Quantum Algorithm for the Collision Problem

  title={Quantum Algorithm for the Collision Problem},
  author={Gilles Brassard and Peter H{\o}yer and Alain Tapp},
  booktitle={Encyclopedia of Algorithms},
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… 
Collision Finding with Many Classical or Quantum Processors
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
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
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
  • H. Yuen
  • Computer Science, Mathematics
    Quantum 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
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
  • Mark Zhandry
  • Computer Science, Mathematics
    Quantum 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
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
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
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.


Tight bounds on quantum searching
A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.
An exact quantum polynomial-time algorithm for Simon's problem
  • G. Brassard, P. Høyer
  • Computer Science, Mathematics
    Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems
  • 1997
It is shown that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical bounded-error probabilistic computer if the data is supplied as a black box.
On the power of quantum computation
  • Daniel R. Simon
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
This work presents here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class.
A fast quantum mechanical algorithm for database search
In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .
Quantum computation and quantum information
This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
Universal Classes of Hash Functions
Minimum Disclosure Proofs of Knowledge
Cryptography - theory and practice
  • D. Stinson
  • Computer Science, Mathematics
    Discrete mathematics and its applications series
  • 1995
Sort L according to the second entry in each item of L
  • Sort L according to the second entry in each item of L
Cr epeau, \Minimum disclosure proofs of knowledge
  • Journal of Computer and System Sciences,
  • 1988