy k that is needed in some security proofs in the quantum random oracle model e.g. Fujisaki-Okamoto transform.

### A quantum lower bound for distinguishing random functions from random permutations

- Computer Science, MathematicsQuantum Inf. Comput.
- 2014

The quantum query complexity of this problem is studied, and it is shown that any quantum algorithm that solves this problem with bounded error must make $\Omega(N^{1/5}/\log N)$ queries to the input function.

### Quantum Multicollision-Finding Algorithm

- Computer ScienceASIACRYPT
- 2017

A new quantum algorithm is proposed, which finds an l-collision of any function that has a domain size l times larger than the codomain size, which matches the tight bound of \(\varTheta (N^{1/3})\) for \(l=2\) and improves the known bounds.

### A note on the quantum collision and set equality problems

- Computer Science, MathematicsQuantum Inf. Comput.
- 2015

It is proved that, as expected, a quantum query complexity of $\Theta(N^{1/3})$ holds for all interesting domain and range sizes, and a new lower bound can be used to improve the relationship between classical randomized query complexity and quantum queries for so-called permutation-symmetric functions.

### Quantum Collision-Finding in Non-Uniform Random Functions

- Computer ScienceIACR Cryptol. ePrint Arch.
- 2017

We study quantum attacks on finding a collision in a non-uniform random function whose outputs are drawn according to a distribution of min-entropy k. This can be viewed as showing generic security…

### N ov 2 00 1 Quantum Lower Bound for the Collision Problem

- Mathematics, Computer Science
- 2018

A lower bound of Ω ( n ) is shown on the number of queries needed by a quantum computer to solve the collision problem with bounded error probability and is given for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements.

### Grover ’ s algorithm

- Physics
- 2017

The first notion of quantum computing was put forward by the great 20th century physicist Richard Feynman [4]. The underlying problem was that for approximating quantum wave functions with N…

### No-iteration of unknown quantum gates

- Computer Science
- 2016

A new no-go theorem is proposed by proving the impossibility of constructing a deterministic quantum circuit that iterates a unitary oracle by calling it only once by way of different schemes.

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- H;1) where H : Y ! f0; 1g denotes the function deened by H(y) = 1 if and only if a pair (x; G(y)) appears in L for some arbitrary x 2 K