# 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…

## 75 Citations

Collision Finding with Many Classical or Quantum Processors

- Computer Science
- 2011

This thesis defines several new models of complexity that take into account the cost of moving information across a large space, and lays the groundwork for studying the performance of classical and quantum algorithms in these models.

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.

4-qubit Grover's algorithm implemented for the ibmqx5 architecture

- Computer Science
- 2018

An implementation of a 4-qubit Grover’s algorithm for the IBM Q computer ibmqx5 is presented and results yield results in line with the theoretically optimal results.

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Sort L according to the second entry in each item of L

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